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ELECTRO-OPTICAL

IMAGING SYSTEM
PERFORMANCE

Gerald C. Holst

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ELECTRO-OPTICAL
IMAGING SYSTEM
PERFORMANCE
Second edition

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Copublished by

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and

SPIE OPTICAL ENGINEERING PRESS

A Publication of SPIE - The International Society for Optical Engineering
Bellingham, Washington USA
Library of Congress Cataloging-in-Publication Data

Holst, Gerald C.

Electro-optical imaging system performance / Gerald C. Holst. — 2nd ed.
p. cm.

Includes bibliographical references and index.

ISBN 0-8194-3701-8 (hard cover) - ISBN 0-9640000-6-7 (hard cover)

1. Electrooptical devices. 2. Imaging systems. 3. Infrared imaging.

4. Infrared detectors. 5. Quantum well. 1. Title

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SPIE Volume PM-84

ISBN: 0-8194-3701-8

Notice:

Reasonable efforts have been made to publish reliable data and
information, but the Author and Publishers cannot assume responsibility for the
validity of all materials or the consequences of their use.

All rights reserved. No part of this book may be reproduced in any form by any
means without written permission from the copyright owner.
Often we do not fully appreciate those
who have significantly affected our lives

This book is dedicated to Cathy
PREFACE to the SECOND EDITION

The first edition was published in 1995. Since then, infrared imagery
applications have exploded. From a modeling point-of-view, the subject of this
book, only a few significant changes have occurred. Uncooled technology is
based upon microbolometer and pyroelectric detectors. Systems with novel
semiconductors such as quantum well detectors are routinely produced. These
detectors, along with their characteristics, are described in this edition.

Although the material in most chapters is the same, the material in all
chapters has been rearranged for conciseness. Where appropriate, it has been
rewritten for clarification. Up-to-date references (1995 to 1999) have been
added. The Preface to the First Edition immediately follows this preface.
Significant changes are listed below.

Chapter 2 lists general detector types (classical semiconductors, novel
semiconductors, and thermal), provides applications, and describes their
operation. NVTherm, a new model that includes sampling effects, has a new eye
model. This comprehensive eye model (Chapter 6) considers the pupil diameter,
retinal response, and eye tremor. Neither the detector nor the optics can be studied
in isolation. The resolution of each is combined in Shade’s equivalent resolution
to create a system-level resolution metric (Chapter 12). This metric illustrates a
gradual change from detector-limited to optics-limited operation as the f-number
changes.

Sampling effects were known since the early days of television (circa
1930). How sampling affected target recognition and identification was not
quantified until 1999. The experimental results prior to 1995 suggested that
there was an optimum sampling rate. As a result, Chapter 14 was originally
entitled, Optimum Sampling. This chapter is now entitled Sampled-data Systems.
An image will appear at least one sample width wider than the sampled object.
Thus sampling can thought of as a blurring function. An increased width in the
space domain appears as a decreased width in the frequency domain. A larger blur
appears as if it the circuitry had a narrower modulation transfer function (MTF).
This apparent decrease (or contraction) in the frequency domain is called a
’’squeeze." The "squeeze" is used in NVTherm.

This edition contains more comparisons between mid-wave infrared
(MWIR) and long-wave infrared (LWIR) systems. Sun glints (Chapter 15) may
alter your selection. Chapter 18 has been completely rewritten. It includes the
specific detectivity (D*) for all the detector types. The noise equivalent
differential temperature (NEDT) is provided for each detector type. System
operation may either be background limited (BLIP) or Johnson noise limited
(JOLI).

FLIR92 only calculates the MTF and the minimum resolvable
temperature (MRT) up to the Nyquist frequency. This limit originally appeared
reasonable. Any spatial frequency above Nyquist frequency is aliased to a lower
frequency. However, the Nyquist frequency limit artificially limited range
performance. Two new models have emerged, NVTherm and TRM3, which no
longer have the Nyquist frequency limit (Chapter 19). Sampling effects are
incorporated through semi-empirical equations.

In 1995, Kennedy described a method for component tolerancing. By
tolerancing each input separately, it is possible to determine which subsystem
has the greatest influence on the MTF, NEDT, and MRT. The specifications of
that subsystem are reviewed. If cost effective, then the specifications are
tightened. Similarly, if the tolerancing on a subsystem shows little effect on the
MRT, then its specifications can be loosened and this implies reduced cost.
Kennedy created a shell around FLIR92. Each input is described by it mean and
standard deviation. The program then performed a Monte Carlo simulation by
selecting random values of the data set. The methods proposed by Kennedy
were commercialized into STADIUM FLIR (Chapter 19).

Target identification is a high-order target discrimination level that is at
the end of a complex process. The first task is searching the field-of-view to find
the target. Search (Chapter 21) may be random or systematic and it varies with
observer training and background. After the target is located, then a static
performance model (1975 NVL model, FLIR92, NVTherm, or TRM3) is used to
predict detection, recognition, and identification ranges.

The author extends his deepest gratitude to all his coworkers and
students who have contributed to the ideas in this book. They are too many to
mention by name. The author especially thanks Harold Orlando, Northrop
Grumman, who read a draft copy of this edition. Although he provided valuable
comments, the accuracy of the text is the sole responsibility of the author. Doug
Marks assisted with the graphics.
PREFACE to the FIRST EDITION

Typical examples of electro-optical systems include electronic still
cameras, image intensifiers, infrared imaging systems, low light level TVs
(LLLTV), machine vision systems, night vision goggles, solid state cameras, and
TVs. These systems have applications in aerospace, airborne reconnaissance,
astronomy, medical imaging, remote sensing, robotics, and spectroscopy.

In the 1950s, Shade predicted the resolution of photographic film and TV
sensors as a function of light level. His approach is the framework of all models
used today. In the 1970s, Rosell and Willson applied Shade’s results to thermal
imaging systems and low light level TVs. Since then, most modeling efforts have
concentrated on infrared imaging systems.

Electro-optical system analysis is based upon the hypothesis that systems
are amenable to spatial and temporal frequency domain analyses. It further
assumes that noise interferes with perception in such as manner that the signal-to-
noise ratio is pertinent. Critical to the analysis is that vision can be modeled
sufficiently accurately so that useful performance predictions can be made.

The advances made in infrared system modeling can now be applied to
all imaging systems with only minor modification. An infrared imaging system is
just a subclass of all electro-optical imaging systems. Different forms exist to
accommodate the terminology associated with different technologies (scanning
versus staring, visible versus thermal imaging systems, etc.). Unless told, the
observer generally will not know the source characteristics nor the sensor's spectral
band pass. Since his primary experience is visual (i.e., 0.4 to 0.7 im), he can only
state if the image on the monitor resembles a visual scene.

Effective modeling requires an orderly integration of diverse
technologies and languages associated with radiation physics, optics, solid state
sensors, electronic circuitry, human interpretation of displayed imagery, and
computer modeling. Each field is complex and is a separate discipline. The system
analyst must be conversant in all these fields.

This book provides the methodology to analyze and optimize systems. It
also includes some practical limits that exist in real hardware. Therefore, this book
is also a practical guide to electro-optical system design. Many aspects of hardware
design can be found in the eight-volume The Infrared and Electro-Optical Systems
Handbook, J. S. Accetta and D. L. Shumaker, eds., copublished by Environmental
Research Institute of Michigan and SPIE Press (1993).
After system optimization, subsystem design goals are established. Each
subsystem is then optimized to meet its goals. However, subsystem optimization
has led to popular rules-of-the-thumb and back-of-the envelope approximations.
These "rules” are not universally valid and must be considered in context with the
system design and application. For example, it is commonly stated that two
samples per detector-angular-subtense are required to maintain image fidelity. This
"rule” applies to those systems whose spatial response is limited by the detector’s
modulation transfer function (MTF). If the system is limited by the optical MTF,
this ’’rule” is inappropriate. But if used, it will increase hardware complexity and
cost.

Range performance is approached from a viewpoint of military tactical
thermal imaging systems. Two standard models are discussed in detail: the US
Army’s Static Performance Model (1975 model) and FLIR92. The results are
applied to visible systems where the minimum resolvable contrast (MRC) is
important. These models are the main analytic tools for deriving system
requirements and predicting performance for military systems. This book shows
the limitations of the current models and offers some methods to overcome the
limitations. It also provides some different MTFs that may be included as an
upgrade to the models.

The book is divided into five parts. The Introduction (Part 1) consists of
five chapters. Although Chapter 2, Principles of Operation, concentrates on the
evolution of thermal imaging systems it also applies to all electro-optical imaging
systems. Specifically, a staring array operating in the infrared region of the
spectrum is functionally the same as one operating in any other spectral band. The
radiometric equations apply to all electro-optical imaging systems.

While Part 2, System MTF, uses infrared imaging system terminology, it
applies to all linear systems. Depending upon the system design (with no
specification on spectral response), the analyst selects the appropriate MTFs for his
particular system.

Part 3, Image Quality, is nonspecific and applies to all electro-optical
imaging systems. Optimum sampling draws heavily upon military applications but
applies to any system used for target detection and recognition. It does not apply to
recognition of alphanumeric characters.

In Part 4, The Environment, the methodology to obtain the atmospheric
transmittance and MTF are extensions from data obtained in the visible region of
the spectrum. There are many references to the literature for other spectral band
analysis. Target signatures, as specified by a temperature differential, are unique to
the infrared imaging bands. For systems that are sensitive to radiation below 3 im,
signatures depend upon reflectivity differences.

Chapter 19 in Part 5, System Analysis, is based upon infrared imaging
system analysis. It is here that the Static Performance Model and FLIR92 are
presented. The three-dimensional noise model was developed to describe noise
processes in infrared imaging systems. This mathematical construct can be
extended to all electro-optical imaging systems. Target discrimination metrics
(Johnson Criteria) were originally developed for image intensifiers and have been
extended to infrared systems.

Electro-optical analysts or system analysts write computer codes, run the
performance models, and interpret the results. However, for success, the analysts
need the support of design engineers, test personnel, managers, and the customer.
All these people must understand the purpose and limitations of the models. In a
global sense, these individuals are part of the analy st team. This book is for the
entire team.

There are many other factors affecting design that cannot be easily
incorporated into any performance model and therefore are not covered in this
book. These include environmental issues, covertness, countermeasures, size,
weight, power consumption, cost, technology maturity, customer demand, and the
-ilities (maintainability, reliability, etc.).

This book does not include automatic target recognizers, target cuers, and
machine vision. Unfortunately, this becomes more difficult as nonlinear image
processing algorithms are incorporated into the systems. Indeed, today's systems
may be called a sensor with a built-in computer. Nonlinear models are beyond the
scope of this book.

The author extends his deepest gratitude to all his co-workers and
students who have contributed to the ideas in this book. They are too many to
mention by name. The author especially thanks all those who read draft copies of
the manuscript: John D'Agostino, US Army Night Vision and Electronic Sensors
Directorate; Herb Huey, Northrop; Howard Kennedy, Texas Instruments; Norman
Kopeika, Ben-Gurion University of the Negev; Harold Orlando, Northrop; Steven
Park, College of William and Mary; and Wendell Watkins, US Army Atmospheric
Sciences Laboratory. Don Davison provided the graphics.
TABLE OF CONTENTS

1. INTRODUCTION....................................................1

1.1. Imaging system nomenclature.................................3

1.2. System modeling.............................................5

1.3. Sensitivity and resolution limits...........................8

1.4. Infrared imaging systems...................................10

1.5. Infrared imaging system modeling...........................13

1.6. Model inputs...............................................14

1.7. References.................................................15

Exercises..................................................15

2. INFRARED IMAGING SYSTEM OPERATION..............................16

2.1. Optics.....................................................17

2.2. Scanners...................................................20

2.3. Detectors and coolers......................................24

2.3.1. Detector classification................................24

2.3.2. Specific detectors.....................................26

2.3.3. Detector operation.....................................27

2.4. Specific systems...........................................29

2.4.1. Line scanners..........................................29

2.4.2. Common module systems..................................30

2.4.3. EOMUX systems..........................................33

2.4.4. EMUX systems...........................................33

2.4.5. Second generation scanning systems.....................36

2.4.6. Staring array systems..................................36

2.5. System magnification.......................................37

2.6. References.................................................38

Exercises..................................................38

3. RADIOMETRY.....................................................40

3.1. Radiative transfer.........................................40

3.1.1. Planck's blackbody law.................................42

3.1.2. Extended source........................................44

3.1.3. Point source...........................................49

3.2. AT concept.................................................55

3.3. Photometry.................................................57

3.4. Normalization..............................................57

3.5. References.................................................60

Exercises..................................................61
4. MTF THEORY..........................................................62

4.1. MTF definition..................................................63

4.2. Linear filter theory............................................65

4.2.1. The EO system as a linear system...........................65

4.2.2. Cascading MTFs.............................................67

4.3. Superposition applied to optical systems........................68

4.4. Phase shifts....................................................71

4.5. References......................................................72

Exercises.......................................................72

5. SAMPLING THEORY.....................................................73

5.1. Sampling theory.................................................75

5.2. Aliasing........................................................75

5.3. Samplers........................................................77

5.3.1. The detector as a sampler..................................77

5.3.2. Spatial sampling...........................................80

5.3.3. A/D converter..............................................83

5.4. Microscan.......................................................84

5.5. Anti-aliasing filter............................................85

5.6. References......................................................86

Exercises.......................................................87

6. COMMON MODULE SYSTEMS...............................................88

6.1. Optics OTF......................................................89

6.1.1. Diffraction-limited OTF....................................90

6.1.2. Central obscuration........................................92

6.1.3. Aberrations................................................94

6.1.4. Defocused optics...........................................95

6.2. Detectors.......................................................97

6.3. Motion..........................................................98

6.3.1. Linear motion..............................................99

6.3.2. Sinusoidal motion.........................................102

6.3.3. Random motion (jitter)....................................104

6.3.4. Nonlinear scan mirror movement............................106

6.3.5. Low frequency motion......................................106

6.4. Electronic MTF.................................................106

6.4.1. Conversion: Electrical frequency to spatial frequency.....107

.6.4.2. Detector time constant...................................113

6.4.3. Amplifiers................................................113

6.4.4. Electrical filters........................................113

6.5. LEDs...........................................................116

6.6. Visual optics..................................................116
6.7. Eye response...................................................116

6.7.1. Conversion: Eye spatial frequency to spatial frequency....117

6.7.2. Eye MTF...................................................118

6.7.3. Noisy images..............................................123

6.8. System design example: Random motion effects..................124

6.9. Central limit theorem..........................................126

6.10. References....................................................127

Exercises......................................................128

7. EOMUX SYSTEMS......................................................130

7.1. Vidicon........................................................130

7.1.1. Conversion: Vidicon lines to spatial frequency............131

7.1.2. Vidicon MTF...............................................131

7.2. Video amplifiers and filters...................................132

7.2.1. Conversion: Video frequency to spatial frequency..........133

7.2.2. Boost circuitry...........................................133

7.2.3. Video amplifiers..........................................135

7.3. Monitors.......................................................135

7.4. System design example..........................................137

7.5. References.....................................................139

Exercises......................................................139

8. EMUX SYSTEMS.......................................................140

8.1. Motion.........................................................141

8.2. Detector.......................................................142

8.2.1. TDI.......................................................143

8.2.2. SPRITE detector...........................................145

8.2.3. Uncooled detectors........................................147

8.3. MTFs in the digital domain.....................................148

8.3.1. Conversion: Sampling frequency to spatial frequency.......149

8.3.2. Sample-scene phase........................................149

8.3.3. Digital filters...........................................150

8.3.4. Zoom......................................................156

8.4. Matching video subsystems to a sampled signal.158

8.4.1. Conversion: Video sampling frequency to spatial frequency..159

8.4.2. Sample-and-hold...........................................159

8.5. Post-reconstruction filter.....................................160

8.6. References.....................................................163

Exercises......................................................164
9. STARING ARRAY SYSTEMS...........................................165

9.1. Motion......................................................166

9.2. Detector Array..............................................166

9.3. Microscan...................................................168

9.4. References..................................................172

Exercises...................................................172

10. LINE SCANNERS..................................................173

10.1. Rectangular aperture.......................................177

10.1.1. Diffraction-limited MTF...............................177

10.1.2. Defocus OTF...........................................179

10.1.3. Ground coverage.......................................182

10.2. Scanner....................................................183

10.3. Motion.....................................................183

10.4. Electronic MTF.............................................183

10.5. AN/AAD-5 CRT MTF...........................................184

10.6. AN/AAD-5 film..............................................185

10.6.1. Conversion: Film response to spatial frequency........185

10.6.2. Film MTF..............................................185

10.7. References.................................................186

Exercises...................................................187

11. VERTICAL MTF...................................................188

11.1. Sampling Rate..............................................190

11.2. Interpolation..............................................194

11.3. References.................................................195

Exercises...................................................195

12. RESOLUTION.....................................................197

12.1. Analog metrics.............................................200

12.2. Sampled data systems.......................................203

12.3. System design based upon resolution........................206

12.4. Shade's equivalent resolution..............................209

12.5. References.................................................215

Exercises...................................................216

13. IMAGE QUALITY..................................................217

13.1. MTF........................................................218

13.2. Equivalent pass band.......................................220

13.3. Perceived signal-to-noise ratio............................220

13.4. Subjective quality factor..................................221

13.5. Square-root integral.......................................224

13.6. References.................................................225
14. SAMPLED-DATA SYSTEMS...........................................226

14.1. Samples per dwell..........................................228

14.2. Pixels on target...........................................231

14.3. Asymmetric sampling........................................236

14.4. Spurious response..........................................237

14.5. MTF squeeze................................................243

14.6. Aliased signal as noise....................................246

14.7. References.................................................246

Exercises....................................................247

15. ATMOSPHERIC TRANSMITTANCE......................................248

15.1. Atmospheric constituenl s..................................251

15.1.1. Water vapor............................................252

15.1.2. Aerosols...............................................255

15.2. Visibility.................................................258

15.2.1. Meteorological range...................................259

15.2.2. Contrast transmittance.................................261

15.3. LOWTRAN, MODTRAN, and HITRAN...............................261

15.4. Spectrally averaged atmospheric transmittance..............263

15.5. Weather conditions.........................................267

15.5.1. Average conditions.....................................267

15.5.2. Probability of occurrence..............................268

15.5.3. Navy model.............................................271

15.6. MWIR versus LWIR...........................................274

15.7. Signal-to-noise optimization...............................280

15.8. Sun glints.................................................283

15.9. Solar scattering...........................................285

15.10. Battlefield obscurants....................................286

15.11. References................................................287

Exercises....................................................288

16. ATMOSPHERIC MTF................................................289

16.1. Cn2........................................................290

16.2. Turbulence MTF.............................................293

16.3. Aerosol MTF................................................299

16.4. References.................................................300

Exercises....................................................302
17. TARGET SIGNATURES...................................................303

17.1. What is AT?....................................................304

17.2. Area-weighted AT...............................................306

17.3. Diurnal variations.............................................308

17.3.1. Solar heating..............................................308

17.3.2. AT cumulative probability..................................310

17.3.3. Environmental modifiers....................................311

17.4. Path radiance..................................................312

17.5. Sky background.................................................314

17.6. Active targets.................................................315

17.6.1. Fuel combustion............................................315

17.6.2. Frictional heat............................................315

17.7. Target signature modeling......................................315

17.8. Thermal structure metrics......................................317

17.9. References.....................................................317

Exercises........................................................319

18. SENSITIVITY and NOISE..............................................320

18.1. Scanning arrays (analog system)................................322

18.1.1. Noise equivalent bandwidth.................................322

18.1.2. Photon noise...............................................324

18.1.3. Johnson noise..............................................325

18.1.4. Amplifier noise............................................325

18.2. Staring arrays.................................................325

18.2.1. Photon noise...............................................326

18.2.2. Dark current...............................................326

18.2.3. Fixed pattern noise........................................327

18.2.4. Multiplexer noise..........................................328

18.3. Detector responsivity..........................................328

18.3.1. Classical semiconductors...................................329

18.3.2. Novel semiconductors.......................................330

18.3.3. Thermal detectors..........................................330

18.4. Specific detectivity...........................................331

18.4.1. BLIP.......................................................331

18.4.2. Johnson noise limited......................................333

18.4.3. D*BB to D*p conversion.....................................333

18.4.4. D*300......................................................3 3 4

18.5. Real systems...................................................335

18.6. System SNR.....................................................337

18.6.1 Scanning systems............................................337

18.6.2 . Staring systems..........................................338
18.7. NEDT.......................................................340

18.7.1. Scanning systems.......................................341

18.7.2. Staring systems........................................342

18.7.3. Background temperature.................................343

18.7.4. Boost..................................................344

18.7.5. Atmospheric transmittance..............................344

18.8. MWIR versus LWIR...........................................345

18.9. NEI........................................................346

18.10. Image reconstruction......................................348

18.10.1. Line-to-line interpolation............................349

18.10.2. Recursive and median filters..........................350

18.11. Sampling and aliasing.....................................351

18.12. References................................................352

Exercises...................................................354

19. SYSTEM PERFORMANCE MODELS......................................355

19.1. Three-dimensional noise model..............................357

19.2. FLIR92.....................................................362

19.2.1. FLIR92 model...........................................364

19.2.2. SNRthandte.............................................368

19.2.3. Head movement..........................................370

19.2.4. 1975 NVL Model.........................................371

19.2.5. Scanning systems (default values)......................371

19.2.6. Staring systems (default values).......................372

19.2.7. Nonuniformity versus ow/uTVH...........................373

19.2.8. Two-dimensional MRT....................................374

19.3. NVTherm....................................................375

19.4. TRM3.......................................................377

19.5. STADIUM FLIR...............................................378

19.6. Minimum resolvable contrast................................380

19.7. General comments...........................................381

19.8. References.................................................383

Exercises...................................................384

20. TARGET DISCRIMINATION..........................................385

20.1. One-dimensional detection..................................386

20.1.1. Johnson criterion......................................386

20.1.2. Extended discrimination................................389

20.1.3. Target transfer probability function...................393

20.1.4. Clutter................................................397

20.1.5. Moderate aspect ratio targets..........................398
20.2. Two-dimensional discrimination...............................400

20.2.1. Pixels on target.........................................400

20.2.2. FLIR92 discrimination requirements.......................403

20.3. Hot spot detection...........................................406

20.4. References...................................................406

Exercises.....................................................408

21. RANGE PREDICTIONS................................................409

21.1. Range prediction methodology.................................411

21.1.1. Atmospheric transmittance................................411

21.1.2. ACQUIRE..................................................412

21.1.3. Range performance probability............................414

21.1.4. Sampling effects.........................................415

21.2. Two fields-of-view...........................................417

21.3. Resolution versus sensitivity................................418

21.4. Line-of-sight stabilization..................................421

21.5. Target size..................................................421

21.6. Electric zoom................................................423

21.7. STADIUM FLIR.................................................423

21.8. MWIR versus LWIR systems.....................................423

21.9. Real Targets.................................................424

21.10. Search.....................................................425

21.11. References.................................................427

Exercises.....................................................428

APPENDIX
f-number...........................................................429

Reference.....................................................430

INDEX

431
1

INTRODUCTION

Electro-optical imaging system analysis is a mathematical construct that
provides an optimum design through appropriate tradeoff analyses. A
comprehensive model includes the target, background, the properties of the
intervening atmosphere, the optical system, detector, electronics, display, and the
human interpretation of the displayed information (Figure 1-1). While any of
these components can be studied in detail separately, the electro-optical imaging
system cannot. Only complete end-to-end analysis (scene-to-observer
interpretation) permits system optimization.

Finding the optimum design is an iterative decision process. Every step in
the design process that has conflicting needs requires a tradeoff analysis. Many
performance parameters can only be increased at the expense of another. For
example, decreasing the resolution can increase sensitivity.

Figure 1-1. Generic sensor operation that applies to all electro-
optical imaging systems. The scanning mechanism will vary
according to the overall design and number of detectors.
Staring arrays do not have a scanner.
Effective modeling requires an orderly integration of diverse
technologies and languages associated with radiation physics, optics, solid state
sensors, electronic circuitry, human interpretation of displayed imagery (human
factors), computer models, and software that might be embedded in the system
Each field is complex and is a separate discipline. The system analyst must be
conversant in all these fields.

Eludson1 stated, System engineering is the discipline that offers an orderly
approach to the design of systems, and, in particular, to systems that are so
complex that no one individual can possibly understand all the pertinent details.
As the system complexity increases and technology advances, it is increasingly
necessary to succinctly define performance requirements for each subsystem to
insure that the overall system requirements are met. This is an ever-increasing
challenge to the system analyst.

System optimization starts with a conceptual design. Then the various
subsystems are designed. Each subsystem will behave slightly differently than the
original design due to hardware constraints (limited availability of components,
space limitations, power consumption constraints, etc.). The analyst must then
modify the model parameters to reflect the current design. As the system is built
up, real components will perform differently due to manufacturing tolerances.
Based upon actual components, the analyst can finalize his model parameters. It is
only now that the analyst can be assured that the predicted performance will match
measured values.

According to Shumaker and Wood,2 a model should answer four basic
questions:

1. What characteristics (specifications) must an electro-optical imaging system
have to do a given task?

2. What design parameters will enable a system to satisfy given specifications?

3. What laboratory performance values will verify that a design provides desired
characteristics?

4. Given an electro-optical imaging system design, how can it best be deployed
and what are the expected results?

There are many factors affecting design that cannot be easily
incorporated into any performance model. These include environmental issues,
covertness, countermeasures, size, weight, power consumption, cost, technology
maturity, customer demand, the -ilities (maintainability, reliability, etc.), and
support issues.
1.1. IMAGING SYSTEM NOMENCLATURE

Due to the atmospheric spectral transmittance, electronic imaging
system design is partitioned into seven generic spectral regions (Figure 1-2) of
which four are associated with thermal imaging systems. The ultraviolet (UV)
region ranges in wavelength from 0.2 to 0.4 pm. The visible spectral region
ranges in wavelength from 0.4 to 0.7 pm. Televisions, electronic still cameras,
and most solid state cameras* operate in this region. The near infrared imaging
spectral region (NIR) spans approximately 0.7 to 1.1 pm. Low light level
televisions (LLLTV), image intensifiers, star light scopes, and night vision
goggles operate in this region. For historical reasons, the UV, visible, and NIR
technologies have developed their own terminologies. The first infrared imaging
band is the short wavelength infrared imaging band (SWIR) which
approximately covers 1.1 to 2.5 pm. The second infrared band is the mid-
wavelength infrared (MWIR) spectral region that covers approximately 2.5 to
7.0 pm. (In older literature, the MWIR band was labeled as SWIR). The third
infrared band is the long wavelength infrared (LWIR) spectral band. It covers
the spectral region from approximately 7 to 15 pm. The fourth infrared band is
the far infrared (FIR) or very long wave infrared (VLWIR) region. It applies to
all systems whose spectral response extends past 15 pm. The MWIR and LWIR
regions are sometimes called the first and second thermal imaging bands.

Wavelength (pm)

Figure 1-2. Representative atmospheric transmittance over a
1-km path length. The transmittance varies with temperature,
relative humidity, and airborne particulates.

* Solid state cameras are popularly called CCD cameras because most contain
charge-coupled device detector arrays.
The word infrared has a different definition that depends upon the
technology. For example, film was invented to produce visible images. If the
film is sensitive to longer wavelengths, it is called infrared film. Infrared film is
sensitive out to about 0.85 pm. Imaging systems were also created to reproduce
visible images. If the spectral response was past 0.7 pm, it was called an infrared
device. Most of these systems used photosensitive detectors whose spectral
response went to about 1 pm. A variety of labels were attached to these devices.
This includes thermal imaging systems, and night vision devices.

Table 1-1 lists the wavelengths associated with specific systems. The
precise spectral response for any system depends upon the design. To say a
system is a LWIR system only means its response is somewhere in the LWIR
region. For example, a LWIR system may have a spectral response from 7.7 to
11 pm or 8 to 12 pm. The atmospheric carbon dioxide (CO2) absorption band
will completely attenuate all target information at 4.2 pm. As a result, a MWIR
system may have a spectral response from 3 to 4.2 or 4.3 to 5.5 pm.

Table 1-1
SPECTRAL NOMENCLATURE
(Continued next page)

Spectral band Wavelength range (pm)
Vacuum ultraviolet 0.05 to 0.20
Short ultraviolet (UV-C) 0.20 to 0.29
Mid-wave ultraviolet (UV-B) 0.29 to 0.32
Long-wave ultraviolet (UV-A) 0.32 to 0.40
Visible 0.40 to 0.70
Violet 0.40 to 0.46
Blue 0.46 to 0.49
Green 0.49 to 0.55
Yellow 0.55 to 0.58
Orange 0.58 to 0.60
Red 0.60 to 0.70
Near infrared (NIR) 0.7 to 1.1
Short wavelength infrared (SWIR) 1.1 to 2.5
Mid-wavelength infrared (MWIR) 2.5 to 7.0
(Labeled as SWIR in older literature)
First thermal imaging band 3.0 to 5.5
Blue spike plume 4.1 to 4.3
Red spike plume 4.3 to 4.6
Long wavelength infrared (LWIR) 7.0 to 15.0
Second thermal imaging band 8 to 14
Table 1-1 (continued)
SPECTRAL NOMENCLATURE

Very long wavelength infrared (VLWIR) > 15.0
Extreme infrared 15 to 100
Near millimeter 100 to 1000
Millimeter 1000 to 10,000

1.2. SYSTEM MODELING

System modeling drives future design, system requirements and quality
assurance specifications (Figure 1-3). Connecting specifications to well understood
physical parameters makes the designer, manufacturer, and customer more
confident that the design objectives have been achieved.

Predictive modeling is a continuous process. The model is used to
optimize system design. As the systems are built with real hardware, new values
are used to refine range predictions. Simultaneously, modeling is used to help
select quality control specifications. Once a model is validated for a particular
design, it is used to develop the next design.

Figure 1-3. System modeling is a continuous process.
System modeling is the mathematical construct that includes system
characteristics, observer experience, scene content, atmospheric transmittance,
monitor settings and a variety of miscellaneous factors. These factors affect the
perceived image quality (Figure 1-4).

Figure 1-4. Image quality contributors. All these factors affect
the perceived quality.

Referring to a list of variables similar to that shown in Figure 1-4, Howe
stated,3 Although the list is by no means complete, it length underscores the
complexity of target acquisition. No single model... could possibly account for all
the factors listed; because of this, many models are specialized to a particular
handful of scenarios and system types. Frequently they incorporate simplifying
assumptions, and they are usually validatedfor only a small portion of the possible
mission or situations. Using a model to predict performance for scenarios where
the model is not validated or specialized can lead to very inaccurate predictions.

Models must be able to relate design parameters, laboratory
measureables, and operational performance. Three levels2 of models contribute to
satisfying these requirements:

Component/phenomenology models

They include the system modulation transfer function (MTFsys),
atmospheric transmittance, target signature, and signal-to-noise ratio.
These models relate quantifiable design and environmental parameters to
higher level summary parameters.
System performance models

These models are built upon component-phenomenology models. They
characterize the total system performance for controlled tasks such as the
detection of a periodic bar pattern. They predict standard measures such
as the minimum resolvable temperature, minimum resolvable contrast,
and minimum detectable temperature.

Operational models

These models combine system models with other models to characterize
overall operational tasks. When operational models include target
signature models, they are used to calculate detection, recognition, and
identification ranges.

Each of the above models can be separated into three categories:

’’Standard” models

These are the models that are used most often by the modeling
community and are readily available.

Special application models

These are models that are unique to a specific application or a specific
design.

Nonlinear models

Nonlinear models cannot be described mathematically in closed-form.

They are used on a case-by-case basis.

Generic modeling equations apply to all imaging systems. Different
forms exist to accommodate the terminology associated with different technologies
(scanning versus staring, visible versus thermal imaging systems, etc.). Unless
told, the observer generally will not know the source characteristics or the sensor's
spectral band pass. Since his primary experience is visual (i.e., 0.4 to 0.7 pm), he
can only state if the image on the monitor resembles a visual scene.

In the 1950s, Shade4 predicted the resolution of photographic film and
television sensors as a function of light level. His approach is the framework of all
models used today. In the 1970s, Resell and Willson extended Shade's work to
encompass thermal imaging systems and low light level televisions. Since then,
most modeling efforts have concentrated on infrared imaging systems.

The perceived signal-to-noise ratio is

SNRp

MTFsys EI x

system noise (eye spatial filter) (eye temporal filter)

(1-1)

where EI is the intensity difference between the target and its immediate
background, A is a proportionality constant that depends upon the aperture
diameter, focal length, and quantum efficiency, and MTFsysEI/(system noise) is
the signal-to-noise ratio at the detector output. The eye's incredible ability to filter
spatial and temporal noise increases the apparent SNR. A threshold value is
selected for SNRp.. This is where the target is just perceived. Then equation 1-1 is
inverted to determine the minimum detectable EI. For systems operating in the
visible or NIR, the minimum level is called the minimum resolvable contrast
(MRC). For systems operating in the MWIR and LWIR, the minimum value is the
minimum resolvable temperature (MRT) or minimum detectable temperature
(MDT).

1.3. SENSITIVITY and RESOLUTION LIMITS

An overwhelming majority of imaging quality discussions center on
spatial resolution or sensitivity. Resolution has been in use so long that it is
thought to be something fundamental that uniquely determines system
performance. It implies something about the smallest detail that can be perceived.
It may be specified by a variety of sometimes unrelated metrics such as the Airy
disk angular size, the detector angular subtense (DAS), or the Nyquist frequency
(specified by an angular sampling rate). Resolution does not include the effects of
system noise. Resolution considerations provide a back-of-the-envelope
approximation from which the target range can be estimated

» » target size

Range = R = —----------

resolution

(1-2)

Sensitivity deals with the smallest signal that can be detected. It is usually
taken as that signal that produces a signal-to-noise ratio of unity at the system
output. Sensitivity is dependent upon the light-gathering properties of the optical
system, the responsivity of the detector, and the noise of the system. It is
independent of resolution. Sensitivity limitations provide another back-of-the-
envelope approximation. The signal-to-noise ratio (SNR) is
SNR =

т M
i atm-ave

system noise

(1-3)

For infrared imaging systems, the target-background intensity difference is
specified by a differential temperature (AT). The system noise is taken as the noise
equivalent differential temperature (NEDT). This approximation only applies to
those targets whose angular subtense is large compared to the system's resolution
at the calculated range. The variable, Tatni_ave , is an average atmospheric attenuation
coefficient. The range predicted by Equation 1-3 can be quite different than that
given by Equation 1-2.

Overall system response depends on both sensitivity and resolution. As
shown in Figure 1-5, the MRT is bounded by sensitivity and resolution
considerations. Different systems (Figure 1-6) may have different MRTs. System
A has a better sensitivity. It has a lower MRT at low spatial frequencies. At mid-
range spatial frequencies, the systems are approximately equivalent and it can be
said they provide equivalent performance. System В has better resolution and can
display finer detail than System A. Figure 1-6 illustrates that neither sensitivity,
resolution nor any other single parameter can be used to compare systems.

Figure 1-5. MRT is bounded by the system's resolution and
the visual sensitivity limit.
Figure 1-6. Two systems with different MRTs. Whether
system A is better than В depends upon the specific
application.

1.4. INFRARED IMAGING SYSTEMS

Infrared imaging is the remote sensing and display of infrared flux
variations. The variations in the displayed image intensity represent apparent
temperature variations across the scene. The detected radiation, which appears to
emanate from a target and its background, consists of self-emission, reflected
radiation, and atmospheric path radiance. To distinguish a target from its
background, the detected radiation must be different.

In 1969, Hudson6 listed over 100 separate applications for thermal
imaging systems. He divided the list into four major categories: military,
industrial, medical, and scientific. Each category was then subdivided into (1)
search, track, and range, (2) radiometry, (3) spectroradiometry, (4) thermal
imaging, (5) reflected flux, and (6) cooperative source. That list is surprisingly
complete.

In early 1970s, the U.S. Department of Defense suggested a modular
approach to thermal imaging system design. This resulted in "common module"
components that have influenced military design ever since. In 1976, the United
Kingdom and France also initiated the common module approach. All three
programs followed the same philosophy: Standardize the basic building blocks.
The parallel scan approach was adopted in the U.S. It used a linear array of 60,
120, or 180 HgCdTe detectors. The U.K. (Thermal Imaging Common Module or
TICM) and France (Systeme Modulaire Thermique) adopted a serial/parallel scan
system. The U.K. has since changed to SPRITE (signal processing in the element)
detectors. Most systems today are derivatives of military systems or at least have
components that were designed for military systems.

Today, two broad categories are in usage: military and commercial. Table
1-2 highlights a few applications in each category. Military and commercial
systems are similar in basic design; but each system is built for a specific purpose.
As a result, military and commercial systems tend to be described by different
performance parameters. Some generic differences are listed in Table 1-3. Imaging
systems have an observer as the image interpreter whereas hardware and/or
software assess machine vision imagery. An important subset of machine vision is
the infrared search and track type systems (IRST). These systems are designed to
detect point sources. The specific system design depends upon the application, the
atmospheric transmittance and availability of optics and detectors. This list is brief.
Detailed information on all phases of electro-optical systems can be found7 in the
eight-volume The Infrared and Electro-Optical Systems Handbook.

Table 1-2

THERMAL IMAGING APPLICATIONS

COMMUNITY APPLICATIONS
MILITARY Reconnaissance Target acquisition Fire control Navigation
COMMERCIAL CIVIL Law enforcement Fire fighting Border patrol
ENVIRONMENTAL Earth resources Pollution control Energy conservation
INDUSTRIAL Maintenance Manufacturing Non-destructive testing
MEDICAL Mammography Soft tissue injury Arterial constriction
Table 1-3

TYPICAL DESIGN REQUIREMENTS

DESIGN AREAS MILITARY COMMERCIAL
Vibration stabilized Required for very narrow fields-of-view Usually not required
Image processing algorithms Application specific (e.g., target detection or automatic target recognition) Menu-driven multiple options
Resolution High resolution (resolve targets at long distances) Typically not an issue since the image can magnified by moving closer
Image processing time Real time Real time not usually required
Target signature (sensitivity requirements) Usually just perceptible (low NEDT) Usually high contrast target (NEDT not necessarily a dominant design factor)

Key to the design of an electro-optical imaging system, is the overall
application. The design depends upon the mission and how the mission is to be
accomplished. Compare the requirements for pinpointing a target with that of
high-speed aircraft navigation. The military wants low noise with maximum
reliability. In the commercial world, low cost and ease of maintenance are
important: they are often willing to accept less performance. However, these
distinctions are getting blurred in today's environment. There is a desire for a
single imaging system to perform all functions. This may be possible as new
image processing algorithms are developed.
1.5. INFRARED IMAGING SYSTEM MODELING

The 1975 NVL model* was developed to predict8 the performance of
U.S. Army thermal imaging systems. It satisfied the Army’s need and it adequately
predicts the minimum resolvable temperature, MRT, at mid-range spatial
frequencies which corresponds to detecting modest sized targets at modest ranges.
The model was essentially one-dimensional and did not incorporate noise sources
other than random noise. To overcome these deficiencies, NVESD created FLIR90
that was subsequently updated9 to FLIR92.

FLIR92 predicts the performance of staring arrays up to the Nyquist
frequency. The Nyquist frequency limit created a ’’brick wall" in range
performance. NVTherm4 accounts for sampling effects beyond Nyquist frequency
by "squeezing" the system MTF.

Models are adequate for comparative analysis but may r£)t predict
absolute performance. If a model does not predict the situation, the following
questions must be asked. Are the underlying assumptions valid? Were the correct
equations used? Were some parameters left out that should have been included?
Were the correct values used (actual hardware values versus design values)? Are
the test results statistically sound? Was the test procedure appropriately controlled?

The biggest problem in modeling and performance predicting is
inadequate documentation. A model is composed of many approximations, each of
which has a limit. Although many models are said to be "validated," the region of
validation may be very restrictive. For example, the 1975 NVL model was
validated for modest-sized vehicles at modest ranges. As the result, this model is
excellent for range predictions for similar sized targets. It is pure extrapolation to
apply the model to the detection or recognition of people, highways, bridges,
aircraft, or aircraft hangars.

* Over the years, the U.S. Army's proponent agency for modeling and testing
infrared imaging systems has changed its name: Night Vision Laboratory (NVL),
Center for Night Vision and Electro Optics (CNVEO), Night Vision and Electro
Optical Laboratory (NVEOL), Night Vision and Electro Optical Directorate,
(NVEOD), and (as of this writing) Night Vision and Electronic Sensor Directorate
(NVESD). As a result, the Static Performance Model8 has been called by a variety
of names: the NVL model, Static Performance Model, Ratches (principal author)
model, CNVEO model, and NVEOL model. This book uses "1975 NVL model."
all the documentation to understand the limitations of the models so that he does
not extend the model past its region of validity.

1.6. MODEL INPUTS

Figure 1-7 lists the inputs required for infrared system modeling. The
MTFs tend to change as the system design changes and each generic system
design is listed in a different chapter.

Although many component descriptions and system level figures-of-
merit to be discussed are devoted to infrared imaging systems, the modeling
methods discussed are general and can be applied to most electro-optical imaging
systems with only minor modifications.

Figure 1-7. Many components are required for a complete
end-to-end analysis. The 1975 NVL model and FLIR92 are
the main analytic tools for deriving thermal imaging system
requirements and predicting performance. Chapters 1 through
5 provide an introduction to modeling concepts. Chapters 12
through 14 discuss various aspects of image quality. The
information flows into the system performance model from
which target range can be predicted.
1.7. REFERENCES

1. R. D. Hudson Jr., Infrared System Engineering, pp. 10-11, John Wiley and Sons, New York (1969).

2. D. L. Shumaker and J T. Wood, ''Overview of Current IR Analysis Capabilities and Problem
Areas," in Infrared Systems and Components II, H. M. Li aw, ed., SPIE Proceedings Vol. 890, pp. 74-
80 (1988).

3. J. D. Howe, "Electro-Optical Imaging System Performance Prediction," in Electro-Optical Systems
Design, Analysis, and Testing, M. C. Dudzik, ed., pg. 60. This is Volume 4 of The Infrared and
Electro-Optical Systems Handbook, J. S. Accetta and D. L. Shumaker, eds., copublished by
Environmental Research Institute of Michigan, Ann Aibor, MI and SPIE Press, Bellingham, WA
(1993).

4 . О. H. Shade, Sr., "Image Gradation, Graininess, and Sharpness in Television and Motion Picture
Systems," published in four parts in J SMPTE\ "Part I: Image Structure and Transfer Characteristics,"
Vol. 56(2), pp. 137-171 (1951); "Part II. The Grain Structure of Motion Pictures - An Analysis of
Deviations and Fluctuations of the Sample Number," Vol. 58(2), pp. 181-222 (1952); "Part III. The
Grain Structure of Television Images," Vol. 61(2), pp. 97-164 (1953); "Part IV: Image Analysis in
Photographic and Television Systems," Vol. 64(11), pp. 593-617 (1955).

5. F. A. Resell and R. H. Willson, "Performance Synthesis of Electro-Optical Sensors," Air Force
Avionics Laboratory Report AFAL-TR-72-229, Wright Patterson AFB, OH (1972).

6. R. D. Fludson Jr., Infrared System Engineering, Chapters 16 to 19, John Wiley and Sons. New York
(1969).

7. The Infrared and Electro-Optical Systems Handbook, J. S. Accetta and D. L. Shumaker, eds.,
copublished by Environmental Research Institute of Michigan, Ann Arbor, MI and SPIE Press,
Bellingham, WA (1993). This eight-volume set contains performance modeling and design methods
for many electro-optical systems.

8. J. A. Ratches, W. R. Lawson, L. P. Obert, R. J. Bergemann, T. W. Cassidy, and J. M. Swenson,
"Night Vision Laboratory Static Performance Model for Thermal Viewing Systems," ECOM Report
ECOM-7043, Fort Monmouth, NJ (1975).

9. FLIR92 Thermal Imaging Systems Performance Model, User's Guide, NVESD document
UG5008993, Fort Belvoir, VA (1993).

10. R. G Driggers, R. Vollmerhausen, and T. Edwards, "The Target Identification Performance of
Infrared Imager Models as a Function of Blur and Sampling," in Infrared Imaging Systems: Design,
Analysis, Modeling and Testing X, G. C. Holst, ed., SPIE Proceedings Vol. 3701, pp. 26-34 (1999).

EXERCISES

1. Explain why the observer's age, IQ, education, training, motivation, personality
and fatigue may affect subjective evaluation of image quality.

2. List three applications for an MWIR system. (Hint: Aircraft engines that are
approximately 800 К emit significant radiation in the MWIR region.)

3. List three applications for an LWIR system. (Hint: Most terrestrial objects that
are at 300 К emit significant radiation in the LWIR region.)

4. Figure 1-2 illustrates typical terrestrial atmospheric transmittance. On the other
hand, deep space has nearly unity transmittance at all wavelengths. A "star wars"
satellite has a detector system that has a spectral response of 4.2 to 4.3 pm. Why?
(Hint: Draw a picture depicting the earth, atmospheric layer and a missile.)
2

INFRARED IMAGING SYSTEM
OPERATION

An electro-optical imaging system consists of many subsystems. Each of
which processes information differently. They may create artifacts or variations in
the processed image that were not present in the original scene. Figure 2-1
illustrates five major subsystems: optics and scanner, detector and detector
electronics, digitization, image reconstruction, and post reconstruction. The
specific design depends upon the number of detector elements and the required
output format. The optics images the radiation onto the detector(s). Scanners
optically move the detector’s instantaneous-field-of-view (IFOV) across the field-
of-view (FOV) to produce an output voltage proportional to the local scene
intensity. In a scanning system, the output of a single detector represents the scene
intensity across a line. With a staring array there is no scanner and adjoining
detector outputs provide scene variations.

Input

Electro-Optical Imaging System

Output

Figure 2-1. Generic electro-optical functional block diagram.
All systems have optics, detectors, detector electronics, and an
output. Staring arrays do not have a scanning mechanism. The
specific electronics design depends upon the detector
configuration and desired output.

The detector is the heart of every electro-optical system because it
converts scene radiation into a measurable electrical signal. Amplification and
signal processing creates an electronic image in which voltage differences
represent scene intensity differences due to the various objects in the field-of-view.

- Each detector will have its own amplifier. The amplifier outputs are
multiplexed together and then digitized. The number of channels multiplexed
together depends upon the specific design. Systems may have several multiplexers
and several A/D converters operating in parallel.
Figure 2-2. The optical system can be considered a single lens.
Pj and P2 are the principal planes. The effective focal length is
measured from the second principal plane. Although shown as
planes, the principal surfaces are typically spheres (see
Appendix).

CosineN0 shading is a geometrical phenomenon that reduces the intensity
reaching off-axis detectors. It depends upon the optical design and the physical
location of apertures and the detectors. With a single detector in a scanning system
(Figure 2-3 a), the detector is always on-axis and there is no cosineN0 shading. It is
the scanner that allows the detector to sense off-axis radiation. With a linear array
of detectors (Figure 2-3b), the cosineN0 variation is in the direction of linear array
(cross scan direction). Staring arrays exhibit radially symmetrical cosineN0 roll-off
as measured from the center of the field-of-view (Figure 2-3 c). Figure 2-4a
illustrates the signals from three different lines for the linear scanning array
depicted in Figure 2-3b. Figure 2-4b illustrates the signals for the staring array
shown in Figure 2-3c. Gain/level normalization minimizes this effect. Figure 2-4c
depicts the normalized signal as a function of line number for a system suffering
from extreme cosineN0 effects.

No CosineN0

(a)

(b)

Figure 2-3. CosineN0 Effect, (a) Single detector, (b) scanning
linear array consisting of 480><l, and (c) staring array
consisting of 480 x 480 detectors.

(c)
Signals are digitized because of the relative ease to manipulate digital
data. Current systems rely heavily upon software for gain/level normalization,
image enhancement, and line-to-line interpolation. To produce a linear input-to-
output system, a gamma correction algorithm removes the CRT-based nonlinear
response. The monitor may or may not be an integral part of the electro-optical
imaging system.

For convenience, targets are labeled as either hot or cold with respect to
the immediate background. For MWIR and LWIR systems, the term thermal is
misleading. Infrared imaging systems with photoconductive or photovoltaic
detectors do not sense warmth or cold (they are not thermometers) but sense the
radiation emitted by an object. All objects emit radiation in the thermal regions.
Hot refers to targets that appear warmer than its immediate background and cold
means the target appears cooler than its immediate background. The choice of hot
objects appearing white and cold objects appearing black is arbitrary. With
electronic polarity reversal available, either ’’white hot’’ or ’’black hot" targets can
be created. With "black hot," hot objects appear black or dark gray against a
neutral background. As the object becomes hotter, its representation on a monitor
becomes blacker. The reverse is true for "white hot." With "white hot," the target
becomes "whiter" as its apparent temperature increases relative to the background.
The output also can be mapped into pseudo-colors. Keeping with human feelings,
cold objects are often represented as blue and hot objects as red.

This chapter highlights the operation of the five major subsystems with
specific attention given to those subsystems that either add noise or modify image
fidelity. Image fidelity is specified by the modulation transfer function (MTF). For
thermal imaging systems, noise is incorporated in the noise equivalent differential
temperature (NEDT) and the three-dimensional noise model. Most systems
operating in the visible region of the spectrum have functional designs similar to
those operating in the infrared.

2.1. OPTICS

Most optical systems are composed of many individual lenses or mirrors.
Each element may have a different refractive index and shape to minimize the
aberrations. For analytic purposes, the optical subsystem is treated as a single
element with an effective focal length (Figure 2-2). The clear aperture is not
necessarily the diameter of an optical element. The clear aperture limits the
amount of scene radiation reaching the detector and is determined by optical
design.
Figure 2-4. CosineN0 effect. Line traces as a function of line
number for (a) a linear scanning array and (b) a staring array,
(c) The normalized output as a function of line number for a
linear scanning array. Gain/level normalization minimizes
cosineN0 and produces a cosmetically pleasing image.

It is well known that an off-axis image will have reduced incidance
compared to an on-axis image. In classical textbooks, the reduction is given as
cosine40. Real systems tend to have a less dramatic roll-off and the reduction is
cosineb0 where 2 <N <4. Whether an image is off-axis depends upon the optical
design. For a single detector system, the detector is always on-axis. It is the
scanner that moves the detector instantaneous-field-of-view around the system
field-of-view. For multi-element arrays, the optical design determines which
detectors are off-axis. Therefore, the cosineN0 factor can only be added on a case-
by-case basis.

Appropriately applied electronic amplification can compensate for the
cosineN0 shading. Here, high contrast images will appear cosmetically appropriate.
However, any amplification also will increase the noise so that the signal-to-noise
ratio remains constant. The NEDT is not affected by this electronic amplification.
2.2. SCANNERS

The function of the scanner is to dissect the image sequentially and
completely. That is, the scanner moves the detector IFOV around the system FOV
in a way that is consistent with the monitor requirements. A large variety of
scanning schemes is available. There is nothing inherently good or bad with any
particular scanning scheme: each has its own advantages and disadvantages. The
most common scanners are rotating drums, polygons, refractive prisms, and
oscillating mirrors. For this text, it will always be assumed that scanning is in the
horizontal direction.

With scanning systems, the detector output creates the image during the
active scan time and the detector output is ignored during the inactive scan time.
The inactive time provides the time necessary for the scanner to come into the
appropriate position for the next frame or scan line.

Detector characteristics may dictate the scan direction. Figure 2-5
illustrates a system with a single detector with unidirectional (raster) scan. SPRITE
(Signal processing m the element) detectors require an unidirectional scan. Figure
2-6 illustrates a linear array that uses bidirectional scanning (parallel scan). U.S.
common module systems employ bidirectional scanners.

Figure 2-5. Single detector scan pattern (serial scan)
employing unidirectional (raster) scanning.
Forward Scan Direction ►

Figure 2-6. Linear array employing bidirectional scan (parallel
scan) and 2:1 interlace.

Figure 2-7a illustrates a series of detectors whose outputs are summed together to
provide time-delay and integration (TDI). Systems with TDI usually require
unidirectional scan. Here, the scan velocity must be matched to the time delay in
the integrating element (Figure 2-7b). Scan linearity is important when using TDI
to avoid geometric distortion and MTF degradation. SPRITE detectors have an
inherent TDI and do not require the multiple delay circuits shown in Figure 2-7b.

The NEDT is inversely proportional to the detector dwell time. The dwell
time is the time it takes for a target edge to be swept across a detector element.
Single element systems will have the highest NEDT for a fixed frame rate. As
more detectors are added (Figure 2-6), the scan speed is reduced, the detector
dwell increases and the NEDT is lowered. Since each detector has a slightly
different responsivity, multiple element systems suffer from fixed pattern noise
that requires electronic correction. The inability to fully compensate produces
streaks in scanning arrays and residual fixed pattern noise in staring arrays.

With TDI, the noise is reduced by the square root of the number of TDI
detectors. A TDI arrangement has the additional advantage that if one detector
element fails, the remaining will still produce an output. With a parallel-scan
system (Figure 2-6), a defective element will produce a line with no information.
This leaves a space in the dissected scene.
-----Scan Direction

Signal

(b)

Figure 2-7. Multiple detectors operating in pure serial scan
mode, (a) Detector configuration and (b) TDI delay elements.
The noise is reduced by the square root of the number of TDI
elements.

Staring arrays, in principle, have the lowest NEDT since the dwell time
may be equal to a frame time. Staring arrays may not reach this theoretical limit
due to residual fixed pattern noise and multiplexer noise.

The scanning mechanism can be either inside or outside the optical
subsystem. When outside (Figure 2-8a), the mirror must be larger than the clear
aperture and provide a scan angle of ± 0scan/2. The mirror can be much smaller
when.inside the optical subsystem but then it must scan a larger angle of ±
Moptics0scan/2. Here, Moptics is the magnification of the afocal telescope (also called
the afocal power).

The relationship between the scan angle and FOV depends upon the
optical design. If the scanner is outside the optical system, the FOV. is 0scan + IFOV.
If the scanner is within the optical subsystem, the FOV may be 9scan + IFOV or tan
(9scan + IFOV). These designs are called f 9 (pronounced f theta) and f tan (9)
(pronounced f tan theta) respectively. For small angles they are equivalent: tan
(9scan +IFOV) »9scan +IFOV.

A full staring array (Figure 2-3 c) does not have a scanner. The field-of-
view depends upon the optical design. If the overall array size is darTay, the FOV
may be 2 tan-1(dan.ay/2 fl) or danay/tl. Again, for small angles, these are equivalent.

Figure 2-8. Location of the scanning mechanism, (a) Outside
the optical subsystem and (b) inside the optical subsystem.
Although shown as an astronomical telescope, it could be any
telescope (Cassegrainian, Galilean, etc.). The scanning
mechanism could be a rotating polygon, refractive prisms, or
(as shown here) an oscillating mirror.
2.3. DETECTORS and COOLERS

Current detector terminology lists detectors as cooled or uncooled. These
are only generic categories. Within each category, there are many different types
of detectors.

COOLED DETECTORS

LWIR photon detectors have to be cooled below 100K with 77K considered a
typical temperature. These temperatures can only be reached with a
mechanical cooler or with liquid nitrogen. Many MWIR detectors can operate
at 200 К and this temperature can be easily achieved with a thermoelectric
cooler (TEC). TECs appear to have an infinite lifetime whereas mechanical
coolers degrade over time. Coolers add cost, bulk, and consume power.

UNCOOLED DETECTORS

Thermal detectors can operate at room temperature and therefore are called
uncooled devices. Although called uncooled, these devices require a cooler to
stabilize the detector temperature. Usually a thermoelectric cooler is used.
Uncooled devices are lightweight, small in size, and easy to use. Because they
use very low power, they lend themselves to hand-held, battery-operated,
devices.

2.3.1. DETECTOR CLASSIFICATION

Detectors are classified as classical semiconductors, novel
semiconductors, and thermal detectors. Detector performance parameters and
therefore system performance parameters were developed for the classical
semiconductor and thermal detectors. More recently, novel semi-conductors such
as the Schottky barrier photodiode and quantum well have been introduced.
Classical theory is no longer appropriate and new measures of performance have
been developed.

There are many different types of infrared detectors. The following lists
the most popular types and some of their features.

CLASSICAL SEMICONDUCTORS

Photoconductive

The photoconductive detector requires a constant bias voltage. An absorbed
photon changes the bulk resistivity, which, in turn, changes the current. The
current change is monitored in an external circuit. Since current is
constantly flowing, the detector dissipates heat. As a result, very large
arrays are difficult to cool.

Photovoltaic

The photovoltaic detector is actually a p-n junction in a semiconductor. An
absorbed photon produces a voltage change that is sensed in an external
circuit. It does not dissipate heat and therefore can be built into very large
arrays. Since minimal current flows, it is relatively easy to couple the
detector to a low noise amplifier.

NOVEL SEMICONDUCTORS

Schottky barrier photodiode (SBD)

The SBD is a photoemissive device that produces a voltage. These detectors
are compatible with silicon fabrication technology. Therefore, it is relatively
easy to fabricate monolithic devices where the detector and readout are
fabricated at the same time. SBDs can be built into very large (5000 х 5000)
arrays.

Bandgap engineered photodetectors (quantum well)

The spectral response of quantum well detectors can be engineered (tuned)
to any wavelength. Unfortunately, it has a very narrow spectral response. Its
current limitation is that it requires significant cooling (below 60K).

THERMAL DETECTORS

Bolometer

As a bolometer absorbs heat, its resistance changes. It requires an external
bias. The change in current (due to the change in resistance) is monitored in
the external circuit. It somewhat difficult to dissipate the heat in large
arrays. Bolometers are usually optically chopped to improve sensitivity and
uniformity.

Pyroelectric

The pyroelectric detector can only sense a changing temperature. The
thermal changes alter the electrical polarization that appears as a voltage
difference. These AC devices create halos around high AT targets. These
systems typically have a chopper (to produce a changing scene) between
the lens system and detector. The chopper is synchronized with the frame
rate of the camera so that the displayed image appears uniform.
2.3.2. SPECIFIC DETECTORS

PHOTON DETECTORS

Silicide Schottky-barrier devices

The most popular is Pt: Si (platinum silicide) which is sensitive in the 1.0 to
5.5 pm region. It requires a filter to limit the spectral response (e.g., to
create a 3 to 5.5 pm system). Because of the low quantum efficiency, Pt:Si
has poor performance when the background temperature is less than 0°C.
Silicon technology is mature and very large arrays can be fabricated very
cheaply. The device is often cooled to 70K to reduce the dark current.

InSb (Indium antimonide)

InSb is a high quantum efficiency MWIR detector. It has generally replaced
Pt:Si in those systems using modest sized arrays (e.g., standard video
format with approximately 640x480 elements). It peak response is near 5
pm. A filter is used to limit the spectral response to the MWIR region.

HgCdTe (Mercury cadmium telluride)

Mercury cadmium telluride is also called merc-cad or MCT. Although
generally labeled as HgCdTe, it is a mixture (Hg^CdyTe). By varying the
ratio, the spectral response can be tailored to the MWIR or LWIR region.
The most popular is the LWIR detector with a peak response near 12 pm. A
filter is used to limit the response to the LWIR region. HgCdTe detectors
are used in all common module systems.

SPRITE (Signal processing in the element)

The SPRITE detector is a stretched HgCdTe filament that provides inherent
time-delay-and-integration (TDI). It was developed in United Kingdom and
has become the U.K. common module detector.

QWIP (Quantum well infrared photodetector)

The QWIP is based upon mature GaAs growth technology. The wells are
created by layers of GaAs/AlGaAs and the response can be tailored from 3
to 19 pm. The LWIR version typically has a spectral from 8.3 to 10 pm.
The responsivity and noise are temperature sensitive so that QWIP devices
are cooled to less than 60K.

THERMAL DETECTORS

As temperature sensors, thermal detectors can sense all wavelengths. Filters
are used to limit the spectral response. Thermal detectors generally have
much lower sensitivity (higher NEDT) than photon detectors. As a result,
they probably will not replace photon detectors in critical, low signal-to-
noise applications. A large variety of materials can be used as bolometer
and pyroelectric detectors.

2.3.3. DETECTOR OPERATION

The detector's electrical output is a temporal representation of target
spatial information. The detector electronics is matched to the detector
characteristics. For mathematical convenience, all noise sources are referred to the
detector output. Noise sources include photon noise, fixed pattern noise, 1/f noise,
detector internal noise, and amplifier or mux noise. The electronics may add
additional noise to the signal.

For many detectors, the specific detectivity, D* (pronounced dee-star), is
a figure-of-merit that combines the responsivity, R, with detector noise current, in\

R jAd kfe
D* = - ——

(2-1)

where A/e is the noise equivalent bandwidth, R is the responsivity, and Ad is the
detector area. Since D is a measure of detector noise, it is incorporated in the
classical NEDT equation. As D* increases, the NEDT decreases. When the
dominant noise is due to the random arrival of photons, the system is said to be
background limited or operating in BLIP (background limited performance) mode.

Real detectors do not achieve this theoretical maximum but have a Z>*
that is a function of operating temperature (Figure 2-9) and detector bias. The
detector temperature depends upon the cooling capacity of the cooler, the ambient
temperature, and the heat load induced by the electronics near the detector.

DETECTOR TEMPERATURE (K)

Figure 2-9. Typical U.S. common module HgCdTe detector
temperature characteristics. Most detectors sensitive to LWIR
radiation are optimized to operate at liquid nitrogen
temperature (77 K).
If the optical FOV is less than the detector FOV (equivalently, the optical
f-number is greater than the cold shield f-number), the detector can sense radiation
from the housing. Since most systems amplify the difference between the target
and background, the veiling glare is not seen on the monitor. However, the glare
introduces photon noise and this increases the NEDT. For staring arrays, the
veiling glare partially fills the charge wells and this limits the available scene
dynamic range.

Each detector/amplifier combination will have a different gain and level
offset (Figure 2-10) and these variations produce fixed pattern noise (FPN). For a
linear array of elements (Figure 2-3b), each line may have a different gain and
offset. This produces FPN in the vertical direction only and will appear as
horizontal streaks. For staring arrays (Figure 2-3c), each detector will have a
different response resulting in a two-dimensional FPN. Electronic gain/level
normalization removes FPN. Incomplete removal results in residual FPN noise.
Single element systems (Figure 2-5) and pure serial systems (Figure 2-7) do not
have FPN.

Figure 2-10. Responsivity for three differences detectors (Db
D2, and D3). (a) Exaggerated nonlinear responsivities and (b)
detector outputs for three different input intensities.
Responsivity variation cause fixed pattern noise.

Figure 2-1 la illustrates a full focal plane array (no scanner) where the
detectors are contiguous. In many arrays, the detectors do not completely fill the
cell area (Figure 2-1 lb). The ratio of active detector element area to the cell area is
the fill factor. If a small object is imaged onto one of these dead areas between the
detectors, there will be no output. As the small object moves, its image will move
on and off the active detector element resulting in a twinkling of the object. As the
detector area decreases, the amount of radiation decreases so that the NEDT
increases. Thus, as the fill factor decreases the NEDT increases. High fill factors
increase the ability to detect point sources.
Full Focal Plane Array
(a)

Fill Factor = —-—
ACell

Low Fill Factor Focal Plane Array
(b)

Figure 2-11. Fill Factor definition, (a) 100% fill factor and (b)
finite fill factor.

2.4. SPECIFIC SYSTEMS

For many years, civilian systems were derivatives of the militarized
systems. However, military systems are usually designed for maximum sensitivity
(very low noise). For many civilian applications, usually there is a strong signal
(large temperature difference) so that low noise is not a major design concern.
Cost, weight, ease of use, and power consumption now drive civilian system
design.

Militarized systems evolved in the following chronological order:

2.4.1. LINE SCANNERS

A line scanner requires platform motion to create a two-dimensional
image. Typically used for reconnaissance, it relies on aircraft or spacecraft motion
to provide the cross-scan dimension. Airborne reconnaissance systems typically
provide wide-angle imagery. The imagery can be obtained either with pushbroom
systems or line-scanners. Since large linear arrays are easily fabricated from
silicon, visible imagery is often obtained with a pushbroom design. Infrared
technology limits the number of detectors that can be placed in an array and
therefore infrared imagery is usually obtained from line-scanners (Figure 2-12).

The very early infrared systems were downward looking line scanners.
When designed to look out the front of an aircraft, they were called forward-
looking infrared (FLIR) systems. The acronym FLIR is used generically to denote
an infrared imaging system.
Electronics

Figure 2-12. Representative line-scanner functional block diagram.

2.4.2. COMMON MODULE SYSTEMS

Figure 2-13 illustrates a diagram of the early common module system.
The detector converts the infrared spatial information into time-varying voltages.
The voltages drive light emitting diodes (LEDs) to create a visible image. By using
the back face of the scanning mirror for the visible image, the visible image is
always synchronized with the infrared image. They were called common modules
because the scanner, detector, pre-amplifiers, post-amplifiers, and light emitting
diodes (LEDs) were manufactured to common specifications. The term now
loosely refers to any LWIR system with bi-directional scan. These systems are
still in existence and lend themselves to small, lightweight man-portable
applications.

Based upon blackbody curves, terrestrial objects emit more radiation in
the LWIR region compared to the MWIR region. Further, for a fixed target-
background temperature difference, the radiance difference (as specified by the
thermal derivative) is about 10 times larger in the LWIR region than the MWIR
region. These facts prompted the design of LWIR common modules rather than
MWIR systems.

The designer's only options were changing the optics to achieve the
desired field-of-view, adding electronic filtering to maximize the signal-to-noise,
and modifying the visual optics to present an appropriate image to the observer.
The pre-amplifier is a low-noise fixed-gain amplifier whereas the post-amplifier
had a variable gain. Each post-amplifier was individually adjusted to compensate
for different detector responsivities and pre-amplifiers gains. Nearly all U.S.
systems today evolved from the common module design.

server

Figure 2-13. Representative common module block diagram.

Example 2-1
EQUIVALENT OPTICAL DESIGN

Design the smallest practical astronomical telescope for a "0.1 mil"
thermal imaging system using U.S. common module elements.

A "0.1 mil" FLIR has a detector-angular-subtense (DAS) of 0.1 mrad.
The common module reimager focal length is 2.67 inches with a f-number of 1.7.
The original common module detector size was 0.002" x 0.002" and is sensitive in
the 8 to 12 pm spectral region. Figure 2-14a provides a stylized optical layout.

As a design starting point, the Airy disk diameter is matched to the
detector size. Equivalently, the Airy disk angular subtense is matched to the DAS:

0 = DAS = 2A4—. (2-2)

D
Using an average wavelength of 10 gm the required clear aperture is Z> = 244 mm
= 9.6 inches. Assuming a f9 design, the detector subtends 0reimager = d/flreimaser =
0.002/2.67 = 0.75 mrad. Since the required system resolution is 0.1 mrad, the
afocal power must be Muptics = 0.75/0.1 =7.5. With an afocal telescope,

^=^ = ^^ = 7.5. (2-3)

For maximum throughput, the afocal telescope and reimager should have

the same f-number. The f-number is F~fl/D. Then

and flo = F D = 1.7 x 9.6 = 16.32 inches (2-4) fl„ = ——— = 16’32 = 2.18 inches . (2-5) Moptics 7.5

The afocal length is flo+fle= 18.5 inches. The equivalent focal length is flreitnagelM=
20 inches. Figure 2-14b illustrates the equivalent design and the detector subtends

d 0.002 _ t

DAS =----------------------------=--------= 0.1 mrad . (2-6)

equivalent focal length 20

(b) Equivalent Design

Figure 2-14. Optical layout, (a) A afocal telescope and (b) the
equivalent single-lens design. Although only three lenses are
shown, the system consists of many elements.
2.4.3. EOMUX SYSTEMS

The common module system permitted only one observer to view the
imagery. To provide multiple observers access, the LED output is scanned by a
vidicon. The vidicon converted parallel scanned infrared information into a serial
data stream consistent with the monitor requirements. As such, the vidicon
provided scan conversion. To offset any degradation in MTF, a boost amplifier
can be added in the video chain. The electro-optically multiplexed (EOMUX)
system added the vidicon, boost, and monitor (Figure 2-15).

Figure 2-15. Representative EOMUX functional block diagram.

2.4.4. EMUX SYSTEMS

The EOMUX system employed LEDs and a vidicon to produce a data
stream consistent with the monitor requirements. This function can be performed
by a digital scan converter (DSC) to create an electronically multiplexed (EMUX)
system (Figure 2-16). Most scanning systems today are EMUX systems.
Gain/level normalization, image processing, and line-to-line interpolation are
typically performed digitally. The conversion from an analog signal to digital data
‘ introduces unique nonlinear effects. Quantization introduces an additional noise
factor whose value depends upon the relative magnitude of the least significant bit
(LSB).
Figure 2-16. Representative EMUX functional block diagram.

For scanning array systems with more than one detector in the vertical
direction, the output of every detector is collected nearly simultaneously at discrete
locations. That is, the scene is digitally sampled in columns as the detectors move
horizontally, In memory, the vertical data is read out horizontally in a format
consistent with monitor requirements. This conversion from a vertical input to
horizontal output is called digital scan conversion and the hardware is called the
scan converter.

In a digitally scan converted system, two different timings exist. The
detector electronics is matched to the scanning parameters in time and active scan
time. The digital data is placed in a digital memory and read out at a rate that is
consistent with monitor line rates.

Image processing algorithms can be used to enhance images, suppress
noise and provide boost. Boost increases the signal amplitude at specific spatial
frequencies but it does not necessarily affect the signal-to-noise ratio. For systems
that are not noise limited, boost may improve image quality. However, for noisy
images, the advantages of boost are less obvious.

Systems with detector arrays or scanning patterns that are not consistent
with analog video formats require image formatting. Monochrome video formats
may either be the U.S. standard1 RS 170 that displays 485 TV lines or may be the
European CCIR standard that displays 577 lines. Generally, infrared imaging
systems manufactured in the U.S. provide 480 active lines with five blank lines to
be consistent with RS 170. An LWIR common module system, which creates
infrared 360 lines, requires vertical interpolation. The interpolator increases the
line rate and may change the video sample rate to provide the number of TV lines
required by the monitor. Interpolation can be achieved by the duplication of lines
or may be a more complex algorithm. The interpolator can dramatically affect
vertical resolution and therefore can greatly affect the ability to resolve vertical
detail.

It is desirable to have a linear system such that if the signal intensity
doubles, then the monitor luminance doubles. However, CRT-based monitors tend
to be nonlinear in terms of output luminance versus input voltage. The slope of this
relationship on a logarithmic scale is the monitor gamma (Figure 2-17). To
provide a linear relationship, the inverse gamma is inserted as an image processing
algorithm (gamma correction). Here, there is no longer a linear relationship
between the analog output video voltage and the input signal intensity. This
nonlinear relationship affects all data collected from the analog video signal and
becomes a measurement problem.2 If a gamma correction algorithm is present,
linearity is only assured when viewing a CRT-based monitor. Flat panel displays
are linear (equivalently, gamma is unity) and therefore no gamma correction is
required.

The D/А converter provides an output that typically has a stair step
appearance. Each step occurs at each digital sample. The post-reconstruction filter
smoothes the data and removes the stair step. Its output is the analog video.

Figure 2-17. Definition of CRT-based monitor gamma.
2.4.5. SECOND GENERATION SCANNING SYSTEMS

The common module system was called a first generation system. In the
late 1980s, it appeared that LWIR scanning systems could be replaced with staring
arrays. At that time staring arrays were called second generation systems.
However, detector technology did not advance as quickly as anticipated so that a
serial/parallel scan mechanism was used - so called second generation scanning
systems. These systems contain arrays that consist of 480*4 TDI detector
elements. These systems tend to use staggered arrays (Figure 2-18).

The 480 elements are consistent with U.S. monitor requirements. Other
detectors available include 288*4 TDI. With interlace, these detectors provide 576
IR lines to be consistent with European formats.

Figure 2-18. Staggered array 480*4 TDI elements. The
staggered configuration increases the sampling frequency in
the vertical direction.

480

2.4.6. STARING ARRAY SYSTEMS

Figure 2-19 illustrates a typical staring array. It does not have a scanner.
Each detector output is digitized by the detector mux. While amplifiers and filters
exist in the mux, they do not process the signal in the same manner as a scanning
system.

Figure 2-19. Representative staring array functional block diagram.
Every detector/amplifier combination will have a different gain
(responsivity) and offset. These variations result in fixed pattern noise or spatial
noise. If large deviations in responsivity exist, the image may be unrecognizable.
As a result, systems employing more than one detector may require gain/level
normalization or nonuniformity correction (NUC) to produce an acceptable image.
Although most literature discusses NUC for staring systems, it applies to scanning
systems that have more than one detector in the cross scan direction (Figure 2-3b).
For good imagery, the individual detector outputs are normalized (made equal) for
several discrete input intensities. These normalization intensities are also called
calibration points, temperature references, or simply points.

Figure 2-10a illustrated the responsivity of three different detectors
before gain/level normalization. The outputs for input intensities Ib I2, and I3 were
shown in Figure 2-10b. Figure 2-20 illustrates the normalized output after
correction at two points. If all the detectors had linear responsivities, then all the
curves would coincide. As individual detectors deviate from linearity, the
responsivity differences become more noticeable.3 It is this variation in
responsivity that creates the residual fixed pattern noise after gain/level correction.

Figure 2-20. Responsivity curves after 2-point correction.
Responsivity deviations from linearity produce residual fixed
pattern noise.

2.5. SYSTEM MAGNIFICATION

There are two different magnifications present. One is the optical
telescope magnification that determines how large the image will be on the
detector plane. The other relates to the magnification seen by the observer. System
magnification is simply the ratio of the horizontal FOV subtended by the observer
and the system horizontal FOV.

M sys

FO Vobserver
FOVsys

(2-7)

The observer's field-of-view changes with the distance from the monitor. Since the
observer can move arbitrarily close to the display,

FO Fobserver

= 2 tan 1

2R,

(2-8)

where R is the distance to the display and W is the width of the display.

2.6. REFERENCES

1. "Electrical Performance Standards - Monochrome TV," EIA Standard RS-170, Electronic Industry
Association, NY, NY.

2. G. C. Holst, Testing and Evaluation of Infrared Imaging Systems, second edition, pp. 57-60, JCD
Publishing, Winter Park, FL (1998).

3. N. Bluzer, "Sensitivity Limitations of IRFPAs Imposed by Detector Nonuniformities," in Infrared
Detectors and Arrays, E. L. Dereniak, ed., SPIE Proceedings Vol. 930, pp. 64-75 (1988).

EXERCISES

1. Referring to Figure 2-9, what D value would you use for the NEDT
calculation? When would you use another value? (Hint: consider ambient
temperature of an air-conditioned laboratory and operation in the desert).

2. An electro-optical imaging system has an FOV of 5ox5o. The user sits 36 inches
from a 14-inch (measured diagonally) display. If the observer moves 2 inches
toward the display, what is the system magnification change? Is system
magnification a useful parameter?

3. What is the smallest astronomical telescope needed for a 0.2 mil thermal
imaging system? A common module reimager is available (/7= 2.67" and F= 1.7).
The detector width is 0.0016 and the system is sensitive in the 8 to 12 pm region.

4. Using the values in Exercise 3, design a telescope for a system operating in the 3
to 5 pm region. Compare the answer to that obtained in Exercise 3. How would the
design change if the system operated in the visible region of the spectrum?

5. Plot the output MTF versus input MTF for a monitor with у = 0.7, у = 1, у = 1.4,
and у = 2. The MTF is (Vntax-Vnйп)/(Vnuif-Vnjif). Let the minimum and maximum
voltages be modified by Voui= VinY.
6. Contrast is C = (Z/ZB)-1 where LT is the target luminance and LB is the
background luminance. Assume that the video output, V, is proportional to the
input luminance and that the monitor has a gamma such that the monitor
luminance is Z = К F7. Derive the relation between the input contrast and output
contrast.

7. Figure 2-20 illustrated residual FPN as a function of input intensity with 2-point
correction. Sketch the output for 1-point, 3-point and 5-point correction.

8. Figure 2-4b illustrated how the signal level decreases according to cosineN0 for
a staring array. The responsivity can be made equal for all the detectors by
increasing the gain of each detector the appropriate amount. What effect would
this have on noise measurements?

9. Describe a scenario in which cosineNG correction is appropriate (Hint: during
low gain operation, the noise is below perceptibility).
3

RADIOMETRY

Radiometry describes the energy or power transfer from a source to a
detector. When the source size is much larger than the projected area of the
detector, the source is said to be resolved or the system is viewing an extended
source. Equivalently, the detector is flood-illuminated. The noise equivalent
temperature differential and minimum resolvable temperature apply to a system
viewing an extended source. As the source size becomes smaller, diffraction limits
the image size. The ratio of the actual response to the ideal response is the target
size function (TSF). As the source size approaches a point, the target size function
approaches a constant value called the point visibility factor, ensquared power
value, or optical shape factor.

The flux emanating from the source is attenuated by the intervening
atmosphere, focussed on to the detector by the optical system and then converted
into a measurable electrical signal by the detector. Planck's blackbody radiation
law describes the source's spectral radiant exitance. For scanning systems, the
power is usually specified in watts whereas for staring systems it is convenient to
use photons per second.

The sun, moon, or starlight illuminates scenes in the visible. Targets are
detected when reflectivity differences exist. The spectral radiant exitances are
typically integrated over the eye's spectral responsivity and photometric units are
used (e.g., lumens).

Although most thermal imaging systems respond to radiant flux
differences, it is convenient to specify the radiant flux difference between a target
and its immediate background by an equivalent temperature difference, AT.

Using average values of the various transmittances simplifies analyses.
However with broad spectral response systems, the average transmittance depends
on the source's spectral radiant exitance, the atmospheric and optical spectral
transmittance, and the detector spectral response.

3.1. RADIATIVE TRANSFER

Radiant sterance, Le, is the basic quantity from which all other
radiometric quantities can be derived (e.g., radiant exitance, radiant flux, or radiant
Radiometry 41
intensity). It contains both the areal and solid angle concept1 that is necessary to
calculate the radiant flux incident onto a system. It is the amount of radiant flux,
дФ, radiated into a cone of incremental solid angle dil from a source whose
incremental area is dAs (Figure 3-1):

Э2Ф w
dAsdCl m2 -sr

(3-1)

Similarly, Lq is the photon sterance in units of photons/(s-m2-sr). The quantities Le
and Lq are invariant for an optical system that has no absorption or reflections.
That is, Le and Lq remains constant as the radiation transverses through the optical
system.

For Lambertian sources, the spectral radiant exitance is related to the
spectral sterance by:

Ме (Л, Т) = лЬе{А, T) —-- (3-2)

m -pm

and the spectral photon exitance is

r photons

Mqi^T) - • (3“3)

s-m -pm

Figure 3-1. Radiant sterance.
42 Electro-optical imaging system performance

3.1.1. PLANCK’S BLACKBODY LAW

The spectral radiant exitance of an ideal blackbody source whose
absolute temperature is T, can be described by Planck's blackbody radiation law:

меа,г>=^-

(с2МЛ

(3-4)

where the first radiation constant is q = 3.74 1 8x108 watt-pm4/m2 and the second
radiation constant is c2 = 14388 pm-K. Figure 3-2 illustrates Planck's spectral
radiant exitance in logarithmic coordinates. Since a photo detector responds
linearly to the available power, linear coordinates may provide an easier
representation to interpret (Figure 3-3).

1E+08 т 4000 К

Q
S

О
111
CL
U)

1E+07

1E+06

1E+05

1E+04

1E+03

1E+02

1E+01

1E+00

0.1 1.0 10.0 100.0

WAVELENGTH (pm)

Figure 3-2. Planck's spectral radiant exitance plotted in
logarithmic coordinates for T = 300, 400, ... , 4000 K. The
units are w/(m2-pm).

WAVELENGTH (pm)

Figure 3-3. Planck's spectral radiant exitance plotted in linear
coordinates. The units are w/(m2-pm).
The spectral photon exitance is simply the spectral radiant exitance
divided by the energy of one photon (hdXy.

<3'5>

where the third radiation constant is c3 = 1.88365xlO27 photons-pm3/s-m2, Planck's
constant is h = 6.626х IO’34 J-s, and the speed of light is c=3x108 m/s. Figures 3-4
and 3-5 provide the spectral photon exitance in logarithmic and linear coordinates
respectively.

WAVELENGTH (pm)

Figure 3-4. Planck's spectral photon exitance for T =300, 400,
... ,4000 K. The units are photons/(s-m2-pm).

WAVELENGTH (pm)

Figure 3-5. Planck's spectral photon plotted in linear
coordinates. The units are photons/(s-m -pm).
When radiation is incident on an object, some of it is transmitted, some
absorbed, and some is reflected. Energy conservation requires that

^transmitted ^absorbed ^reflected Ф incident • (3-6)

Expressed as a ratio,

т(Л) + а(Л) + р(Л) = 1, (3-7)

where т(Л), а(Л), and р(Л) are the transmittance, absorptance, and reflectance
respectively. The emissivity, reflectivity, and absorptZvzTy can be calculated from
Maxwell's equations and represent the values for an ideal material. Real materials
deviate from the ideal properties and have emittance, reflectance, and absorptance.

Real blackbodies do not emit all the radiation described by Equation 3-4
but only emit a fraction of it. The ratio of actual radiant sterance to the theoretical
maximum, МВВ(Л,Т) is the emittance

,n_ MacluralO,O £{Л 1 — , мвво,т) (3-8)

By Kirchkoff s law, when an object is in thermal equilibrium with its environment,
absorptance is equal to emittance:

а(Л) = £ (Л). (3-9)

This leads to the popular saying "Good absorbers are good emitters." For ideal
blackbodies, e(2) = 1.

3.1.2. EXTENDED SOURCE

If an imaging system is at a distance Rr from a source (Figure 3-6), the
radiant flux incident onto the optical system of area Ao is

д Ф — т £_ д т lens -у s*sL atm * Ri (3-10)

where the small angle approximation was used and Tntm is the intervening
atmospheric transmittance. The on-axis radiant flux reaching the image plane is:
Radiometry 45

ф. - J ____9- л T . т

image e 2 5 optics * atm ">

(3-11)

where xop6cs is the system's optical transmittance. By definition, the image of an
extended source is larger than the detector area (At-» Af/). That is, the detector is
flood illuminated.

Infrared Imaging
System Entrance
Aperture Area, д

Image of
Source
Area Aj

\ Detector

Figure 3-6. An imaging system directly viewing a source.

The radiant flux incident onto the detector is simply the ratio of the areas

detector

= ф .

image

(3-12)

Using the small angle approximation for paraxial rays

^s _

(3-13)

The radiant flux becomes

Ф detector

____________^e^o^d________________________________
,/Z2(l+4/npfa)2

optics ? atm

(3-14)

The optical magnification is Mop,ics=R2IRr Here, Rt and R2 are related to the focal
length,/?, by
1 1 1

+ R2 ~ fl '

(3-15)

Assuming a circular aperture and defining the f-number as F=fl/D (see Appendix)

detector

Л LeAd

4 F2(l + M„ptics)1

r . T
optics л atm ">

(3-16)

The voltage produced by a detector is proportional to the detector's
responsivity, Rd by

^d ^d detector

(3-17)

The system output, Vsys is simply Vd multiplied by the system gain, G. Since all of
the variables are a function of wavelength and the source spectral radiant sterance
depends on temperature

^2 Л ГЯ 1

Vsys = G |\, (Я) - g( ’ - ?optics (Я)^ (Я)<U. (3-18)

2 0 * Moptics)

Off-axis responses may be reduced by cosineN0

What is of interest is the signal difference produced by a source (target) at
temperature Tr and its immediate background at temperature TB

\Vsys = G (Я)E(Л’T WTatm(Я)</Я.(3-19)

2 4F1(1 + Mpptics)1

If the source and background are ideal blackbodies with unity emittance, then

л2

A^=G рг„(Я)

Л)Д („0)

4F2 (1 + M optics)2

When 7?!» R2 or equivalently when M—> 0,

= G рг,(Я) 1Д/еО>Гг) .(3-21)
When the wavelength is less than 3 im, targets are differentiated by
reflectance differences. Here, both the target and the background are assumed to
be illuminated by the same source (sun, moon, nightglow, or starlight)

Л г/ _ r f n / ; x [pr(^) PbW]Le(A,Tul)Ad

^^sys G I vO 2 ✓-« яж \2 opUcs^y^atm (3-22)

J 4Г2(1 + М0/,и„)2

The source radiant sterance is used since the illuminating source may not be an
ideal blackbody.2 The variables p7(2)and PbW are the spectral reflectances of the
target and background, respectively. If Planck’s blackbody law cannot describe the
source's spectral exitance, then the actual spectral radiance sterance must be used
in the appropriate radiometric equations. This is especially true for starlight and
nightglow conditions.2

The T-number is sometimes useful for evaluating different optical
designs. It is the f-number divided by the square root of the optical transmittance:

(3-23)

Then the extended source radiometric equation (Equation 3-21) becomes

ЛГ -r fff (n[^eO,Tr)-MeO,rB)l4d

△ *sys I d 1 atm (Л)аЛ . (3 -24)

System output does not infer anything about the source other than that of
an equivalent blackbody of a certain temperature would provide the same output.
This is true no matter what output units are used (volts, amps or any other arbitrary
unit). These units, by themselves, are not very meaningful for system-to-system
comparison. For example, can be increased by increasing the system gain G.
As such, it is dangerous to compare system response based upon only a few
numbers.
Example 3-1

\Vsys for a LWIR IMAGING SYSTEM

What is the expected output voltage difference for a US common module
system? The system is viewing an extended 3I0-K target against a 300-K
background in a collimator. The collimator optically places the target at infinity so
that Equation 3-21 is appropriate. Assume the HgCdTe detector has a peak
responsivity of 20,000 v/w, = 1, Toptics(f) = 0.70 (all independent of
wavelength), F= 3 and G= 10,000. Common module detectors are band limited
from 8 to 12 pm. Outside this region the response is zero. The detector element is
square with each side equal to 0.002 inches (5.08 x I0'3 cm) so that the detector
area is Ad = 2.58 x 10'9 m2. For convenience, the target and its background are
considered ideal blackbodies. Assume that the system focal length is 18 inches.

The responsivity of most photoconductors follows

W ~ ~ Rpeak •
peak

(3-25)

Then the output voltage difference for an on-axis detector is:

т AH 2

A Vsvs = G [-----\me (2,310) - Me (2,300)]</2. (3-26)

4F2 J 2 k

This can be approximated by

12 Л

LVsys = O.Ol^------Л/Д2,Д7’)Д2, (3-27)

8 Apeak

where - Mff,300). Using Simpson's rule, the integral is

evaluated numerically at the center of each increment. For example, when АЛ = 1,
the interval [8 pm, 9 pm], is evaluated at 8.5 pm (Table 3-1).
Table 3-1

AV CALCULATION
________________

Wavelength Qum) Z/ Xp Mc(X,310) Mc(X,300) X/kp-Mc(X, AT)
8.5 0.708 36.0 30.0 4.25
9.5 0.792 36.8 31.2 4.44
10.5 0.875 35.7 30.8 4.29
11.5 0.958 33.5 29.2 4.12
sum =17.1

Then &Vsys = (0.01)(17.1) = 171 mv. The wavelength interval, Az = 1 gm, was
selected for illustrative purposes only. For numerical integration, at least 20
intervals should be chosen.

Historical note: The early common module detectors were 0.002 inches by 0.002
inches. The size has changed to 0.0016 x 0.0024 inches. Since the array is still
linear and contains 180 elements, it is still called a common module detector.

3.1.3. POINT SOURCE

As the source area approaches zero, the source becomes an ideal point
source. Geometric optics predicts that the image size also will approach zero.
However, diffraction and aberrations will limit the minimum image size. The
differential system output, AVsys, depends on the relative size of the blur diameter
to the detector size. If the source solid angular subtense (A JR2) is much less than
the detector solid angular subtense (AJR22), the source is considered a point
source.

If the blur diameter is much less than the detector size,

ф . . t = ф .

detector image

(3-28)

The detector will sense radiant flux from the point source and the radiant flux from
the immediate background within the DAS. In Figure 3-7, Adas is the projected
area of the detector angular subtense m object space. The radiant flux is
proportional to

Ф detector = V, 77 )AS + Le(Л, TB ){Adas - As )] . (3-29)
Figure 3-7. Small target inside the projected area of a DAS.
The detector senses radiation from the target and the back-
ground.

The radiant flux difference between this detector and an adjoining detector
location is

ДФ = к {\be (Л, TT )As + Le (Л, TB )(Adas - As )] - Le (Л, TB )Adas } (3-30)

△Ф = к \Le a,TT)~Le (Л, Tb)]as .

By definition,

Adas

(3-32)

and assuming a circular aperture and ideal blackbodies with unity emittance

/^2

=G (ДДЯ) ЦА f торас!!(Л)Та,т(Л)(1Л , (3-33)

i bF\l + Moptiay Adas

where LJf, AT) = -LffTfy Figure 3-8 illustrates the ideal (geometric)
image area as a function of source area and where Equations 3-19 and 3-33 are
applicable. The slope of the line is the ratio AJAdas. When As = Adas the two
equations are identical.
Source Area A s

Figure 3-8. Geometric relationship between source and image
areas. When As - Adas, Equations 3-19 and 3-33 are identical.

Diffraction and aberrations will limit the smallest image size that can be
achieved. Figure 3-9 illustrates three cases: (a) the geometric approach where there
is no diffraction, (b) when the blur diameter is less than the diameter of a detector,
and (c) when the blur is larger than a detector. Figure 3-10 illustrates the
relationships depicted in Figure 3-9. Curve A is the ideal case and is identical with
Figure 3-8. Curve В illustrates a system where the diffraction blur area is smaller
than the detector area. This case is equivalent to Figure 3-9b and Equations 3-19
and 3-33 are applicable as shown in Figure 3-8. For curve C, diffraction produces
a blur diameter that is larger than the detector. Here, Equation 3-19 is only valid
when curve C asymptotes to the geometric curve.

When diffraction is important, Equation 3-33 must be modified by the
aperiodic transfer function, ATF:

A Vsys =G $RdW A TF Wdx (3_34)

2 (1 + Moptics )

Equation 3-34 combines Equation 3-19 and 3-33. When the source is resolved,
ATF~ 1 and Equation 3-19 is obtained. When Ai<A(b Equation 3-33 is obtained.
Figure 3-11 illustrates the ideal and system ATFs and the applicable regions for
the equations. The ATF is the input-to-output transformation versus As
normalized to unity. Calculation of the system ATF is beyond the scope of this
book. However, the limiting case, where the image size becomes independent of
As, is of interest for this is the case when the system is viewing a point source.
Figure 3-9. Various image sizes, (a) No diffraction, (b) blur
diameter less than detector size, and (c) blur diameter greater
than detector size.

Source Area As

Figure 3-10. Relationship between image size and source size
for cases (a), (b) and (c) given in Figure 3-9.
Figure 3-11. Aperiodic transfer function. Equation 3-34
asymptotes to Equation 3-19 for extended sources and to
Equation 3-33 for point sources.

The aperiodic transfer function depends on target size. For most range
predictions, the source is either considered a point source (IRST applications) or
resolved (general imagery). For point sources, the ATF is the point visibility factor
and for resolved sources, the ATF is one. The change from a point source to a
resolved source is gradual and depends upon range. Poropat3 predicted range
performance over these intermediate values.

The target size function (TSF) is

ATFSVS

TSF =-------= A TFtm

A TF y

‘ ideal

lDAS

when As < Adas (3-35)

and

TSF = A TF^ when As> ADAS .

(3-36)

Substitution yields

/1*2

A Vsys= G U TSF roptics{X)Talm (Я)dA. (3-37)

J 4F\1 + Moplicsy Adas

As As becomes smaller, the TSF approaches a constant (the point visibility factor
or PVF). The PVF is also called the blur efficiency, ensquared power,4 and optical
shape factor.5 As As approaches zero, it is appropriate to represent Le(A,AT)As by
the source radiant intensity /е(2,Л7) = /е(2,Тг) -IffTf). Point sources are assumed
to exist at a long distance from the imaging system so that M—> 0. Then

Л г/ — f J? < I PFV fQAT

AJ/yys ~ Fr I Rtf (/I) .2 a optics atm ' (3-38)

r 4F Adas

Substituting Equation 3-32 into Equation 3-38 provides the more familiar form of
the point source equation:

Д Vvs = G Гядл) PFV Topacs WTaan (Л)ал. (3.39)

, Ry

The maximum value of the PVF occurs when the image is centered on a
detector. As the image moves off-center, the PVF decreases and reaches a
minimum when the image center is exactly between detectors. While this phasing
effect is present in all systems, it is most noticeable in staring arrays. The PVF
refers only to the relationship between the detector size and the blur diameter. The
measured signal may be further modified in width and amplitude by electronic
MTFs.

The point visibility factor is omitted in some point source derivations.
This is appropriate if the detector size is small compared to the image size (PVF =
1). Phasing effects can significantly affect the PVF. However, the PVF can only be
calculated on a case-by-case basis since it depends upon the optical blur diameter,
detector size and the detector-to-detector spacing.

Example 3-2
POINT SOURCE

The system described in Example 3-1 is viewing a small target (As = 2x 10 9 m2)
in a collimator with flcol=60 inches. The point visibility factor is 0.65. What is the
expected output?

The projected detector area in a collimator focal plane is

flioi

DAS ~
fl

(3-40)
Using Equation 3-38 and A/e(2,A7>4s=Ze(2,A7),

дк (—ме(л,лт) ал (3-41)

4^ Adas £ Ap

which is approximated by

ЛИ
sys

G ^icJ^Rp^k у (A, дПдя

Л5РРГ
^DAS

(3-42)

From Example 3-1, the first bracket is 171 mv and the second bracket is 0.045.
Then AUSJS = 171 x (0.05) =8.6 mv. The output is significantly smaller than that
obtained for an extended source of the same temperature.

3.2. AT CONCEPT

It is convenient to express the small radiant exitance difference between a
target and its background by the temperature difference between the target and its
background. That is, it is desirable to express Af y,5 as a function ofJT and call the
proportionality constant the signal transfer function (SiTF). The SiTF is a
laboratory measure that does not include the atmospheric transmittance \Tatm(X) =
1]. For extended sources,

AK5J5 = SiTF AT.

(3-43)

This approach assumes that the target and background are ideal blackbodies with
unity emittance. The difference between two values can be expressed as a Taylor
series expansion

Ме(Л,Тв+ЛГ)-Ме(Л,Тв) =

(3-44)

аме(я,рв)

ЛТ +

а2ме(я,тв)

дТ2

ДГ2

2
For small values of AT, the first term is used and then the radiant exitance
difference is proportional to the partial derivative of Planck's law with respect to
temperature ("thermal derivative")

where

дМе(Л,Тв)
дТ

= Ме(Л,Тв)

ЛТ\еС11хт -V)

(3-45)

(3-46)

The SiTF is

SiTF — G optics •

J dT 4Fl

(3-47)

The thermal derivative is a function of wavelength and the background
temperature (Figure 3-12). As a result, all performance predictions that use AT as
an input will be affected by the background temperature and system spectral
response. The AT concept is a matter of convenience. Infrared imaging systems
that have photo detectors do not measure temperature but respond to radiance
differences. While the AT concept may be useful for thermometers, it does not
uniquely specify system performance unless both the system spectral response and
the background temperatures are specified.

Figure 3-12. Thermal derivative of Planck's blackbody
radiation law for three different background temperatures. The
units are w/(m2-pm-K)
The 3 to 5 pm region is more sensitive to background changes than the 8
to 12 gm region. Since the MRT and NEDT are inversely proportional to the SiTF,
the MRT6'8 and NEDT increase as the ambient temperature decreases. The linear
approximation given by Equation 3-43 is only valid for small excursions about the
background temperature. Departures from linearity are apparent when AT> 10°C
in the LWIR and AT> 5° C in the MWIR. The nonlinearity can be avoided if all
calculations and measurements used watts or photons rather than AT. Since the AT
concept depends upon the system’s spectral response, different LWIR systems
(e.g., 7.5 to 11.5 pm and 8 to 12 pm) will produce different responsivity curves
when plotted as a function of AT. Nevertheless, AT has become an industry-wide
standard and the nonlinearity exists in all input-to-output transformations (e.g.,
SiTF, NEDT, MRT, and MDT).

3.3. PHOTOMETRY

Photometry describes the radiative transfer from a source to a detector
where the units of radiation have been normalized to the spectral sensitivity of the
eye. It applies to all systems that are sensitive to visible radiation (0.4 to 0.7 pm).
The luminous flux emitted by a source is

0.72

F = 683 eO,T)As dX lumens, (3-48)

0.38

where F(2) is the eye’s photopic responsivity9 normalized to unity at 0.555 pm.
Numerous photometric units exist (cgs, mks, and English).10

Photometric calculations may be carried out exactly as radiometric
calculations using the appropriate equations provided in this chapter.

3.4. NORMALIZATION

"Normalization .... is the process of reducing measurement results as
nearly as possible to a common scale.’’11 Normalization is essential to insure that
appropriate comparisons are made. Figure 3-13 illustrates the relationship between
the spectral response of a system to two different sources. The output of a system
depends upon the spectral features of the input and the spectral response of the
imaging system.
WAVELENGTH

Figure 3-13. Sources with different spectral outputs can produce
different system outputs. T2 provides more radiant flux than
The system output will be higher when viewing T2.

Variations in output can also occur if ’’identical” systems have different
spectral responses (Figure 3-14). Spectral mismatch is a major contributor to fixed
pattern noise in focal plane arrays when each detector has a different spectral
response.12 Equation 3-21 is integrated over the interval [2 b 2 2] for System 1 and
over the interval [23, 24] for System 2. Because of the spectral mismatch, Ф
bVsy*.* For example, an infrared imaging system whose spectral response is 8 to 12
pm may have a different responsivity than a system that operates 7.5 to 11.5 pm
although both systems are labeled as LWIR systems. Systems can be made to
appear as equivalent or one can be made to provide better performance by simply
selecting an appropriate source or atmospheric condition.

Figure 3-14. Different spectral response systems can produce
different outputs when viewing the same source.
It is sometimes useful to discuss average responsivity or average
transmittance. The term average represents the mean value of a function. If/(x) is
weighted by another function g(x), the average of/(x) over interval [a,b] is

ave

(3-49)

Using this methodology, the average optical transmittance becomes

(3-50)

Similar equations exist for the average atmospheric transmittance. Equation 3-50
indicates that the average response depends on the source's spectral characteristics
(e.g., the source temperature) and its relationship to the detector spectral
responsivity.

The extended source radiometric equation becomes

Д1/ =Gt

r sys w L optics-ave

JRd (Я) Me (Л’ АГ)А-а ТаОя (Л) dA. (3-51)

J 4F2(1 + Mopdcs)2

In general, it not permissible to "pull out” each variable as an average value. This
back-of-the-envelope approximation is only valid if the parameters are
independent of wavelength over the spectral region of interest. As AX becomes
smaller, the approximation becomes more accurate and is correct for very narrow
band systems such as laser systems:

лг ~ г в Ме(Л,ЛТ)Аа т

^'sys optics-ave ^d-ave .r2z1 м \2 1 atm-ave '
(1 AZ Optics )
Example 3-3

AVERAGE OPTICAL TRANSMITTANCE

What is the average optical transmittance of a lens system for the LWIR
system described in Example 3-1? The optical spectral transmittance, Toptics(X), is
given in Table 3-2.

The average optical transmittance, Toptical.ave, is approximated by
^2

L Toptics (Л)Ме АЛ

t . = .h______________________________ {

optics-ave '

^«rf(A)Me(2,AT)AA

Я1

In Table 3-2, Simpson's rule is used to numerically evaluate the integral at the
center of each interval (Ai =1). The values for Me(f ЛТ) are given in Table 3-1.

Table 3-2

Toptical_ave OBTAINED by NUMERICAL INTEGRATION

X AMC ^optics O'") R„0.) МАЛТ) t011UcA)
8.5 0.708 60 0.8 33.98 42.48
9.5 0.792 56 0.8 35.48 44.35
10.5 0.875 49 0.7 30.01 42.87
11.5 0.958 43 0.6 24.71 41.19
sum = 124.2 sum = 170.9

The average transmittance is 124.2/170.9 = 0.727. The average transmittance could
be substantially different if the blackbody source had a different absolute
temperature. As with the previous examples, zfi= 1 im was chosen for illustrative
purposes. Smaller increments should be selected when performing the calculation.

3.5. REFERENCES

1. C. L. Wyatt, Radiometric System Design, Chapter 3, Macmillan Publishing Co. New York, NY
(1987).

2. D. Kiyskowski and G. H. Suits, "Natural Sources," in Sources of Radiation, G. J. Zissis, ed., pp.
151-209. This is Volume 1 of The Infrared and Electro-Optical Systems Handbook, J. S. Accetta and
D. L. Shumaker, eds., copublished by Environmental Research Institute of Michigan, Ann Arbor, MI
and SPIE Press, Bellingham, WA (1993).
3. G. V. Poropat, "Effect of System Point Spread Function, Apparent Size, and Detector Instantaneous
Field of View on Infrared Image Contrast of Small Objects," Optical Engineering, Vol. 32(10), pp.
2598-2607 (1993).

4. L. M. Beyer, S. H. Cobb, and L. C. Clune, "Ensquared Power for Circular Pupils with Off-center
Imaging," Applied Optics, Vol. 30(25), pp. 3569-3574 (1991).

5. J. M. Lloyd, "Fundamentals of Electro-Optical Imaging System Analysis," in Electro-Optical
Systems Design, Analysis, and Testing, M. C. Dudzik, ed., pg. 18 This is Volume 4 of The Infrared
and Electro-Optical Systems Handbook, J. S. Accetta and D. L. Shumaker, eds., copublished by
Environmental Research Institute of Michigan, Ann Arbor, MI and SPIE Press, Bellingham, WA
(1993).

6. American Society of Heating, Refrigeration and Air-conditioning Engineers, "Applications of
Infrared Sensing Devices to the Assessment of Building Fleat Loss Characteristics," ANSI/ASHRAE
Standard 101-1981, ASHRAE, Atlanta, GA (1983).

7. Y. M. Chang and R. A. Grot, "Performances Measurements of Infrared Imaging Systems used to
Assess Thermal Anomalies," in Thermal Imaging, I. R. Abel, ed., SPIE Proceedings Vol. 636, pp. 17-
30 (1986).

8. G. В McIntosh and A. F. Filippone, "Minimum Resolvable Temperature Difference (MRTD)
Testing: Equipment Specifications for Building Performance Diagnostics," in Thermosense IV, R. A.
Grot and J. Wood, eds., SPIE Proceedings Vol. 313, pp. 102-111 (1981).

9. The photopic response curve can be found in many textbooks. See, for example, W. J. Smith,
Modern Optical Engineering, second edition, pp. 222-227, McGraw-Hill, New York (1990)

10. G. C. Holst, CCD Arrays, Cameras, and Displays, second edition, pp. 23-27, JCD Publishing,
Winter Park FL (1998).

11. F. E. Nicodemus, "Normalization in Radiometry," Applied Optics, Vol. 12(12), pp. 2960-2973
(1973).

12. N. Bluzer, "Sensitivity Limitations of IRFPAs Imposed by Detector Nonuniformities," in Infrared
Detectors and Arrays, E. L. Dereniak, ed., SPIE Proceedings Vol. 930, pp. 64-75 (1988).

EXERCISES

1. Using the approach in Example 3-1, calculate AE,lv for a source temperature of
100°C above an ambient background of 27° C. The SiTF is AF^/AT. Discuss the
difference in this calculated SiTF compared to the values given in Example 3-1.

2. For the system described in Example 3-1, calculate the SiTF for a target
temperature of 280 К on a 270-K background.

3. Using the approach in Example 3-1, calculate AEVJ4. for an MWIR system whose
spectral response is from 3 to 5 pm for a 320-K target against a 300-K
background. Assume 1^^ = 30,000 v/w, ro/,ftcv = 0.8, F=4, 6’ = 20,000, Ad=4 x 10 s
cm2, and fl= 10 inches. What is the SiTF?

4. Calculate Toptics_ave using the data given in Example 3-2 but assume the target
temperature is 800 K. Compare the answer with that obtained in Example 3-3.

5. The system described in Example 3-1 will be used as a spectroradiometer. It will
have band pass filters that are 1 pm wide centered at 8.5, 9.5, 10.5 and 11.5 pm.
Assume the noise voltage is 100 pv. What is the SNR for each band if the target is
at 310 K? What is the SNR for a target that is 800 К against a 300-K background?
Assume that each filter has a transmittance of 75%.

6. Depending upon the definition (see Appendix) either 1/(4F2+ 1) or 1/4F2 may be
used in the radiometric equations. Plot both factors as a function of the f-number.
4

MTF THEORY

The optical transfer function (OTF) plays a key role in the theoretical
evaluation and optimization of an optical system. The modulation transfer function
(MTF) is the magnitude and the phase transfer function (PTF) is the phase of the
complex-valued OTF. When an ideal system is viewing incoherent illumination,
the OTF is real-valued and positive so that the OTF and MTF are equal. When
focus errors or aberrations are present, the OTF may become complex valued.
Electronic circuitry also can be described by an MTF and PTF. The combination
of the optical MTF and the electronic MTF creates the electro-optical imaging
system MTF. The MTF is the primary parameter used for system design, analysis
and specifications.

When coupled with the three-dimensional noise parameters, the MTF and
PTF uniquely define system performance. The MTF and PTF are measures of how
the system responds to spatial frequencies. They do not contain any signal
intensity information.

The system MTF and PTF alter the image as it passes through the
circuitry. For linear-phase-shift systems, the PTF is of no special interest since it
only indicates a spatial or temporal shift with respect to an arbitrarily selected
origin. An image where the MTF is drastically altered is still recognizable whereas
large nonlinearities in the PTF can destroy recognizability. Modest PTF
nonlinearity may not be noticed visually except those applications where target
geometric properties must be preserved (i.e., mapping or photogrammetry).
Generally, PTF nonlinearity increases as the spatial frequency increases. Since the
MTF is small at high spatial frequencies, the nonlinear-phase-shift effect is
diminished.

Four conditions must be met to use MTF theory: (1) the radiation is
incoherent, (2) the signal processing is linear, (3) the image is spatially invariant,
and (4) the system mapping is single-valued (non-noisy and not digitized). These
last three conditions are violated by most electro-optical imaging systems. The
optical system will be spatially variant if its impulse response varies from the
center to the edge of the field-of-view due to aberrations. Each detector is noisy
and this violates the one-to-one mapping requirement. The analog electronics may
be noisy and nonlinear image processing may also be present.
MTF analysis is not strictly applicable to electro-optical imaging systems.
Electronic shaping may cause MTFs greater than unity and this causes
normalization (defmition) problems. Scanning systems typically have a low
frequency cutoff to reduce 1/f noise. With these systems, the MTF at zero spatial
frequency is zero. This also causes normalization problems. Finally the system has
frames of data and the image is not continuous in time. But this can be neglected.
MTF does not include dynamic (time-variant) phenomena.

A linear system merely modifies the amplitude and phase of the target.
No new frequencies are generated in the process. Sampling can create new
frequencies. The detector array samples the scene in the vertical direction.
Depending upon the design, scanning devices may sample in the horizontal
direction. Staring devices sample the scene in both directions. These effects are
discussed in Chapter 5, Sampling Theory.

In spite of these disclaimers, electro-optical systems are treated as quasi-
linear over a restricted operating region to take advantage of the wealth of
mathematical tools available to analyze linear systems. For modeling purposes,
electro-optical systems are characterized as linear spatial-temporal systems that are
shift-invariant with respect to both time and two spatial dimensions. Although
space is three-dimensional, an imaging system displays only two dimensions.

4.1. MTF DEFINITION

Modulation is the variation of a sinusoidal signal about its average value
(Figure 4-1). It can be considered as the AC amplitude divided by the DC level.
The modulation is

MODULATION = M = = —

5max + Bmin DC

(4-D

where Bmax and Binin are the maximum and minimum signal levels respectively.
The modulation transfer function is the output modulation produced by the system
divided by the input modulation at that frequency:

MTF ~ OUTPUT MODULATION
INPUT MODULATION

(4-2)

The concept is presented in Figure 4-2. Three input and output signals are plotted
in Figures 4-2a and 4-2b and the resultant MTF is shown in Figure 4-2c. As a
ratio, the MTF is a relative measure whose values range from zero to one.
Intensity

Figure 4-1. Definition of Target Modulation. The variable d is
the extent of one cycle. For optical systems, d is measured in
angular space and the spatial frequency is fx = Md. For
electronic circuitry, d is measured in time and the electrical
frequency isfHz = Md.

Input (Object) Output (Image)

Figure 4-2. Modulation transfer function, (a) Input signal for
three different spatial frequencies, (b) output for the three
frequencies, and (c) MTF is the ratio of output-to-input
modulation.
4.2. LINEAR FILTER THEORY

Linear filter theory was developed for electronic circuitry and has been
extended to optical, electro-optical, and mechanical systems. Linear filter theory
forms an indispensable part of system analysis.

Let 5{ } be a linear operator that maps one function, f(x), into another
function, g(x):

s{f(x)}=g(x). (4-3)

Let the response to two inputs,/^) and f2(x), be gr(x) and g2(x):

s{fi(x)} = gi(x) and s{f2(x)}=g2(x). (4-4)

For a linear system, the response to a sum of inputs is equal to the sum of
responses for each input acting separately. For any arbitrary scale factors, the
superposition principle states

S{alfl(x) + a2f2(x)} = aigl(x') + a2g2(x). (4-5)

4.2.1. THE EO SYSTEM as a LINEAR SYSTEM

An object can be thought of as the sum of an infinite array of impulses
(Dirac delta functions) located inside the target boundaries. Thus, the object can be
decomposed into a series of weighted Dirac delta functions, д(х-хо), д(у-уоУ

co co

Tj Т1О(Х°,У°№Х~Х°^У~У°^°&У°' <4’6)

*<,-«> Л-°°

As Axy and byo decrease in magnitude, Equation 4-6 becomes an integral. Figure
4-3 illustrates one-dimensional decomposition. Similarly, an electronic waveform
in time can be decomposed into a series of weighted Dirac delta functions.

Using the superposition principle, A{ } operates on each individual input.
The individual outputs are added to produce the image (Figure 4-4).
Mathematically,

co co
Figure 4-3. An electronic waveform or object can be
decomposed into a series of closely spaced impulses whose
amplitude is equal to the waveform at that value.

Figure 4-4. The linear operator, 5{ }, transforms each input
Dirac delta into an output point spread function. The sum of
all the point spread functions creates the image.

The object O(xo,yo) can be considered the weightings, afo in Equation 4-5.
That is, the input has been separated into a series of functions affx) + af2(x) +...
For small increments, this becomes the convolution integral

co co

/h>>’)= \ jo(x0,y0)S{3(x-x0)3(y-y0)}dx0dy0 (4-8)

— CO—CO

and is symbolically represented by
where * indicates the convolution operator. 5{ } is the system's response to an
input impulse. The impulse creates the point spread function for the optical
systems and impulse response for the electronic circuitry.

If the image is passed through another linear system, the superposition
principle is applied again:

(4-10)

Z(x, у} = O(x, y) * (x, y) * S2 (x, y). (4-11)

4.2.2. CASCADING MTFs

Time and spatial coordinates are treated separately. For example, optical
elements do not generally change with time and therefore are characterized only
by spatial coordinates. Similarly, electronic circuitry exhibits only temporal
responses. The detector provides the interface between the spatial and temporal
components and its response depends on both temporal and spatial quantities. The
conversion of two-dimensional optical information to a one-dimensional electrical
response assumes a linear photo-detection process. Implicit in the detector MTF is
the conversion from input flux to output voltage.

Time filters are different from spatial filters in two ways. Time filters are
single-sided in time and must satisfy the causality requirement that no change in
the output may occur before the application of an input. Optical filters are double-
sided in space. Electrical signals may be either positive or negative whereas optical
intensities are always positive. As a result, optical designers and circuit designers
use different terminology.

Symbolically, the optics and detector spatial responses are given by

OTFspatial(fx,fy)=MTFspatial(fx,fy) eJPTF^’O. (4-12)

Analog electronic circuitry operates on a serial stream of data (assumed to be in
the horizontal direction:

H elee (fxe) = MTFdec (fxe)e jPTF^>. (4-13)

With imaging processing algorithms (including line-to-line interpolation), the
circuitry can affect the vertical MTF:
H elec (.fye ) = MTFetec (fye ) e iPTF^ .

(4-14)

If multiple components exist in either the spatial or electronic domains, the
individual MTFs can be multiplied together. Equivalently, multiple convolutions
in space or time is equivalent to multiplications (or cascading) in the frequency
domain. Note that while individual lens elements have a unique MTF, the MTF of
the lens system is not usually the product of the individual lens-element MTFs.
This occurs because one lens may minimize the aberrations created by another
(i.e., the lenses are not decoupled).

With appropriate scaling, the electrical frequency can be converted into
spatial frequency. This is symbolically represented by fxe -+fx and fye fy. The
combination of spatial and electronic responses is some-times called the system
OTF. For independent (decoupled) subsystems, the system MTF is

MTF Sys (Jx * f у )

MTFsfaaal (fx, fy ) MTFelec (Jxe -> fx )MTFelec (fye -> fy )

(4-15)

and the system PTF is

PTFsys(.fx^fy) = e ^IPTF(f‘+PTF(S~-'F>+PTF(f„-yf11)]_

(4-16)

4.3. SUPERPOSITION APPLIED to OPTICAL SYSTEMS

If the MTF of a system is known, one can compute the image for any
arbitrary object. First, the object is dissected into its constituent spatial frequencies
(i.e., the Fourier transform of the object is obtained). Next, each of these
frequencies is multiplied by the system MTF at that frequency. Then the inverse
Fourier transform provides the image.

To illustrate the superposition principle and MTF approach, we will show
how an ideal optical system modifies an image. An ideal optical system is, by
definition, a linear-phase-shift system. The most popular test target consists of a
series of bars - typically three or four bars although more may be used. For
illustrative purposes, the periodic bars are assumed to be of infinite extent. A one-
dimensional square wave, when expanded into a Fourier series about the origin,
contains only odd harmonics:

1 .

—sin
n

2ЯПХ

where n = 1, 3, 5..., (4-17)

о
and Xo is the period of the square wave. The fundamental frequency fo is \!XO.
Taking the Fourier transform of the square wave provides discrete spatial
frequencies 1/XO, 3/Xo, 5/Xo, ... whose amplitudes are 4/л, 4/3tt, 4/5tt, ... ,
respectively.

Let a circular optical system image the square wave. The OTF for a
circular, clear aperture, diffraction-limited lens is

OTF<liffOx> = — cos

, (4-18)

when/^. < foco and foco, is the optical cutoff Since this optical OTF is always

positive, it is popularly called the MTF.

By superposition, the optical MTF and square wave amplitudes are
multiplied together at each spatial frequency

ЛЛ ) = MTFdiff (fx ) O(fx ). (4-19)

Taking the inverse Fourier transform provides the resultant image. Equivalently,

I(x) = MTFdiff(fo)

—sin(2^x f0)

(4-20)

MTFlUff(3f0)

—sin(6^x/„)
Зя

If fo is greater than f„J3, only the fundamental of the square wave will be
faithfully imaged by the optical system. Here, the square wave will appear as a
sine wave. As fo decreases, the image will look more like a square wave (Figure 4-
5). Since the optics does not pass any frequencies above foco, the resultant
waveform is a truncation of the original series modified by the optical MTF. This
results in some slight ringing. This ringing is a residual effect of the Gibbs
phenomenon.
RELATIVE OUTPUT RELATIVE OUTPUT RELATIVE OUTPUT

(a)

Figure 4-5. Square waves imaged by a circular aperture. As
the square wave fundamental frequency increases, the edges
become rounded. Eventually, the square wave will appear as a
sinusoid when/0>/„„/3. (a)/«=/«</40, (b) fo=fKJ 10, and (c)
fo '
4.4. PHASE SHIFTS

A phase shift represents a displacement from an arbitrarily selected
origin. A phase shift moves the image in the optical plane and provides a time
delay in the electronic world. To use the superposition theorem, each input
waveform is decomposed into a Fourier series

(4-21)

For a linear-phase-shift system, 0n = n0o where 0o is the distance from the
origin. For optical systems, the shift can be in two directions representing
translation and rotation. For electronic circuitry, a phase shift is simply a time
delay from an arbitrarily selected origin. With a linear-phase-shift system, by
redefining the origin location, f)o becomes zero in the new coordmate system.

An optical system is isoplanatic if the translation or rotation of an object
produces a proportional translation or rotation of the image. An isoplanatic region
is that region within the field-of-view where the optical transfer function may be
considered shift-invariant within measurement accuracy for all spatial frequencies
of interest. Since aberrations do not generally change radically, it is reasonable to
treat a small area as isoplanatic or to call this area an isoplanatic patch. Most
optical systems are rotationally symmetrical, have minimal aberrations, and may
be considered isoplanatic. Tsoplanatism applies only to the optical subsystem.
Electronic circuitry is simply labeled as shift-invariant.

The ideal common module system is a linear-phase-shift system.
However, the mechanical scanning method emphasizes the electronic time delay.
Common modules use a bidirectional scan technique. The scene signal is
"reversed" in time (Figure 4-6). Since electronic circuitry has a phase shift, the
output will be delayed in time. This delay is constant in time. But since the scene is
"reversed" a vertical bar will be serrated.' In common module systems, this is
compensated for by physically moving a lens (phase shift lens) for the forward and
interlace scans. For EMUX systems, the timing is altered for each field so that the
detector outputs are summed together properly. Here, phasing effects that were
emphasized by hardware are compensated for in hardware.
Figure 4-6. Electronic delay in bidirectional scanning. The
time delay is "reversed" for the interlace scan. This serrates
vertical bars.

4.5. REFERENCES

1. J. M. Lloyd, Thermal Imaging Systems, pp. 103-108, Plenum Press, New York (1975)

EXERCISES

1. The MTF of a rectangular diffraction-limited aperture is MTF(f) = 1 -fjfoco.
Plot the image of a square wave where/, = /co/40, fo and fo =fJ3.

Compare these plots to Figure 4-5.

2. Assuming a phase shift of ^ = 7i/4, use Equation 4-19 to plot the image of the
three cases shown in Figure 4-5.

3. Instead of a linear phase shift of n0o, assume a nonlinear phase shift of nOJ2.
Plot the image of the three cases shown in Figure 4-5 where 0o=ti /4.

4. An electronic single-pole, low pass filter has an MTF of

MTF =

Plot the output when the input is a square wave with fo = f3dBlME ft = fW10, and
ft=A//ft3.

5. Assume that an ideal electronic low pass filter has an MTF of unity when f <
f3dB and zero elsewhere. Plot the output the input is a square wave with fo = f3dB!^,
fo = and/, = W3-

6. Discuss the differences between the plots obtained in Exercise 4 with the plots
obtained in Exercise 5. Which is better - the ideal lowpass filter or the single-pole,

low pass filter? Why?
5

SAMPLING THEORY

Sampling (digitization) is an inherent feature of all imaging systems.
Staring systems, because of detector location symmetry, tend to have equal
sampling rates in both the horizontal and vertical directions. With scanning
systems, the detector output in the scan direction can be electronically digitized at
any rate whereas in the cross scan direction, the detector locations define the
sampling rate. Therefore, in a scanning system, the sampling rate may be different
in the horizontal and vertical directions.

In most scanning systems, the detector output is a continuous analog
signal that is then digitized (sampled and quantized) by the system's internal A/D
converter. If the sampling rate is sufficiently high and the analog signal is band-
limited, the reconstructed signal will replicate the analog signal in frequency,
amplitude and pulse width. However, to conserve on memory requirements and
minimize data rates, scanning systems tend to operate at marginal sampling (clock)
rates. Consequently, the output signal amplitude and pulse shape will be modified.

Signals can be undersampled or oversampled. The highest frequency that
can be faithfully reconstructed is one-half the sampling rate. Any input signal
above the Nyquist frequency, fn, (which is defined as one half the sampling
frequency,^) will be aliased down to a lower frequency. That is, an undersampled
signal will appear as a lower frequency after reconstruction (Figure 5-1).
Undersampling is a term used to denote that the input frequency is greater than the
Nyquist frequency. It does not imply that the sampling rate is inadequate for any
specific application. Similarly, oversampling does not imply that there is excessive
sampling. It simply means that there are more samples available than that required
by the Nyquist criterion.

After aliasing, the original signal can never be recovered. Undersampling
creates moire patterns (Figure 5-2). Diagonal lines appear to have jagged edges or
"jaggies." Aliasing is not always obvious when viewing complex scenery and, as
such, is rarely reported during actual system usage although it is always present.
However, it may become apparent when viewing periodic targets such as fences,
plowed fields, and railroad tracks. Whether this is bothersome depends upon the
task. Photo interpreters rely on aliasing to emphasize railroad tracks!
Figure 5-1. An undersampled sinusoid will appear as a lower
frequency after reconstruction. T is the sample-to-sample
time. The sampling frequency is/ = MT.

Figure 5-2. Moire pattern. A raster scan system creates moire
patterns when displaying wedges or star bursts.

According to the sampling theorem, any frequency,/, sampled at fs > 2f,
can be uniquely recovered. However, the square wave is the most popular test
target. When expanded into a Fourier series, the square wave consists of an infinite
number of frequencies. Although the square wave fundamental may be
oversampled, the higher harmonics will not. Aliasing can distort periodic test
patterns such as the four-bar MRT target. Aliasing can increase the MRT at
selected frequencies.1 The appearance of the square wave after reconstruction
depends upon the relative values of the optical, detector, and electronic MTFs.
This is a system test problem.
5.1. SAMPLING THEORY

In a sampled-data system, the sampling frequency interacts with the
signal to create sum and difference frequencies. Any input f will appear as nfs ± f
after sampling (л = -co to +oo). Figure 5-3 illustrates a band-limited system with
frequency components replicated by the sampling process. The baseband (~fh to fh)
is replicated at nfs. To avoid distortion, the lowest possible sampling frequency is
that value where the baseband adjoins the first side band (Figure 5-3c). This leads
to the sampling theorem that a band-limited system must be sampled at twice the
highest frequency (fs > 2fh) to avoid distortion in the reconstructed image. The
sampling theorem applies to any periodic input frequency. The units can be either
spatial frequency or electrical frequency.

After digitization, the data resides in data arrays with nonspecific units.
The user assigns units to the arrays during image reconstruction. That is, the data is
read out of the memory in a manner consistent with monitor requirements so that
an image can be displayed. If the post-reconstruction filter limits frequencies to fn
and if/,, > fh, then the reconstructed image can be identical to the original image.

5.2. ALIASING

As the sampling frequency decreases, the first side band starts to overlap
the baseband and the power spectrums add (Figure 5-4). The overlaid region
creates distortion in the reconstructed image. This is aliasing. Once aliasing has
occurred, it cannot be removed. Within an overlapping band, there is an ambiguity
in frequency. It is impossible to tell whether the reconstructed frequency resulted
from an input frequency of/or nfs ± /

All frequency components above fn are folded back into the baseband so that the
baseband contains

О baseband ~~ ± f) where nfs±f< f„.

п=-<л

(5-1)

To avoid aliasing, the signal may be passed though a low pass filter (anti-
aliasing filter) to insure thatfh<fn.
RELATIVE FREQUENCY
(c)

Figure 5-3. Sampling replicated frequencies at nfs ± f. (a)
Original band-limited signal, (b) frequency spectrum after
sampling, and (c) when fs = 2fh, the bands just adjoin.
Figure 5-4. Aliasing alters both the signal and noise baseband
spectra. In a real system, signals and noise are not necessarily
band-limited and some aliasing may occur.

5.3. SAMPLERS

Three samplers may exist in an electro-optical imaging system: (1) For
scanning systems, the detector acts as a sampling aperture, (2) the discrete location
of the detectors spatially samples the scene, and (3) if the detector output is
digitized (as in most scanning systems), the A/D converter is the third sampler.
Any one of these samplers may limit performance.

5.3.1. THE DETECTOR as a SAMPLER

The detector OTF cannot exist by itself. Rather, the detector OTF must
also have the optical OTF to make a complete system. In one-dimension, the OTF
of a rectangular detector is

OTFdetector(fx) =

sin (тга fx)
ла fx

(5-2)

where a the detector angular subtense (DAS). Figure 5-5 illustrates the OTF in one
dimension. The OTF is equal to zero when fx = к/a. The first zero (k = 1) is
considered the detector cutoff, fko , because any higher frequency will not be
faithfully reproduced. Any input spatial frequency above detector cutoff will be
aliased down to a lower frequency. When A 1, there is precisely one cycle across
the detector. It is customary to plot the OTF only up to the first zero (Figure 5-6).
Since the OTF is positive in this region, it is also called the MTF.

NORMALIZED SPATIAL FREQUENCY

Figure 5-5. Detector OTF as a function of normalized spatial
frequency fx /fdco. Detector cutoff is fdco = l/«. The negative
OTF values represent contrast reversal: periodic dark bars will
appear as light bars.

Figure 5-6. Typical detector OTF representation as a function
of normalized spatial frequency fx !fdco. The OTF is usually
plotted only up to the first zero and is identical to the MTF in
this region.
О

The detector MTF representation from fx = 0 to fx = l/« has become so
wide spread that it often forgotten that the detector can respond to higher spatial
frequencies. It has become common practice to imply there is no response above
fdco . For a band-limited system, the sampling rate should be twice the highest
frequency present. This leads to the common parlance that there should be two
samples per DAS to avoid aliasing. This is only true if the optical signal is limited
to l/<z. That is, fdco > foco.
The detector's Nyquist frequency is fn = l/« and its effective sampling
rate is fs = 2/«. Any signal above fn will be aliased to a lower frequency according
to

О baseband

n=-<x>

ship а (я/, +Л)]
^«(«Л ±Л)

where nfs ± fx <fn . (5-3)

Figure 5-7 illustrates the replicated spectrum for the first side bands (n =
±1). This aliasing occurs at the detector and can only be eliminated by assuring
that the detector cutoff is greater than the optical cutoff (flc0 > foco). That is, if no
signal exists above foco, there is no signal to be aliased. Since f. = 2f(lco, when fdco >
foco, the optical system is sampled at twice the highest frequency present and the
Nyquist criterion is met for the optical system. All the frequencies imaged by the
optical system will be faithfully detectediffdeo^foeo-

Figure 4-5 illustrated how an optical system modified an infinite square
wave. The image contains all the spatial frequencies up to optical cutoff. The
detector then converts the spatial frequencies into electrical frequencies. Whether
the detector aliases these frequencies depend upon the relationship between the
optical cutoff and the detector cutoff. Aliasing occurs as fdco is reduced. However,
the aliased signal will be reduced in amplitude because the optical MTF reduces
the amplitude.

n=-1 n=0 n=1

NORMALIZED SPATIAL FREQUENCY

Figure 5-7. The detector acts as a sampling aperture where the
sampling frequency is fs = 2fdco. The baseband and first side
bands are illustrated. The optical MTF reduces the aliased
signal amplitude.
5.3.2. SPATIAL SAMPLING

Although not stated, the detector output shown in the previous figures
can only be obtained with a scanning device operating in an analog mode. Staring
arrays and the cross scan direction in a scanning array place the detectors at
discrete locations. These discrete locations sample the image at an effective spatial
sampling rate equal to the inverse of the effective detector center-to-center
spacing. If the detector center-to-center spacing is dcc, then the angular sampling
rate \sScc = dJfl.

Staring arrays can reproduce signals up to fn = l/25cc . Although the
individual detectors can reproduce higher spatial frequencies, the spectrum is
sampled &fs= \!SCC (Figure 5-8). Staring arrays are inherently undersampled when
compared to the detector spatial frequency cutoff (Figure 5-9).

The highest spatial frequency that can be faithfully reproduced is the
system cutoff frequency. For oversampled systems, it is where the MTF
approaches zero. For undersampled systems, it is the Nyquist frequency. Systems
can detect signals whose spatial frequencies are above cutoff but cannot faithfully
reproduce them. For example, a high frequency (above system cutoff) 4-bar
pattern may appear as one low contrast blob in an oversampled system. For an
undersampled system, patterns above the Nyquist frequency are aliased to a
frequency below Nyquist and a 4-bar pattern may appear as a distorted 3-bar
pattern. From a design point of view, the MTF should be "high" over the spatial
frequencies of interest; this range of frequencies is application specific.

Example 5-1
SYSTEM CUTOFF

A staring array consists of detectors that are 40x40 pm in size. The
detector pitch is 60 pm. The effective focal length is 30 cm. The aperture diameter
is 15 cm and the average wavelength is 10 pm. What is the system cutoff?

System cutoff is defined as the smaller of the optical cutoff, detector
cutoff, or Nyquist frequency. The optical cutoff is foco=DIX = 15 cycles/mrad. The
DAS is 40x10'6/0.3 = 0.133 mrad and the detector cutoff is 1/DAS = 7.5
cycles/mrad. The detector pitch provides sampling every 60x10"6/0.3 = 0.2 mrad
for an effective sampling rate at 5 cycles/mrad. Since the Nyquist frequency is
one-half the sampling frequency, the system cutoff is 2.5 cycles/mrad. For staring
arrays, the Nyquist frequency is always smaller than the detector cutoff and is
typically also the system cutoff
NORMALIZED SPATIAL FREQUENCY
(a)

(b)

Figure 5-8. Two arrays with different center-to-center spacing.
The DASs are the same for both. The spatial frequency axis is
normalized to fxlfdco. (a) Scc = «. This represents a 100% fill
factor staring array and a scanning array with one
sample/DAS. (b) A finite fill factor (fill factor of 50%) staring
array and the cross scan direction in a scanning array where
the overscan ratio is 0.5.
NORMALIZED SPATIAL FREQUENCY

Figure 5-9. The sampling frequency is inversely proportional
to the detector center-to-center spacing. For 100% fill factor
arrays, fs = l/2«. The base band and first side bands are
illustrated. See Figure 5-7 for comparison.

Example 5-2

STAGGERED ARRAY CUTOFF

The staggered elements, as illustrated in Figure 5-10, increase the vertical
spatial sampling rate. If the effective focal length is 40 cm, what is the system
cutoff?

The vertical DAS is 40><10‘6/0.4 = 0.1 mrad representing a 10-cycles/mrad
capability. The sampling is every 25*10’6/0.4 = 0.0625 mrad or at a rate of 16
cycles/mrad. The highest spatial frequency that can be faithfully reproduced is 8
cycles/mrad. This represents 16/10 = 1.6 samples per DAS. This system is
considered (erroneously) undersampled since common parlance states that the
Nyquist criterion requires two samples per DAS. The system is undersampled only
if the optical system passes spatial frequencies above 8 cycles/mrad.
Figure 5-10. The vertical Nyquist frequency can be increased
with a staggered array. Staggered arrays are only used in
scanning systems.

5.3.3. A/D CONVERTER

In principle, the clock rate of an A/D converter can be set at any rate.
However, with real hardware, weight, power consumption, and size may limit the
clock rate to just satisfy Nyquist frequency.

Example 5-3

SYSTEM NYQUIST FREQUENCY

What are the horizontal and vertical Nyquist frequencies for an LWIR
common module system? The DAS is 0.2 mrad horizontally by 0.2 mrad
vertically. The field-of-view is a square format with 72 mrad on a side. Each
horizontal line (scan direction) is electronically digitized at two samples per DAS.
The optical cutoff is 15 cycles/mrad.

An LWIR common module system consists of 180 detectors that are
interlaced to produce 360 infrared scene lines. In the horizontal direction, there are
72/0.2 = 360 independent DASs. With two digital samples per DAS there are 720
digital samples in the horizontal direction. The Nyquist frequency is one-half the
sample frequency. Equivalently there are 360 cycles in the horizontal direction and
the Nyquist frequency is 720/(2x72) = 5 cycles/mrad. This is identical with 1/DAS.
In the vertical direction, there are only 360 independent samples (the number of
detectors) representing 180 cycles. Thus the vertical Nyquist frequency is
360/(2x72) = 2.5 cycles/mrad. Line interpolation may further degrade this value.

Example 5-4
SAMPLING CUTOFF

The LWIR system described in Example 5-3 is electronically digitized at
four samples per DAS. What is the horizontal system cutoff7

With four digital samples per DAS, there are 1440 digital samples in the
horizontal direction and the A/D converter Nyquist frequency is 10 cycles/mrad.
However, the highest spatial frequency that can be faithfully reproduced by the
detector is 5 cycles/mrad Oversampling minimizes phasing effects.

The required sampling rate depends upon the optical MTF, detector
MTF, and the amount of aliasing tolerated. Only a detailed study will determine if
the aliasing is unacceptable.

5.4. MICROSCAN

Staggered arrays increase the vertical sampling frequency in scanning
systems (Figure 5-10). Microscan2"5 effectively increases the sampling frequency
for staring arrays. With microscan, the detector line-of-sight is moved a fraction of
a DAS. This increases the sampling rate by reducing the effective center-to-center
spacing. While this improves image fidelity (Figure 5-11), it does have some
disadvantages. The detector integration time decreases for each microscan location
and this increases the NEDT. It also increases hardware complexity. The monitor
may limit the maximum amount of microscan. If the monitor can only display 480
lines, then the microscan should only create 480 lines. An array that consists of
640x480 detector elements should only be microscanned if the monitor can
display 1280x960 pixels.

worst phase

best phase

3x3 microscan

random phase

3x3 microscan

Figure 5-11. Microscan improves image fidelity. In-phase and
out-of-phase are also called best-phase and worst-phase
respectively. The array consists of 64x64 elements. The 3x3
microscan produces 192x192 pixels that must be converted
into a monitor format. (From reference 5).

5.5. ANTI-ALIASING FILTER

An anti-aliasing filter is a low pass filter that band-limits the system
before sampling to avoid aliasing. Optical band-limiting can be achieved by using
small diameter optics or by defocusing. Unfortunately, these two approaches also
degrade the MTF in the baseband and typically are considered undesirable.

In electro-optical imaging systems, the signal is band-limited by the
optics and detector. Subsequent electronics is designed to maintain signal fidelity.
On the other hand, noise is assumed to be generated in the detector. The noise is
assumed to be white (all frequencies are present). By selecting electronic filters
that minimize noise and maximize the signal, the signal-to-noise ratio is
maximized.
If the filter is just before the A/D converter, and its purpose is to
minimize aliasing, then the filter is called an anti-aliasing filter. These filters
cannot remove the aliasing that has taken place at the detector. They can only
prevent further aliasing that might occur in the A/D converter. Figure 5-12
illustrates the desired cutoff features of an ideal filter. The ideal filter is, of course,
unrealizable.

Figure 5-12. An ideal anti-aliasing filter passes all the signals
below fn and no signal above fn . For an appropriately sampled
scanning system, fn =foco.

5.6. REFERENCES

1. G. C. Holst, Testing and Evaluation of Infrared Imaging Systems, second edition, pp. 347-351,
JCD Publishing, Winter Park, FL (1998).

2. D. J. Bradley and P N. J. Dennis, "Sampling Effects in CdHgTe Focal Plane Arrays," in Infrared
Technology and Applications, A. Baker and P, Masson, eds., SPIE Proceedings Vol. 590, pp. 53-60
(1985).

3. R. J. Dann, S. R. Carpenter, C. Seamer, P. N. J. Dennis, and D. J Bradley, "Sampling Effects in
CdHgTe Focal Plane Arrays - Practical Results," in Infrared Technology XII, I. J. Spiro and J.
Mollicone, eds., SPIE Proceedings Vol. 685, pp. 123-128 (1986).

4. E. A. Watson, R. A. Muse, and F. P. Blommel, "Aliasing and Blurring in Microscanned Imagery," in
Infrared Imaging Systems: Design, Analysis, Modeling, and Testing III, G. C. Holst, ed., SPIE
Proceedings Vol. 1689, pp. 242-250 (1992).

5. F. P. Blommel, P. N. J. Dennis, and D. J. Bradley, "The Effects of Microscan Operation on Staring
Infrared Sensor Imagery," in Infrared Technology XVII, B. F. Andresen, M. S. Scholl, and J. Spiro,
eds., SPIE Proceedings Vol. 1540, pp. 653-664 (1991).
EXERCISES

1. List five different objects seen on television that appear jagged.

2. Describe the appearance of the objects listed in Exercise 1 as they move slowly,
at a moderate rate, and moving quickly.

3. A staring array consists of 512x512 detector elements. If the effective focal
length is 40 cm and the detectors are 50 x50 pm on 75 pm centers, what is (a) the
detector DAS, (b) the system FOV, and (c) the system Nyquist frequency?

4. For the system described in Example 5-3, calculate the optical resolution,
detector resolution, and system resolution both horizontally and vertically. Assume
that resolution is the inverse of the cutoff frequency.

5. Using the results from Exercise 4, calculate the "detection” range for a 2x2 m
target both vertically and horizontally. Discuss the differences in the "detection"
ranges. Let

„ target size

Range = —-----------.

resolution

6. Using the results from Exercise 4, calculate the horizontal and vertical
"detection" ranges for a U5 m and 5X1 m target both vertically and horizontally.
Discuss the differences in the ranges.

7. FLIR92 uses a two-dimensional approach where the effective system cutoff is
feff= Calculate the system "detection” ranges for a 1 x5 m and 5x 1 m target.
Discuss the advantages of the two-dimensional approach.

8. A staring array consists of 64x64 detector elements. The detectors are 40x40
pm on 80 x 80 pm centers. The focal length is 50 cm. What is the system cutoff
frequency? What is the cutoff with (a) 2x2 microscan, (b) 3x3 microscan, (c) 4x 4
microscan, (d) 5x5 microscan, (e) 6x6 microscan, (f) 7x7 microscan, (g) 8x8
microscan, (h) 9 x9 microscan, and (i) 10x10 microscan? What limits the highest
practical microscan?
6

COMMON MODULE SYSTEMS

Figure 6-1 illustrates a functional electro-optical block diagram of an
early common module system. The components were called common modules
because the scanner, detector, pre-amplifiers, post-amplifiers, and light emitting
diodes (LEDs) were manufactured to common specifications. The pre-amplifier is
a low-noise fixed-gain amplifier. The post-amplifier had a variable gain and was
adjusted for the desired displayed contrast.

The common module concept was initiated in the 1970s to reduce cost
through the economy of scale and competition. These systems are still in existence
and lend themselves to small, lightweight man-portable applications. The
designer’s only options were changing the optics to achieve the desired field-of-
view, adding electronic filtering to maximize the signal-to-noise ratio, and
modifying the visual optics to present an appropriate image to the observer. Nearly
all systems today evolved from the common module. Although a misnomer, the
derivatives are also called common module systems. For example, both EOMUX
and EMUX systems are sometimes called common modules to indicate that they
contain a scanning scheme similar to the original common design. Common
modules systems operate in the LWIR regions and employ HgCdTe detectors.

Figure 6-1. Common module components considered for
analysis.

For analysis purposes, the common module system consists of optics, a
stabilization element, detector, amplifiers, electronic filters, LEDs, visual optics,
and the eye. The detector response consists of both the spatial and temporal
responses that are characterized by OTFdetector, and MTFtconstant respectively. The
OTF is applied to spatial coordinates and the MTF is applied to time coordinates.
Using linear system theory, the OTFs and MTFs of the individual subsystem are
multiplied together to create the system OTF. The subscripts are self-explanatory.

OTFsys = OTFo^sOTF^nOTF^^MTFK<m^t

x MTFampMTFfilterMTFLEDMTFvoptiaMTFeye .

Since the spatial OTF is positive for most applications, the usual representation is

MTFsys = MTFopticsMTFmotlon MTFdetector MTFtconstant

x MTFampMTFр^МТР^рМТР^р^МТРеуе . ( '2)

All OTFs and MTFs are a function of frequency (which is omitted for equation
compactness). The optical OTF, OTFoptics is for the overall lens system. It is not the
product of the individual lenses since these OTFs may be correlated. While not
truly separable, for mathematical convenience the optical OTF is separated into
three components (diffraction, aberrations, and defocus)

OTFoptics * OTFdiffOTFaberratlonOTFdefocus . (6-3)

Motion consists of three different forms: linear, sinusoidal and random

OTFmotion -OTFlinearOTFsinusoidalOTFrandom . (6-4)

For modeling purposes, the system response is usually limited to those spatial
frequencies where the OTF is positive. For this limited region, the OTF and MTF
are identical.

All the MTFs apply to similarly designed imaging systems operating in
all spectral bands ranging from the UV to infrared. The components do not
"know" what created the signal.

6.1. OPTICS OTF

Lens systems consist of several elements with varying focal lengths and
varying indices of refraction. Different elements are used to minimize aberrations
and to control beam wander during scanning. Since one lens offsets the aberrations
of another, linear system theory cannot be applied to the individual elements.
While individual element OTFs are appropriate for specifying each lens, the lens
system must be considered in its entirety. For modeling purposes, the optical
system will be treated as a single lens that has the same effective focal length and
aberrations as the lens system.

While OTFdiff is exact, OTFabcnation and OTFdefocus are mathematical
approximations. The actual lens OTF can be computed from standard optical
design codes. Most performance models (limited to positive OTFs or, equivalently,
the MTFs) including FLIR92, allow the user to directly enter optical MTF values
and thereby bypass the approximations. The sagittal MTF is used as a horizontal
MTF and the tangential MTF is a vertical MTF component. If the system analyst is
creating his own performance model, he may wish to use a curve fit to the actual
MTF values.

Optical spatial frequency is two-dimensional with the frequency ranging
from -co to +co. By convention, the optical and detector OTFs are shown in one-
dimension with the spatial frequency ranging from zero to the cutoff frequency.
The highest spatial frequency that can be reproduced is limited by the optical
cutoff.

6.1.1. DIFFRACTION-LIMITED OTF

The diffraction-limited OTF (FLIR921 ARG-1) for a circular aperture is

OTF<njf=z:

cos

when f

When f x> foco, OTIdlff = 0. Since the OTF is positive, it is identical to the MTF.
The optical cutoff, foco, is

foco

Do _ aperture diameter

2 wavelength

(6-6)

Figure 6-2 illustrates MTFdiff as a function o£fx/foco. If Do has units of millimeters
and 2 has units of micrometers, then foco will have the usual units of cycles/mrad.
This equation is only valid for monochromatic light and the cutoff frequency is
dependent upon the wavelength. The extension to polychromatic light is lens
specific. Most lens systems are color corrected (achromatized) and therefore there
is no way to predict the MTF. As an approximation to the polychromatic MTF, the
average wavelength, 2Я1,е, is used in Equation 6-5. This average wavelength is
determined by the spectral content of the incident radiation and by what the
detector senses.
Figure 6-2. MTFdiff for a circular aperture normalized to fjfoco.

Since most systems are designed to measure the radiation difference
between a target and its background, the spectrally weighted average wavelength
is

p Optics UK (л)ме (л, at) ал

_ A______________________________

ave ~

р<,рЛж(я)я(г(я)л/г(я,дт)<м

(6-7)

If ToPtics(^), ^/2), and Me(l,AT) are constant over the interval [21? 22], the average
wavelength may be approximated by

Q ~ +
^ave ~ -

(6-8)

The slope of MTFdifr is 4/я at zero spatial frequency. A straight line fit to Equation
6-4 provides the back-of-the-envelope approximation

MTF a# (J0*1 - ~ 4^-
71 foco

(6-9)

An ideal lens creates a diffraction pattern that consists of a series of
concentric rings with a bright center (resembles a bull's eye target). The central
portion is called an Airy disk. This radially symmetric pattern is described
mathematically by a Bessel function. Sometimes, the Airy disc intensity is
approximated with a Gaussian function for mathematical convenience. While this
approach maybe useful for intensity mapping, it is not prudent to use a Gaussian
MTF for MTFdiff.

6.1.2. CENTRAL OBSCURATION

A Cassegrainian telescope has a central obscuration (Figure 6-3). For an
optical system with a central obscuration, the diffraction-limited OTF is

a + в + c

where

OTFdiff ~

obs

(6-10)

(6-П)

where Do is the overall diameter and is the obscuration diameter. Let

Л ... -V -i|1 + R2 - 4X2 | ,o.

X =-------, Y =—, and 6z = cos 1 ------------------- . (6-12)

foco К L 27? J

The variables A, B, and C are

A = r* I cos'1(X)-xVl-A'2 A = 0 elsewhere when 0 < 5 X< cl
B = со8-1(У)-у71-У2 71 when 0 cl

В = 0 elsewhere ,

(6-14)

С = -2Я2 when 0<A'< ——

„ 2R . 1 + й2 2(1-Я2)

C =---sin a +-----a-----------tan

when

1 + 1?

1-1?J

-2Я2

\-R v 1 + 7?

2 2

(6-15)

C = 0 when X > -- -

When JoZ)V-0, Equation 6-10 is identical to Equation 6-5. Figure 6-4 illustrates the
diffraction limited MTF for various obscuration diameters.
Do

Figure 6-3. Cassegrainian telescope. The aperture diameter is
Do and the obscuring diameter is dobs. The primary mirror is
usually a parabaloid and the secondary mirror is usually a
hyperboloid.

MTF

0 0.2 0.4 0.6 0.8 1

NORMALIZED SPATIAL FREQUENCY

Figure 6-4. MTFdiff as a function of the obscuration ratio. The
obscuration ratio is R = dobsJDo. The spatial frequency is
normalized tofx /foco.
6.1.3. ABERRATIONS

It is mathematically convenient to represent an aberrated optical system
by two OTFs

OTFoptics^OTFd{ffOTFaberration .

(6-16)

Shannon2 developed an empirical relationship that encompasses most aberrations
of real lens systems

OTFaberration ~ I

1 - 4

when f

oco

(6-17)

where Wrms is the rms wave-front error expressed as a fraction of waves and A =
0.18. Marechal suggested3 that rms wave-front error is related to the peak-to-peak
wave-front error by Wrnts = WPP/3.5. Although ^^=0 could be used for diffraction
limited optics, apractical choice would be WPP= 0.25 or, equivalently, Wrms = 1/14.
This simulates the wave-front error that typically occurs during manufacturing.
Figure 6-5 illustrates the aberrated MTF for two different wave-front errors for a
clear aperture optical system. This approximation is valid for small wave-front
errors (ИрГ<0.5) which is a reasonable limit for well-designed optics.

Figure 6-5. MTFODtics for WPP = 1/2 and = 1/4 wave front-
errors. Shannon’s approximation was used for MTFaberration.

Aberrations can be approximated (FLIR92 ARG-2) by a Gaussian OTF

r aberration

@ (ifactorf )

(6-18)
Since a Gaussian response is always positive, the OTF is usually given as a MTF.
Although vdfactor has the dimensions of a blur diameter, it is not a measurable
quantity. It accounts for the degradation associated with the manufacturing quality
(material, design, fabrication and mounting). Here, OTFaberTation simply provides a
convenient mathematical representation that relates OTFoptlcs to the actual
(measured) data

OTFaberraUm = OTF^red = MTF„KasuKd _

OTFdiff MTFdiff

The approximation for aberrations is only valid for modest values. Any
computer program that includes the approximation must specify the region of
applicability.

6.1.4. DEFOCUSED OPTICS

Fixed focus systems can be grossly out of focus if the target is outside the
depth of field. The OTF for a circular, defocused, diffraction-limited lens4 is

OTF

'z' optics

cos

(6-20)

where

(6-21)

The focus shift, Af, in units of wavelength. When A; = 0, Equation 6-20 is identical
with Equation 6-5. Figure 6-6 illustrates the defocused MTF for a variety of errors.
There is a typographical error in Levi and Austing's original paper that has
propagated throughout the literature. The correct equation (shown above) has 4/я
in front of the integral rather than the erroneous 4/n a. This error was noted in their
errata.4

The OTF can be approximated by

OTFoptics « OTF^OTF^ .

(6-22)
NORMALIZED SPATIAL FREQUENCY

Figure 6-6. Theoretical OTFs for a diffraction-limited circular
aperture as a function of focus error, A-. The spatial frequency
is normalized to fx lfoco. Negative OTFs represent a phase
reversal. Here, targets appear reversed in contrast: periodic
white bars appear black and black bars appear white.

Shannon5 reported an approximation that is good up to 2.2 waves

MTFdefocus(fx)

UWpp

(6-23)

where Jf ) is the first-order Bessel function. For wave-front errors less than 0.75
waves, Jfx) can be approximated by the first five terms of the infinite series

3 5 7 9

X X X X X

2 ~ 1? + 384 ~ 18432 + 1474560 '

(6-24)

If the defocus errors are less than 0.5 waves peak, then the first four terms provide
a good approximation. The approximation for defocus is valid for modest values.
Any computer program that includes this approximation must specify the region of
applicability. See also Section 10.1.2., Defocus MTF. Rather than use the
approximations, the complete Bessel functions can be used for defocus.6
6.2. DETECTORS

The detector OTF cannot exist by itself. Rather, the detector MTF must
also have the optical OTF to make a complete system. In one-dimension, the
spatial response of a rectangular detector (FLIR92 ARG-3) is

ZZ A • Z Z A Sin(^/x)

OTFdetector (fx ) SinC (ОС f x )

(6-25)

This assumes that the active area is rectangular and that the response is uniform
across the active area. Using the small angle approximation, a, the detector angular
subtense (DAS), is

~ „ 1 d detector size

a = DAS =-------- — =----------------------.

fdco fl effective focal length

(6-26)

The instantaneous-field-of-view (IFOV) is the angular cone over which the
detector senses radiation. It includes both the optical blur diameter and the DAS.
When the blur diameter is small compared to the DAS, the IFOV and DAS are
approximately equal. Using the DAS in Equation 6-25 does not neglect the blur
diameter. It is already included in MTFdiff.

Figure 6-7 illustrates the OTF in one dimension. The OTF is equal to
zero when fx = kla. The first zero (A'=l) is considered the detector cutoff, fdco,
because any higher frequency will not be faithfully reproduced. Any input spatial
frequency above detector cutoff will be aliased down to a lower frequency. When
k= 1, there is precisely one cycle across the detector.

It is customary to plot only the OTF up to the first zero (Figure 6-8).
Unfortunately, this representation may lead the analyst to believe that there no is
response above fdco. The absolute highest spatial frequency that can be faithfully
reproduced is limited by the optical system cutoff. The detector cutoff may be
either higher (optically limited system) or lower (detector limited system).

The detector temporal response is considered separable from the spatial
response. The detector's temporal MTF is presented in Section 6.4.2.
Figure 6-7. Detector OTF from 0 to +oo as a function of
normalized spatial frequency fx lflco. The negative OTF values
represent contrast reversal where periodic dark bars will
appear as light bars.

Figure 6-8. Typical detector OTF representation as a function
of normalized spatial frequency fjfdco. Since the OTF is
positive up to the first zero, it called the MTF.

6.3. MOTION

Motion produces a blurred image. Motion blurs detail and therefore the
net effect is to decrease the range at which target detail can be resolved. The
effects of motion during the entire integration and interpretation process must be
considered.7 If the motion is slow compared to the integration time or processing
time, the target appears to move.
Linear motion includes both target movement (relative to the imaging
system) and motion of the image across the detectors. Unbalanced scanning
mirrors may introduce a sinusoidal motion. Motors or turbines on aircraft and
vehicles also can induce sinusoidal motion through mechanical vibrations.
Mountmg structures may have resonant frequencies that induce sinusoidal motion.
High frequency random motion is simply called jitter.

Typically, an imaging system is subjected to linear, sinusoidal, and
random motion simultaneously

OTFmolil,„ = OTFUnear OTFslnusoidal OTFrmdom . (6-27)

The spectrum of these frequencies extends from very low frequency to high
frequencies. The magnitude of each motion must be quantified so that the
appropriate equations are used. In a laboratory, the system may be mounted on a
vibration-isolated stabilized table. Here, no motion is expected so that OTFmotion =
1. Systems may contain a stabilized mirror that reduces OTFmotion. The values used
for the various motional OTFs should be the values expected after stabilization is
in operation and not the movement of the platform.

Motion affects the ability to resolve detail. It does not necessarily affect
the ability to detect a target. The eye can follow a slow moving blob. However,
with all motion, the edges become fuzzy. As the speed increases, the edge
becomes less distinct.

6.3.1. LINEAR MOTION

The OTF degradation due to linear motion (FLIR92 ARG-7) is

OTFlinear (fx ) = sine (a{fx ) , (6-28)

where a, is the angular distance moved and is equal to vrtint. The variable, vr is the
relative angular velocity between the sensor and the target in the horizontal
direction. The integration time is tint . The precise value to be used for the
integration time depends upon the system application. When an observer views the
image, the eye blends many frames of data. For common module systems, the eye
integration time should be used. Although the exact eye integration time is
debatable, FLIR92 documentation recommends 0.1 s. The eye integration time is
further discussed in Section 19.2.2., SNRfa and te. For other applications, different
integration times may be appropriate. See Sections 8.1, 9.1, and 10.3, all entitled
Motion.
Figure 6-9 illustrates the OTF due to linear motion as a fraction of a
DAS Linear motion only affects the OTF in the direction of the motion. As the
motion increases, the OTF becomes negative. A negative OTF indicates that phase
reversal has occurred: periodic black bars appear white and periodic white bars
appear black.

NORMALIZED SPATIAL FREQUENCY

Figure 6-9. OTFlinear as a function of normalized spatial
frequency fxlf(ko. The linear motion is normalized to «//DAS.
In the laboratory a, is usually zero.

Multiplying the detector OTF and linear motion OTF provides the
composite OTF shown in Figure 6-10. As a rule-of-the-thumb, when the linear
motion is less than about 20% of the DAS, it has minimal effect on system
performance. When linear motion becomes large, phase reversal can occur (Figure
6-11). Figure 6-12 illustrates a test pattern smeared by linear motion. At cutoff
(where the system OTF approaches the first zero), the bar pattern cannot be
resolved. Above cutoff, phase reversal is seen. The ability to see imagery above
cutoff is called false resolution or spurious resolution.

Blurring due to linear motion well known in photography. Movement of
the camera or the scene blurs the image. The shutter speed determines the
integration time. Staring arrays acts in a manner similar to still cameras. Scanning
systems do not.

With interlace scanning, continuously moving (linear motion) targets will
be at different locations from one field to another. That is, scanning takes time and
the target is constantly moving during this period. Here straight lines will appear
disjointed. For pure serial scan systems, a vertical line will appear as a diagonal
line. The offset from the vertical depends upon the relationship between the linear
motion and the frame time. For common module systems with bidirectional
scanning, time is "reversed” during the interlace field. Here, a vertical line may
appear serrated from line-to-line. These effects are not present with random or
sinusoidal motion since the target’s average location never changes.

Figure 6-10. OTFdetectorOTF,inear as a function of normalized
spatial frequency/^ /flco. When linear motion is 20% of a DAS,
OTFlinear is 0.935 at/x//Zc0=l.

NORMALIZED SPATIAL FREQUENCY

Figure 6-11. OTFdetectorOTFlinear as a function of normalized
spatial frequency fx /fdco for large movements. The movement
is expressed as a percentage of a DAS. The negative OTF
represents phase reversal.
< ц
ж ж ж

ORIGINAL IMAGE

LINEAR MOTION

Figure 6-12. A test pattern image when linear motion is
present. Since the motion is horizontal, the vertical resolution
is maintained. When the bar target frequency is equal to the
first zero of the OTFlinear, the bars blend into a blob. Higher
spatial frequency bars illustrate phase reversal: There appears
to be two black bars on a gray background. (From reference 8,
by courtesy of G. L. Conrad).

6.3.2. SINUSOIDAL MOTION

When sinusoidal motion is described by

в = as sin (cot}.

(6-29)

The image blur is twice the amplitude of the oscillatory motion as. If the total
viewing time is large such that many cycles are viewed,

(6-30)

Then the OTF degradation (FLIR92 ARG-9) is
OTFsinusoidal (fx ) = Jo (iHsrfx ) >

(6-31)

where Jo( ) is the zero-order Bessel function. Sinusoidal motion only affects the
OTF in the direction of the motion. Figure 6-13 illustrates sinusoidal peak-to-peak
motion as a fraction of the DAS. For x < 3, Jo( ) can be approximated by the first
five terms of the infinite series

4 + 64 2048 + 147456 '

(6-32)

If the OTF is plotted only up to fdco then the approximation can be used when 2as <
DAS. The approximation for sinusoidal motion is valid for modest values. Any
computer program that includes this approximation must specify the region of
applicability. Rather than use the approximations, the complete Bessel function
can be used for sinusoidal motion.6

Figure 6-14 illustrates OTFsinusoidalOTFdetector as a function of the
normalized spatial frequency fjfdco. As with linear motion, large movements can
produce negative OTFs. As a rule-of-the-thumb, when the peak-to-peak motion is
less than about 20% of a DAS, system performance is not significantly affected.

NORMALIZED SPATIAL FREQUENCY

Figure 6-13. OTFsinusoidal as a function of 2a/DAS. In the
laboratory as is usually zero.
NORMALIZED SPATIAL FREQUENCY

Figure 6-14. OTFsjnusoidalOTFdetector as a function of normalized
spatial frequency fx !flco. Movement is expressed as a
percentage of2«v/DAS.

6.3.3. RANDOM MOTION (JITTER)

High frequency random motion can be described (FLIR92 ARG-8) by a
Gaussian OTF

OTF н (f } = е~2(тУг/х}
random \ J x ' c

(6-33)

where or is the rms random displacement in mrad. It is assumed that the image has
moved often during the integration time, tint, so that the central limit theorem is
valid. The central limit theorem says that a Gaussian distribution can describe
many random movements. Figure 6-15 illustrates the random motion OTF. Since
this OTF is always positive, it is given as an MTF.

Multiplying the detector OTF and random motion OTF provides Figure
6-16. As a rule-of-the-thumb, when the rms value of the random motion is less
than about 10% of a DAS, system performance is not significantly affected.
MTF MTF

Figure 6-15. MTF degradation due to high frequency random
(Gaussian) movement as a function of or fx. In the laboratory,
or is usually zero.

NORMALIZED SPATIAL FREQUENCY

Figure 6-16. OTFrandomOTFdetector as a function of normalized
spatial frequency fx lflco. The rms movement is expressed as a
percentage of <rr./DAS. MTFrandom is always positive so that
phase reversal is not seen.
6.3.4. NONLINEAR SCAN MIRROR MOVEMENT

Since all calculations are performed in object space, any nonlinear scan
mirror motion must be referred to object space. Let the scanner be between the
afocal telescope and re-imager (see Figure 6-1). If the afocal telescope has
magnification Pafocai then the high frequency mirror movements in object space are

„ _ ^Imirror

al-m~ p
rafocal

„ — as-mirror _ аг-mirror o/i\

as-m-—----------, and crr-m=—---------• (6-34)

*afocal ^afocal

The linear and sinusoidal components of nonlinear movements are additive and the
random movements add in quadrature. The total motional values are

V2 2

<7r + a rm . (6-35)

The variables at_tot , as_tot, and vr_tot are used in Equations 6-28, 6-31, and 6-33,
respectively.

6.3.5. LOW FREQUENCY MOTION

Image blurring due to low frequency vibrations is a random process.
Quantification9'14 depends upon the initial phase of the motion and the exposure
time. Low frequency MTFs can only be handled statistically since no closed-form
solution is possible. Even though edges may be blurred, the eye can easily follow
low frequency movement.

6.4. ELECTRONIC MTF

While the optical OTF is two-dimensional, electrical circuits are causal
and one-dimensional in time. The Fourier transform is appropriate for optical
systems but the Laplace transform is more convenient for analyzing electrical
circuits.
Common module systems 107

6.4.1. CONVERSION: ELECTRICAL FREQUENCY TO
SPATIAL FREQUENCY

The electrical frequency,/,, is related15 to the spatial frequency by the
scan velocity,

fe~v

(6-36)

where r must have units of mrad/s to convert cycles/mrad (£) to Hertz (/). The
scan velocity is the ratio of the DAS to the detector dwell time

dwell

(6-37)

where Tdwel) is the time it takes a target edge to scan across the detector element.
Using the small angle approximation and assuming constant scan velocity, it is

~ dwell

Na/3risc

(6-38)

where N is the number of detectors operating in parallel (vertical direction), a is
the DAS in the horizontal direction, p is the DAS in the vertical direction, A is the
HFOV, В is the VFOV, Fr is the frame rate, i]os is the overscan ratio, and t]sc is the
scan efficiency.

The overscan ratio is given by

„ _PNscene /3 NR

os В В '

(6-39)

where Nscene is the number of scene lines per frame and R is the interlace ratio.
When t]os = 1, the entire scene is perfectly dissected by the detector array. When t]os
< 1, areas of the scene are missed and when tjos> 1, overlapping occurs (Figure 6-
17). The relationship between the overscan ratio and sampling rates is discussed in
Chapter 11, Vertical MTF. Substituting Equations 6-38 and 6-38 into 6-37 yields

A F N

г7 scene

(6-40)

Note that Nscene may not be the same as number of lines seen on the monitor.
EMUX, EOMUX, and staring array systems may employ line-to-line interpolation
to achieve monitor compatibility. Line-to-line interpolation can affect the vertical
MTF (discussed in Chapter 11, Vertical MTF).

Figure 6-17. Definition of overscan ratio. When *]os= 1, the
entire scene is perfectly dissected by the detector array. When
t]os < 1, areas of the scene are missed and when tjos > 1,
overlapping occurs.

The vertical field-of-view is specified by the effective vertical extent of the array,
dyarr(ly, and the effective focal length

FFOK = B = 2taii 1

у-array

(6-41)

where

dy-array = dcc (TV - 1) + dy + (/? - 1)^ , (6-42)

and dcc is the center-to-center spacing of the detector elements and dy is the
detector dimension in the vertical direction. The horizontal field-of-view is
determined by the optical design and the scan mirror angular movement.

The scan efficiency is the ratio of the amount of time that the detectors
actively sample the field-of-view in one frame, tactive, to the total time taken for one
frame, tfnme

t frame
Example 6-1
SCAN EFFICIENCY

What is the scan efficiency for an LWIR common module system?

The U.S. common module system contains 180 detectors that are
interlaced by an oscillating mirror to create 360 infrared scene lines (Figure 2-6).
As illustrated in Figure 6-18 the detector senses radiation from the FOV when the
scanner is between -5° and +5°. This is the active scan time. The scan mirror
requires time to decelerate, stop, reverse direction and accelerate to a linear
velocity as shown in Figure 6-19. The common module system operates at a frame
rate of 30 Hz. Each frame lasts 1/30 sec and each field lasts 1/60 sec. If the active
scan time is 12.5 ms for each field, then the scan efficiency is 12.5/16.7 = 75%.
During the inactive time (0 < -5° and 0 > +5°), the detector senses radiation from
parts of the scene and the housing. For background limited systems, this radiation
introduces scan noise if it reaches the detector during the active scan time.
Bidirectional oscillating mirrors typically have a scan efficiency over 75%.

An observer viewed the output of the original common module systems
(Figure 6-1). The frame rate must be sufficiently high such that the eye does not
perceive flicker. U.S. commercial televisions employ a frame rate of 30 Hz and
this value has become standard for nearly all imaging systems used in the U.S. The
scan frequency must be 30 Hz if the output is displayed on a standard monitor
(EOMUX, EMUX, and staring systems) but need not be used for common module
systems when viewing LEDs.

Figure 6-18. Beam location as a function of scan angle when
employing an oscillating mirror, (a) Scan angle = +5°, (b) scan
angle = -5°, and (c) mirror looking outside the field-of-view.
Figure 6-19. Scan angle as a function of time for an oscillating
mirror. The ratio of active scan time to one field time is the
scan efficiency.

Example 6-2
SCAN VELOCITY

A common module system employs 180 parallel detectors with 2:1
interlace and operates at 30 Hz. The scan efficiency is 0.80 and HFOV = 2.75°.
The 2:1 interlace provides 360 independent IR lines (t]os= 1). If the DAS is 0.1
mrad, what is the scan velocity and detector dwell time?

2.75°

v =------

1745™™f
deg )

(30 HzX360 lines)

(180 detectors^.^^)

^^mrad д
= 3600—-, (6-44)

and

a 0.1 __ _o

I dwell ~ — — ’

v 3600

(6-45)
Example 6-3
ROTATING POLYGONS

Figure 6-20a depicts a system that has a four sided rotating polygon for
horizontal scanning and a moving reflective mirror for vertical scanning. The
detector array is illustrated in Figure 6-20b. With a 1-m effective focal length, the
FOV is 8x8 mrad. The DAS in both the vertical and horizontal directions is 25
prad. What is the required polygon rotational speed and detector dwell time? The
monitor operates at 30 Hz.

Vertical Scanner

(a) (b)

Figure 6-20. Polygon scanner, (a) Scanning concept and (b)
the detector array contains 40 elements.

The monitor requirements determine the overall timing requirements.
The scanners must trace out the entire field-of-view within 1/30 sec. Precisely how
that is done depends upon the scanner and the scan efficiency. The scan speed and
scan efficiency drive the electronic filter design. With no overscan (t]os = 1), the
number of individual IR lines required to fill the VFOV is 8/0.025 = 320. With 40
elements in the array, there must be 320/40 = 8 sweeps to cover the VFOV or that
there must be 8:1 interlace. The rotating polygon and vertical mirror must be
synchronized to achieve the 8:1 interlace. Two options are available: (a) the
polygon can have all four faces parallel and the vertical scanning mirror provides
the interlace or (b) the polygon faces are cut at slight angles so that each face
moves the line-of-sight down one DAS (4:1 interlace). Then the vertical mirror
moves to provide another 2:1 interlace. The polygon must provide one full rotation
for each location of the vertical scanning mirror. Figure 6-21 illustrates the scan
pattern for the latter case.
The required rotation rate is

_ revs
rotation rate = 2------

frame Jy

3Q frames
s

- 60 rps - 3600 rpm . (6-46)

Rotating polygons typically have a scan efficiency of 50%. The active scan time is

I active

6O/7W facets j

(o.s)= 2.08 ms.

(6-47)

The scan velocity is 12/2.08 = 5760 mrad/s and r(hven = 0.1/5760 = 17.4 ps. The
frame time affects the scan velocity and detector dwell time. Usually the scan
efficiency is not compatible with monitor requirements. The data stream must be
reformatted in either a digital scan converter (EMUX system) or by a vidicon
(EOMUX system) to achieve monitor compatibility.

Detector

Polygon
Effect

Detector

Figure 6-21. Scan pattern.
6.4.2. DETECTOR TIME CONSTANT

The detector temporal response is assumed to follow a simple RC
network response where Td is the detector electrical time constant (FLIR92 ARG-
H)

MTFfconstant (fe ) “

_____1

71 + (2^/J2

The 3 dB frequency is f(i3dB=^^Td • The total detector MTF is the combination of
its spatial response and its electrical response.

6.4.3. AMPLIFIERS

The common module pre-amplifier and post-amplifier responses can
each be described by a single-pole, low pass filter whose fe3dB frequency is
between 100 kHz and 500 kHz. The 3-dB frequency is often called the amplifier
bandwidth.

^TFamp(fe) =

(6-49)

Since common modules have both pre-amplifiers and post-amplifiers, the total
response is

MTF - MTF MTF

amps л pre- amp post-amp •

(6-50)

6.4.4. ELECTRICAL FILTERS

The common module amplifier bandwidth is usually much broader than
that required by the signal. The signal is assumed to be band-limited by the
detector spatial frequency cutoff. Low pass filters are added to limit the noise
bandwidth and, ideally, match it to the signal bandwidth. These filters can be in
many forms. A system may have just one or may have several in series. The
common module design typically has a filter on the pre-amplifier and one on the
post-amplifier. The exact location of the filters in the circuit is at the discretion of
the designer.
With the desire to amplify all signals and minimize noise, a rule-of-the-
thumb suggests that the "ideal" low pass filter should be matched to the detector
cutoff. Here it is assumed that the signal is limited by the detector cutoff, fdco
(Figure 6-22). The "ideal" filter is

filter

when fe< —
a

(6-51)

MTFfll{er = 0 elsewhere .

While the ideal filter is useful for illustrative purposes, it is not physically
realizable. The MTF for a N-pole, low pass filter (also called a N-pole Butterworth
filter) (FLIR92 ARG-13) is

MTFfilter{fe) =

(6-52)

When N—>co, it becomes an ideal filter. The precise filter selected depends upon
the application. Since the eye tends to ignore high frequency noise, a higher cutoff
may be used. On the other hand, electronic detection is very sensitive to noise and
a lower cutoff may be appropriate.

Figure 6-22. Ideal low pass filter as a function of normalized
spatial frequency fx !fdco . The electrical response has been
referred to object spatial frequency. The ideal filter does not
attenuate any signal passed by the detector.

Common module systems are AC coupled to remove the large
uninformative background. The difference between the target and its background
is amplified and displayed. AC coupling is achieved with a high-pass filter, The
AC cut on frequency should be as low as possible to minimize AC coupling
artifacts (e.g., droop and overshoot). For a N-pole, high-pass filter used for AC
coupled systems (FLIR92 ARG-12)

MTFfiUer{fe) =

(6-53)

Example 6-4

FILTER BAND WIDTHS

The purpose of the low pass filter is to limit the noise bandwidth without
significantly affecting the signal. The ideal filter, shown in Figure 6-22, will pass
the entire signal. Physically realizable circuits do not have the ideal shape and
therefore there is a tradeoff between signal fidelity and noise bandwidth.

Many texts suggest that a bandwidth equal to one-half of the reciprocal of
the detector dwell time will pass most of the signal energy. However, to faithfully
reproduce the signal temporal characteristics, a wider bandwidth is required. When
using a single-pole, low pass filter, a range of bandwidths may be bounded by

(6-54)

Usmg the system described in Example 6-2, r(hvell = 27.78 ps,/eC/ = 18 kHz and/eC2
= 72 kHz. Figure 6-23 illustrates the response, MTFdetectorMTFfllter, for these single-
pole filters. An electrical filter response is usually plotted in log-log coordinates.
Since linear coordinates are used, the shape of the MTF curve looks different than
that shown in electrical engineering texts (where log-log plots are used). While a
single-pole filter with feC2 = 72 kHz does not significantly affect the signal, it does
produce an excessive noise bandwidth because the noise bandwidth is proportional
to MTFfllter Vortman and Bar-Lev16 provided a formal approach to filter
optimization. High order Butterworth filters can approximate the ideal filter
characteristics. But these filters increase circuit complexity and cost.
Figure 6-23. MTFdetectorMTFfllter for two single-pole, low pass
filters. When fe3(lB = 72 kHz, the signal is only slightly
degraded but the noise bandwidth may be excessive.

6.5. LEDS

In a common module system, the LEDs are modeled (FLIR92 ARG-16)
as rectangular shaped emitters

OTFLED (Л ) = sine (aLED fx),

(6-55)

where ar Fn is the LED angular subtense that is determined by the LED size and the
effective focal length of the visual optics lens system.

6.6. VISUAL OPTICS

The visual optics couples the output of the LEDs to the eye. It effectively
places the LEDs at infinity for comfortable viewing. The appropriate functional
form of OTFoptics is used for the visual optics. Here, Xave is the average wavelength
of the LED radiation.

6.7. EYE RESPONSE

The sine wave response (SWR) is used as an approximation to the eye-
brain MTF. The SWR depends on diffraction by the pupil, aberrations of the lens,
finite size of the photoreceptors, ocular tremor, and neural interconnections within
the retina and brain. Diffraction and aberrations vary with overall luminance,
monitor brightness, and chromatic composition of the light. Since the retina is
composed of rods and cones of varying densities, the location and size of the
object significantly affect the SWR. The usual measurement is to determine the
minimum contrast required to just perceive a target. The sine wave response is the
inverse of the contrast demand curve normalized to unity. It is sometimes called
the human visual system MTF or HVS-MTF.

The SWR ignores spatial noise, background luminance, angular
orientation, and exposure time. Each of these parameters significantly affects the
interpretation of image visibility. The MTF in the purest sense is noise
independent but the eye's response is very sensitive to spatial and temporal noise.
Therefore, the SWR is only an approximation to the true response. Furthermore,
the overall population exhibits large variations in response. Any MTF
approximation used for the eye therefore is only a crude approximation and
probably represents the largest uncertainty in the overall MTF analysis approach.
Factors that affect observer performance include angular subtense of the monitor
(usually selected to reduce raster effects), minimum subtense of the target, target-
background contrast, brightness of the background immediately around the target,
overall illumination, gamma of the monitor, observation time, and dynamic range
of the monitor. Visual psychophysical data can be found in texts by Farrell and
Booth17 and Biberman.18

6.7.1. CONVERSION: EYE SPATIAL FREQUENCY to
SPATIAL FREQUENCY

The spatial frequency presented to the eye depends upon the monitor
size, W, and the distance the observer is from the monitor, R, and the electronic
zoom, Z (if present). In the visual psychophysical literature, feye has units of
cycles/deg

A x

2 tan-1 -------------

I 2x17.45хЯ^Их Л
\ Л-

(6-56)

The factor, 17.45, converts degrees to mrad. The small angle approximation may
nofbe valid since the observer can get arbitrarily close to the monitor. However, it
is appropriate when the eye is centered on the monitor center and the target is
small. Here,
eye

17 AS* HFOV

I R)

(6-57)

where W/R is the visual angle subtended by monitor. The system magnification is

1 sys

FOV subtended by observer W Z
HFOV ” R HFOV *

(6-58)

Then, for small angles

feye

1 sys

(6-59)

Although early common module systems did not have electronic zoom, it is
included here for completeness. Electronic zoom may be used in any system that
digitally processes the video signal. These conversions assume that there is no
distortion in either the imaging system field-of-view or the monitor.

6.7.2. EYE MTF

The eye's detection capability depends upon the visual angle subtended
by the target size and the distance from the monitor to the observer. As shown in
Figure 6-24, in the absence of noise, the eye's contrast threshold is characteristi-
cally J-shaped. The eye is most sensitive to periodic targets whose spatial
frequencies19 range between 3 and 8 cycles/deg. The decrease in sensitivity at low
frequencies is due to the eye's inhibitory signal processing component.

Spatial Frequency (Cycles/Deg)

Figure 6-24. Representative observer contrast threshold curve.
LT and Ln are the target and background luminances
respectively.
Various researchers modeled the inverse of the contrast threshold curve
and labeled this the MTF (Figure 6-25). MTF values for spatial frequencies less
than 0.5 cycles/deg are simple extrapolations and should not be considered as
accurate. Nill20 recommended

0.19 + 0.81

(6-60)

Schulze21 recommended

-0.1138Д,е _ е-0.325Д,в

(6-61)

the Campbell-Robson data22 can be approximated by

-1-4 logj„

MTFeye(feye)^10 L

J eye
fpeak

(6-62)

and deJong and Bakker21 used

-.2

M7^(AjJ~sin2

(6-63)

Since there is significant interest is in detecting small targets, or
equivalently, seeing detail, only the high frequency (feye > fpeak) has been modeled
by several researchers. That is, they neglected the low frequency inhibitory
response. Komfeld and Lawson24 (FLIR92 ARG-25) suggested that when ocular
tremors are included, the MTF can be represented by

y-, feye

MTFeye(feye)*e~ ™ , (6-64)

where Г is a light-level dependent eye response factor that was presented in tabular
form25 by Ratches et. al. Figure 6-26 shows the data and a 3rd order polynomial fit
given by

г = 1.444 - 0.344log(s) + 0.039log2 («)+ 0.00197 log3 (s), (6-65)

and В is the monitor brightness in foot-lamberts.
Figure 6-25. Various mathematical approximations to the
eye's MTF.

Figure 6-26. Gamma variation. The squares are the data25
given in Ratches et. al. and the smooth curve is the 3rd order
approximation.

LLoyd26 suggested the eye's line spread function can be modeled as a
Gaussian function whose MTF is

MTF^ 2

(6-66)
where 0.0115 < < 0.0172. Figure 6-27 illustrates these representations.

Considering the large variation in observer responses, these curves have a similar
shape when f^e > fpeak. The original NVL 1975 model25 and FLIR92 use the
Kornfeld-Lawson eye model. This model was validated by the U.S. Army when
detecting tank-sized targets at modest ranges. This translates into modest spatial
frequencies.

SPATIAL FREQUENCY (cycles/deg)

Figure 6-27 Gaussian, Kornfeld-Lawson, and Mill's eye
model. The Gaussian curves illustrate veye = 0.0115 and veye =
0.0172. For mid to high spatial frequencies (6 to 16
cycles/mrad) the curves are similar to within a multiplicative
factor.

With the introduction of NVTherm29 (which accounts for sampling
effects), NVL suggested an eye model introduced by Overington’ and used by
Barten.31 The overall eye response depends upon several elements: pupil, retina,
and tremor:

MTF = MTF , MTF MTF
ivi 11 eye ivi 11 retinaivii l tremor •

(6-67)

The pupil MTF is

where

MTFpupil{feye) = enp -

2.504/eje

pupil

f pupil = exP (3.663- 0.026lD2upil logpupil) ’

(6-68)

(6-69)

and
N =
pupil

0.7155 +

0.277

’D pupil

(6-70)

The pupil diameter depends upon the display brightness В (units of foot-lamberts).
For monocular viewing

logg

Dpupil = -9.011 +13.25 e 21-082 .

(6-71)

For binocular viewing, the pupil diameter is reduced by 0.5 mm

DpupU = -9.511 + 13.25/2' »82 . (6-72)

The retina and tremor MTFs are

MTF • (f )-e °-01179-f^1,21

11 retina eye ) e

(6-73)

and

MTF ( f ) - e °'001458Л^
1 J tremor \J eye'

(6-74)

Figure 6-28 compares the NVTherm model to the other models.

Figure 6-28. NVTherm eye model (binocular viewing with В
= 50 Ft-L) compared to Nill, Kornfeld-Lawson, and Gaussian
(aeye= 0.0172) models.
The MRT should follow, in part, the shape of the eye's threshold curve.
Measured MRT data exhibits departures from the predicted MRT when using the
NVL 1975 model and FLIR92 for low and high spatial frequencies. As a result
deJong23 and others27,28 suggested that better MRT predictions are possible when
using an MTF similar to that shown in Figure 6-25.

The Komfeld-Lawson and the NVTherm eye model appear to follow the
other eye models (to within a multiplicative factor) for mid-range spatial
frequencies. This multiplicative factor becomes a normalization issue for the MRT.
Insufficient data exists to say with certainty which eye model is best. Because of
this uncertainty, range predictions cannot be placed on an absolute scale. All
analyses, no matter which eye model is used, must only be used for comparative
performance purposes.

Two operational scenarios are possible: the observer is allowed to move
his head or the head is fixed in space. Since the eye's detection capability depends
upon the angular subtense of the target, head movement may provide different
results than if the head is fixed in space. In the laboratory, the distance to the
monitor is not usually specified or limited in any way. To maximize detection
capability (stay on the minimum of the contrast threshold curve), an observer
subconsciously moves toward the monitor to perceive small targets and further
away to see larger targets. By allowing the observer to adjust his viewing distance
to the monitor, he apparently optimizes several interrelated detection criteria that
include striving for apparent edge sharpness and maximizing his perceived signal-
to-noise ratio. This apparently results in an equal detection capability for all spatial
frequencies such that eye's contrast sensitivity approaches a constant. This results
in a nearly constant MTF that is called the "non-limiting eye MTF" (FLIR92
ARG-24)

MTFeye (feye ) = 1 when movement is allowed . (6-75)

6.7.3. NOISY IMAGES

Although the contrast threshold is reported as J-shaped, the actual shape
depends upon the noise power spectral density.32'34 If the noise is restricted to
certain spatial frequencies, then the detection of targets of comparable spatial
frequencies becomes more difficult. The observer’s ability to see a specific spatial
frequency target depends upon the noise content in the neighborhood of that
spatial frequency (Figure 6-29). Low spatial noise frequency components will
interfere with detecting low frequency targets (large objects). Mid-spatial
frequency noise increases the contrast threshold curve at mid-frequencies and so
on. These noise factors are included in the FLIR92 model via the three-dimensi-
onal noise model. As such, they are not included in the eye MTF.
Frequency Noise

Noise Only

High Frequency
Noise Only

No Noise

о
о

Mid Frequency
Noise Only

100

Spatial Frequency (cycles/deg)

Figure 6-29. Effects of spectral noise on the contrast
threshold. The MRT should approximately follow the shape of
the contrast threshold curve.

6.8. SYSTEM DESIGN EXAMPLE: RANDOM MOTION
EFFECTS

A common module FLIR (0.1 mrad) has the following characteristics: an
effective focal length = 20 inches, input aperture = 10 inches, or = 50 grad rms,
scanner efficiency = 80%, frame rate = 30 Hz, interlace ratio = 2:1, detectors array
= 180x1 elements, detector element size of 0.002x0.002" on 0.004" centers, low
pass filter with a 3 dB point of 21.6 kHz, and horizontal field-of-view = 48 mrad.
What is the system’s horizontal MTF at the output of the pre-amplifier? What
component limits the system MTF?

SinceXco=Z>o/2 = (25.4 mm/10 pm) = 25.4 cycles/mrad,

The detector DAS is a = 0.002/20 = 0.1 mrad and

MTFdetector (fx ) = sinc (0-1 fx ) •

(6-77)

The 2:1 interlace ratio provides 360 IR lines. The scan velocity is
AF N
rJ scene

^sc

48 x 30 x 360 _ mrad

= 3600
180x0.8------------5

(6-78)

Then ffco =fe3dBN=21600/3 600 = 6 cycles/mrad and

MTFamp(fx) =

(6-79)

The system MTF is plotted in Figure 6-30. Random motion is the
limiting MTF. As shown in Figure 6-16, when the rms value of the random
motion is greater than 10% of the DAS, the system MTF is affected. It is
essential to reduce jitter for system improvement. If the random motion is
reduced to 10 grad though appropriate hardware design, the system MTF
increases (Figure 6-31). While further reduction in random motion will improve
the MTF, it may not be cost-effective to do it. It is usually cost-effective to
make all subsystem MTFs equal over the spatial frequencies of interest. Here the
random motion need only be 20 grad (Figure 6-32). Since each component
provides roughly the same MTF, it is now prudent to increase the MTF by
making the change that is least costly.

Figure 6-30. Component and system MTFs. cr = 50 grad.
Stabilization (high frequency jitter) is the limiting MTF. The
detector cutoff at 10 cycles/mrad is assumed to be the system
cutoff.
SPATIAL FREQUENCY (cycles/mrad)

Figure 6-31. Component and system MTFs. vr = 10 prad.

SPATIAL FREQUENCY (cycles/mrad)

Figure 6-32. Component and system MTFs. o; = 20 grad. The
subsystem MTFs are approximately equal over the region of
interest: zero to fdc0.

6.9. CENTRAL LIMIT THEOREM

The central limit theorem suggests that the product of a series of
monotonically decreasing subsystem MTFs will tend toward a Gaussian shape.
Specifically, the MTFs need to be roughly equivalent with approximately the same
cutoff values. This applies to MTFs such as MTTontic„ MTFtconstant, and MTTfi|ter. It
jl jl uptivs" iCAjiibidiii? niter

does not apply to MTFdetector. When the system is not detector-limited the Gaussian
representation may be adequate for back-of-the-envelope calculations

where asys is simply a system constant and has no physical meaning.
6.10. REFERENCES

1 FLIR92 uses many of the equations given here. ARG-1 refers to Equation 1 in FLIR92 Thermal
Imaging Systems Performance Model, Analyst's Reference Guide, Document RG5008993, Ft. Belvoir,
VA, January 1993.

2. R. R. Shannon, "Aberrations and their Effect on Images," in Geometric Optics, Critical Review of
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3. M Bom and E. Wolf, Principles of Optics, 3rd edition, pp. 468-469, Pergamon Press, New York
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4. L. Levi and R. Austing, "Tables of the MTF of a Defocussed Perfect Lens," Applied Optics, Vol.
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6. W. H. Press, В. P. Flannery, S. A. Teukolsky, and W. T. Vettling, Numerical Recipes, pp. 172-173,
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9. D. Wulich and N. S. Kopeika, "Image Resolution Limits Resulting From Mechanical Vibrations,"
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10. M. Fisher, O. Hadar, and N. S. Kopeika, "Numerical Calculation of Modulation Transfer
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15. W. N. Peters and D. Schribner, "Transformation for the Unified Linear Analysis of Optics and
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16 J. Vortman and A. Bar-Lev, "Optimal Electronic Response for Parallel Thermal Viewing Systems,"
Optical Engineering, Vol. 23(4), pp. 431-435 (1984).

17. R. J. Farrell and J. M. Booth, Design Handbook for Imagery Interpretation Equipment, Reprinted
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20. N. Nill, "A Visual Model Weighted Cosine Transform for Image Compression and Quality
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Proceedings Vol. 1342, pp. 317-327, (1990).
22. F. W. Campbell and J. G. Robson, "Application of Fourier Analysis to the Visibility of Gratings,"
J. Physiol. Vol. 197, pp. 551-566 (1968).

23. A. N. deJong and S. J. M. Bakker, "Fast and Objective MRTD Measurements," in Infrared Systems
- Design and Testing, P. R. Hall and J. S. Seeley, eds., SPIE Proceedings Vol. 916, pp. 127-143 (1988)

24. G. H. Komfeld and W. R. Lawson, "Visual Perception Model," Journal of the Optical Society of
America, Vol. 61(6), pp. 811-820 (1971).

25. J. Ratches, W. R. Lawson, L. P. Obert, R. J. Bergemann, T. W. Cassidy, and J. M. Swenson, "Night
Vision Laboratory Static Performance Model for Thermal Viewing Systems," US Army Electronics
Command Report ECOM Report 7043, pg. 11, Ft. Monmouth, NJ (1975).

26. J. M. Lloyd, Thermal Imaging, pg. 125, Plenum Press, New York (1975).

27. M. A. Karim, M. L. Gao, and S. H. Zheng, "Minimum Resolvable Temperature Difference Model:
a Critical Evaluation," Optical Engineering, Vol. 30(11), pp. 1788-1796 (1991).

28. J. G. Vortman and A. Bar-Lev, "Improved Minimum Resolvable Temperature Difference Model
for Infrared Imaging Systems," Optical Engineering, Vol. 26(6), pp. 492-498 (1987).

29 R. G. Driggers, R. Vollmerhausen, and B. O'Kane, "Sampled Imaging Sensor Design using the
MTF Squeeze Model to Characterize Spurious Response," in Infrared Imaging Systems: Design,
Analysis, Modeling, and TestingX, G. C. Holst ed., SPIE Proceedings Vol. 3701, pp. 61-73 (1999).

30.1 . Overington, Vision and Acquisition, Pentech Press, London (1976).

31. P. Barten, "The SQRI as a Measure for VDU Image Quality," Society of Information Display 92
Digest, pp. 867-870 (1992).

32. S. Daly, "Application of a Noise Adaptive Contrast Sensitivity Function in Image Data
Compression," Optical Engineering, Vol. 29(8), pp. 977-987 (1990).

33. H. Pollehn and H. Roehrig, "Effect of Noise on the Modulation Transfer Function of the Visual
Channel," Journal of the Optical Society of America, Vol. 60, pp. 842-848 (1970).

34. A. Van Meeteren and J. M. Valeton, "Effects of Pictorial Noise Interfering With Visual Detection,"
Journal of the Optical Society of America A, Vol. 5(3), pp. 438-444 (1988).

EXERCISES

1. Plot the back-of-the envelope approximation OTFdifr (Equation 6-9) and the
theoretical OTFdiff (Equation 6-5) on the same graph. For what spatial frequencies
is the approximation valid?

2. Plot Equation 6-22 using the approximation given in Equation 6-23. Assume a
peak-to-peak wave-front error of 0.75 waves. Compare the plot with Figure 6-6.

3. Plot MTFopticsMTFdetector for when//cy/,co equals 0.2, 0.5, 0.75, 1.0, and 2.0. Is
there an optimum ratio? Is there an advantage to using an infinitely small detector
or is there a reasonable limit on detector size?

4. Why would you design a system where fiJfoco = 0.5? (Hint: review Section
3.1.2., Extended Source).

5. List three causes for (a) linear motion, (b) sinusoidal motion, and (c) random
motion.
6. A common module system operating in the LWIR spectral region has an 8-inch
aperture and a DAS of 0.1 mrad. Calculate MTFdiffMTFdetectorMTFrandom at 4, 6, and
8 cycles/mrad. Let ur vary from 0 to 50% of the DAS in increments of 5%. Plot the
MTF for each spatial frequency as a function of ur in both linear and logarithmic
coordinates. Is there an optimum value (maximum acceptable value) for <7r?

7. Repeat Exercise 6 but assume the optical diameter is 4 inches. Comment on the
differences between the results.

8. What is the required scan velocity for an array consisting of 480x4 elements.
The frame rate is 30 Hz and the scan efficiency is 0.80. The HFOV is 48 mrad.
The DAS is 0.1 mrad. What is the detector dwell time?

9. Plot the MTF for a N-pole Butterworth filter (Equation 6-52 for N = 2, 4, 6, 8,
and 10. Also plot the ideal filter (Equation 6-51). How many poles (what value of
N) are required to approximate the ideal filter? Assume both have the same cutoff
fe3dB'

10 The high pass filter (Equation 6-53) is often used to reduce detector 1/f noise.
The 1/f noise knee may vary from a few hertz to 1 kHz. Plot
MTFopticsMTFdetectorMTFfllter for the system described in Example 6-2. The optical
aperture is 10 inches. Let fe3dB be 10, 100, 300, 600, and 1000 Hz. What effect does
fe3dB have on the system MTF?
7

EOMUX SYSTEMS

The common module system described in Chapter 6 permitted only one
observer to view the imagery. To provide multiple observers access, a vidicon
scanned the LED output. The vidicon converted parallel scanned infrared
information into a serial data stream consistent with the monitor requirements. As
such, the vidicon provided scan conversion. To offset any degradation in MTF, a
boost amplifier can be added in the video chain. MTFoptics, MTFmotion, MTFdetector,
MTFtconstant, MTFamp, MTFflIter, MTFled , MTFvoptics , and MTFeye are identical to
those described in Chapter 6. The electro-optically multiplexed (EOMUX) system
added the vidicon, boost and monitor (Figure 7-1). The pre-amplifier, post-
amplifier, filters, and LEDs are matched to the detector and scan velocity whereas
the vidicon and boost amplifier responses are matched to the monitor
requirements.

Careful electronic design insures that the circuitry has linear phase over
the frequencies of interest. By using an MTF approach to model each subsystem, it
is assumed that each subsystem has linear phase. The equations provided in this
chapter are found in many text books. Real circuit responses may deviate from
these theoretical responses. The system analyst must consult with the electronic
designers to insure that the circuits are modeled correctly.

Telescope Detector

Optics

Figure 7-1. Electro-optically multiplexed (EOMUX) system.

7.1. VIDICON

The vidicon scans the LED outputs in a manner consistent with monitor
timing requirements. The vidicon converts the number of IR lines, Nscene, into the
numbers of lines displayed on the monitor, Nmonitor. Implicit in the MTF is the
conversion from input brightness on the vidicon to output voltage.
7.1.1. CONVERSION: VIDICON LINES TO SPATIAL
FREQUENCY

Vidicon tube response is typically given in units of lines referenced to the
picture height (lines/РН). There are 2 TV lines per cycle. The conversion from
lines, f^, to spatial frequency is:

fM=2VFOVfx .

(7-1)

If the resolution is given simply as lines, it is assumed that the units are lines/PH.

7.1.2. VIDICON MTF

Many imaging tubes can be modeled1 as

MTFvidicon

(7-2)

where Fra is the value where MTF = e-1. The MTF is experimentally obtained for a
variety of frequencies. A plot of -log(MTF) versus frequency becomes a straight
line in logarithmic coordinates. Both Nv and Fres depend upon the particular device
selected. Nv typically ranges from 1.1 to 2.1. When Nv = 2, the MTF is Gaussian
distributed. Figure 7-2 illustrates three vidicon MTFs. Figure 7-3 provides the
same MTF curves in linear coordinates. Vidicons typically have a gamma of about
0.6 and therefore are nonlinear. As a result, the MTF is only valid for small
signals.

Vidicons sometimes have a nonsymmetric MTF about Fres. Here, two
exponents are used: Nvl and Nv2:

/ f

— I J tvl I

MTFvidicon = e when f^d <Frea, (7-3)

and

/ f \^v2

I J tvl

MTFvidicon = еГ'е' whenftvl > Frea • C7”4)

When fh>[=Fres, MTF = e1 and the MTF is independent of Nv.
Figure 7-2. -Log(MTFvidicon) versus frequency (lines/РН) for
three different tubes and a Gaussian distribution (Nv = 2). The
values in parentheses are (Nv, Fres}.

SPATIAL FREQUENCY (lines/PH)

Figure 7-3. MTFvidicon versus frequency (lines/PH) for three
different tubes and a Gaussian distribution. The values in
parentheses are (Nv, Fres).

7.2. VIDEO AMPLIFIERS and FILTERS

Both active circuits and tuned circuits can provide boost. Boost simply
refers to any circuit whose MTF is greater than unity. It can compensate for almost
any MTF degradation caused by other subsystems. In the early literature, each
subsystem was called an aperture. Therefore, any circuit that improved a
subsystem MTF provided "aperture correction." Each electronic filter must be
carefully designed to provide a linear phase over the range of frequencies of
interest. Nonlinearities will cause signal distortion.
7.2.1. CONVERSION: VIDEO FREQUENCY TO SPATIAL
FREQUENCY

Boost amplifiers and video filters operate on the video signal. The
bandwidth of these filters must be consistent with monitor requirements. The video
frequency,/,, is related to spatial frequency by

HFOV
t J'
* line

(7-5)

where tline is the active line time of the monitor. Table 7-1 gives some nominal
values. The actual values may vary by 10% from the nominal value.

Table 7-1

STANDARD VIDEO TIMING

Standard Number of active lines, N monitor Active line time, tlme
525 lines (RS 170) 485 52.25 ps
875 lines 809 31.1 ps
625 lines (European standard) 577 53 ps

7.2.2. BOOST CIRCUITRY

The boost amplifier can be an active analog circuit whose peaking
compensates for any specified MTF roll-off. Originally, boost was achieved with a
time-delay feedback circuit because it had a linear phase. Today, a variety of
circuits are available. Its MTF (FLIR92 ARG-14) is:

MTFboost

1 - COS Л

(7-6)

K \

1N >

where К is the boost amplitude, N is the boost order (usually 7V=1) and/oorf is the
boost frequency. When K= 1, no boost occurs.
Figure 7-4 illustrates the boost MTF for several values of К with 7V= 1.
When used with the all the other subsystem MTFs, the resultant MTFsystem is
typically less than one for all spatial frequencies. If the boost amplitude is too
large, ringing may occur. Since the boost circuitry modifies a serial data stream,
ringing will only occur in the horizontal direction.

Tuned filters can also provide boost:

(7-7)

where Q is the quality factor and equal to boost amplitude when/, = fboost. A net
lead circuit can also provide boost. The boost frequency is determined by the
relationship between the flead and the other subsystem MTFs.

(7-8)

NORMALIZED SPATIAL FREQUENCY

Figure 7-4. MTFboost for several different boost amplitudes as a
function of fjfboost with N = 1. Excessive boost will cause
ringing on edges in the horizontal direction.
7.2.3. VIDEO AMPLIFIERS

The video may have other filters to limit the bandwidth. Section 6 4.4.
Electrical Filters listed several different types. For example, in the video domain,
the N-pole Butterworth is

MTFvid =

(7-9)

The 3-dB electrical frequency is often called the video bandwidth.

7.3. MONITORS

The monitor MTF is a composite MTF that includes both the internal
amplifier response and the CRT phosphor response. Implicit in the MTF is the
conversion from input voltage to output monitor brightness. The same functional
form can describe vidicon and monitor MTFs:

ftvl

MTF = e
л monitor

res

(7-Ю)

Although not explicitly stated, Equation 7-2 and Equation 7-8 imply radial
symmetry: The MTF is the same in both the vertical and horizontal directions. If
Fres is measured in the vertical direction, then it should be multiplied by the
monitor aspect ratio to obtain the resolution in the horizontal direction.

If the monitor is spot size limited and spot size is Gaussian then

MTF = e

* monitor c

2-.^. raster

I mnm
nlf/rn r-p
( Zoom

(7-И)

When the electronic zoom is unity, the equation reduces to FLIR92 ARG-21.
Manufacturers of monitors use different methods to characterize monitor
resolution. They include (1) TV limiting response where the MTF is equal to
0.025, (2) the value where the MIT is 10%, (3) when the luminance is down 3 dB,
(4) shrinking raster, (5) when the output drops to 50% of the amplitude, (6) the
value where the MTF is 50%, and (7) where the amplitude is e'1. Table 7-2
provides the approximate relationships between these measures. For example, if
the resolution is expressed as the TV limiting resolution, 7Vft,, then апюп = 1/(2.31
With televisions, each line is created by an electron beam and the
phosphor brightness is assumed to be Gaussian. If the individual lines are
separated, there are dark bands between the lines that create a modulation whose
frequency is fraster (raster frequency). Using the Nyquist criterion, fraster requires two
times more lines than the number available on the monitor. This fictitious number
is often used for monitor calculations. The literature relates monitor resolution to
this fictitious number ("half-cycles"). Therefore, there is a factor of two difference
between the values shown in Table 7-2 and the literature.2

Table 7-2

MONITOR CONVERSION FACTORS
LINES to a GAUSSIAN DISTRIBUTION

Resolution metric (lines) Conversion
TV limiting 2.-? 1 ^ШОП
MTF1TO 2.9 j crmon
TV3dB 3. j3 ctmon
Shrinking raster
Amplitude = 50%
mtf50% □ .3 J crпюп
Amplitude = e-1 5.66 ctmon

If the TV limiting resolution is not known, it is assumed to be equal to
the number of monitor lines

/V = N

1 1 tv 11 monitor

(7-12)

and/„„„= VFOVfx. Table 7-1 provides some typical Nmonitor values.

For EOMUX, EMUX, and staring systems, the user typically supplies the
monitor. Thus, at least two calculations are necessary. One for laboratory
measurements and one for the monitor used during actual operation. If other
monitors are used for field testing, then those monitor characteristics must be used
for performance predictions. The system should not be monitor-limited. High
resolution monitors do not offer any "extra" resolution for the entire system.
Rather, high resolution monitors just insure that all the information available is
displayed.
A side issue here is whether the raster is visually obvious. It may be
better to accept poorer vertical MTF to suppress the raster pattern. An obvious
pattern is disturbing to the observer and he will claim the image quality is poor.

7.4. SYSTEM DESIGN EXAMPLE

The system described in Section 6-8 has been EO multiplexed. The
LEDs, relay lens, vidicon, boost filter, and monitor MTFs are unity over the spatial
frequencies of interest. A stabilized mirror reduces the high frequency jitter to is
20 prad. The 14-inch square monitor has a brightness of 10 fL. The observer is
initially at a distance of 29 inches but then moves to 13.45 inches to see more
detail. What is the effect of observer viewing distance on the horizontal MTF?

, . FOV subtended by observer / \ z

Magnification =-------------------------[electronic zoom) (7-13)

System FOV

Zoom.

u sys

(7-14)

At the 29 inch viewing distance,

0eye - 2 tan

' D

480 шгш/ and 0sys=48mrad . (7-15)

With no electronic zoom, Msys= 10. Using Equation 6-65, when В = 10, Г = 1.07.

Using the Komfeld-Lawson eye model (Equation 6-64)

(7-16)

Figure 7-5 illustrates all the subsystem components considered. Here, the
observer is too far from the monitor and his eye limits his ability to see detail (high
spatial frequencies). Three choices are available to improve system performance:
(a) move closer, (b) use a larger monitor, or (c) use electronic zoom. Electronic
zoom is equivalent to moving closer to the monitor. However, excessive zoom
may create an unacceptable blocky image
SPATIAL FREQUENCY (cycles/mrad)

Figure 7-5. Observer 29 inches from monitor. MTFoptics,
MTFt,ctccl0„ MTFoptics, MTFmotion, and MTFfilKr are identical with
the MTFs given in Figure 6-33. The observer's eye MTF limits
the system MTF.

At 13.45 inches,

eye

= 2 tan 1

= 960 mrad

(7-17)

and Msys = 20. As shown in Figure 7-6, the MTFs of all the subsystems are
approximately equal. Therefore to improve performance further, modify the
subsystem that is the least costly to change. Figures 7-5 and 7-6 assume that the
observer is at a fixed distance from the monitor. With head movement MTFeye = 1
and the overall system MTF is shown in Figure 6-32.

SPATIAL FREQUENCY (cycles/mrad)

Figure 7-6. Observer 13.45 inches from monitor.
7.5. REFERENCES

1. C. B. Johnson, "A Method for Characterizing Electro-Optical Device Modulation Transfer
Functions," Photographic Science and Engineering, Vol. 14(6), pp. 413-415 (1970).

2. L. M. Biberman, "Image Quality," in Perception of Displayed Information, L. M. Biberman, ed.,
pp. 16-19, Plenum Press, New York (1973).

EXERCISES

1. A vidicon manufacturer reported that the MTF is 0.8, 0.5, and 0.2 at 200, 500,
and 1000 lines respectively. What are Nv and Freft

2. A vidicon response is Gaussian. What is a if the MTF equals e'1 when the input
is 800 lines?

3. An imaging system contams an array of 480x480 detector elements. The output
is RS 170. Table 7-1 indicates that RS 170 displays 485 lines. What value should
be used in Equation 7-10?

4. A common module system produces 360 independent IR lines. Its output is
interpolated to produce 480 TV lines. What value should be used in Equation 7-
10?

5. A boost circuit has been added to the system described in Figure 7-5. The boost
amplitude, K, is 4. Plot the resultant MTFsys when fboost is 1.84 MHz, 3.67 MHz,
5.51 MHz, 7.35 MHz, and 9.19 MHz. The HFOV is 48 mrad and the output is in an
RS 170 format. Is there an optimum boost frequency?

6. An observer is free to move his head when viewing the imagery. Here, the non-
limiting eye MTF is used. Repeat Exercise 6 but assume that the eye MTF is unity.
Is there an optimum boost frequency?

7. Discuss the differences and similarities between the plots obtained from
Exercise 6 and Exercise 7.

8. Using the system described in Figure 7-5, add a net lead circuit (Equation 7-6).
Let fead be 0.92 MHz, 1.84 MHz, 3.67 MHz, 5.51 MHz and 7.35 MHz. Is there an
optimum value for feaf>

9. Does boost alter the signal-to-noise ratio? If so why? If not, why not? What is
the advantage of boost?