##### Document Text Contents

Page 1

PRENTICE HALL SIGNAL PROCESSING SERIES

Alan V. Oppenheim, Editor

ANDREWS AND HUNT Digital lmage Restoration

BRIGHAM The Fast Fourier Transform

BRIGHAM The Fast Fourier Transform and Its Applications

BURDIC Underwater Acoustic System Analysis

CASTLEMAN Digital Image Processing

COWAN AND GRANT Adaptive Filters

CROCHIERE AND RABINER Multirate Digital Signal Processing

DUDGEON AND MERSEREAU Multidimensional Digital Signal Processing

HAMMING Digital Filters, 3IE

HAYKIN, ED. Array Signal Processing

JAYANT AND NOLL Digital Coding of W a v e f o m

KAY Modern Spectral Estimation

KINO Acoustic Waves: Devices, Imaging, and Analog Signal Processing

LEA, ED. Trends in Speech Recognition

LIM Two-Dimensional Signal and Image Processing

LIM, ED. Speech Enhancement

LIM AND OPPENHEIM, EDS. Advanced Topics in Signal Processing

MARPLE Digital Spectral Analysis with Applications

MCCLELLAN AND RADER Number Theory in Digital Signal Processing

MENDEL Lessons in Digital Estimation Theory

OPPENHEIM, ED. Applicatiom of Digital Signal Processing

OPPENHEIM, WILLSKY, WITH YOUNG Signals and Systems

OPPENHEIM AND SCHAFER Digital Signal Processing

OPPENHEIM AND SCHAFER Discrete-Time Signal Processing

QUACKENBUSH ET AL. Objective Measures of Speech Quality

RABINER AND GOLD Theory and Applications of Digital Signal Processing

RABINER AND SCHAFER Digital Processing of Speech Signals

ROBINSON AND TREITEL Geophysical Signal Analysis

STEARNS AND DAVID Signal Processing Algor i thm

TRIBOLET Seismic Applications of Homomorphic Signal Processing

WIDROW AND STEARNS Adaptive Signal Processing

PROCESSING

JAE S. LIM

Department of Electrical Engineering

and Computer Science

Massachusetts Institute of Technology

PRENTICE HALL PTR, Upper Saddle River, New Jersey 07458

Page 2

Library of Congress Cataloging-in-Publication Data

Lim, Jae S.

Two-dimensional signal and image processing 1 Jae S. Lim

p. cm.- rentic ice Hall signal processing series)

~ i b l i o ~ r a ~ h ~ : p.

Includes index.

ISBN 0-13-935322-4

1. Signal processing-Digital techniques. 2. Image processing-

Digital techniques. I. Title. 11. Series.

TK5102.5.L54 1990

621.382'2-dc20 89-33088

CIP

EditoriaYproduction supervision: Raeia Maes

Cover design: Ben Santora

Manufacturing buyer: Mary Ann Gloriande

O 1990 Prentice Hall PTR

Prentice-Hall, Inc.

Simon & Schuster I A Viacom Company

Upper Saddle River, New Jersey 07458

All rights reserved. No part of this book may be

reproduced, in any form or by any means,

without permission in writing from the publisher.

Printed in the United States of America

ISBN 0-13-735322-q

Prentice-Hall International (UK) Limited, London

Prentice-Hall of Australia Pty. Limited, Sydney

Prentice-Hall Canada Inc., Toronto

Prentice-Hall Hispanoamericana, S. A , , Mexico

Prentice-Hall of India Private Limited, New Delhi

Prentice-Hall of Japan, Inc., Tokyo

Simon & Schuster Asia Pte. Ltd., Singapore

Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro

TO

KYUHO and TAEHO

Page 179

Figure P5.19

(a) Let a sequence c(n, , n,) denote the inverse of b(n , , n,); that is,

Is c(n, , n,) a well-defined sequence; that is, is C ( w , , w,) a valid Fourier transform?

Explain your answer.

(b) Can you recover a(n, , n,) from b(n , , n,)? If so, express a(n , , n,) in terms of

b(n , , n,). If not, explain why a(n , , n,) cannot be recovered from b(n, , n,).

5.20. Suppose an image x(n , , n,) is degraded by blurring in such a way that the blurred

image y(n , , n,) can be represented by

where b(n, , n,) represents the impulse response of the blurring system. Using the

complex cepstrum, develop a method to reduce the effect of b(n , , n,) by linear filtering

in the complex cepstrum domain. This type of system is referred to as a homomorphic

system for convolution. Determine the conditions under which the effect of b (n , , n,)

can be completely eliminated by linear filtering in the complex cepstrum domain.

5.21. Consider a 2 x 2-point first-quadrant support sequence a(n , , n,) sketched in the

following figure.

Infinite Impulse Response Filters Chap. 5

Figure P5.21

Let b(n, , n,) be a 2 x I-point sequence that is zero outside 0 5 n , 5 1 , n, = 0.

Determine b(n , , n,), the planar least squares inverse of a (n , , n,).

5.22. In this problem, we study one significant difference between spatial and frequency

domain methods. For simplicity, we consider the I-D case. The result extends

straightforwardly to the 2-D case. Let the desired impulse response hd(n) be given

by

hd(n) = ( f )"u(n) + 8(4)"u(-n - 1). (1)

Let the desired frequency response Hd(w) be the Fourier transform of hd(n). We

wish to design an TIR filter with the system function given by

Let h(n) denote the impulse response of the filter.

(a) In a spatial domain design method, we assume that the computational procedure

is given by

We estimate the filter coefficients a , b , and c by minimizing

Error = 2 (hd(n) - h(n) )2 .

" = -r

Show that a = -4, b = 0, c = 1 , and h(n) is given by h(n) = (4lnu(n).

(b) Is the system in (a) stable?

(c) In a frequency domain design method, we assume that

We estimate a , b , and c by minimizing

' j H a w ) - H(w)12 dw. Error = - 27T w = - =

Chap. 5 Problems

Page 180

Show that a = - 912, b = 2, c = - 7, and h(n) obtained by inverse Fourier

transforming H(w) is given by

Note that the filter is stable, but it does not have the same region of support as

h(n) in (a), and it is not even recursively computable.

(d) Suppose we use the a, b, and c obtained in (c) but use the computational procedure

in (3). Determine h(n) and show that the system is unstable.

(e) Suppose hd(n) = (f)"u(n). Show that the impulse responses obtained from the

spatial domain design method in (a) and the frequency domain design method in

(c) are the same.

Even though Parseval's theorem states that the two error criteria in (4) and (6) are

the same, they can lead to different results in practice, depending on how they are

used.

5.23. One error criterion that penalizes an unstable filter is given by (5.127). If we let the

parameter a in (5.127) approach a , is the filter designed guaranteed to be stable? Is

I

minimizing the Error in (5.127) while letting a approach a good way to design an

IIR filter?

5.24. In one design method by spectral transformation, a 2-D IIR filter is designed from a

I-D IIR filter. Let H(z) represent a 1-D causal and stable IIR filter. The filter is

a lowpass filter with cutoff frequency of 312. We design two 2-D filters H,(z,, z,)

and H2(z1, 2,) by

(a) What is the approximate magnitude response of H,(z,, z,)?

(b) Is H,(zI, 2,) stable and recursively computable?

(c) What is the approximate magnitude response of [email protected],, z,)?

(d) Is H2(z1, 2,) stable and recursively computable?

(e) We design another 2-D filter HT(zI, 2,) by

What is the approximate magnitude response of HT(zI, z2)? : I

5.25. Let H(z,, z,) denote a system function given by 1

(a) Sketch a signal flowgraph that realizes the above system function using a direct

form.

(b) Sketch a signal flowgraph that realizes the above system function using a cascade

form.

5.26. Consider the following computational procedure:

Draw a signal flowgraph that requires the smallest (within a few storage elements

from the ~ i n i m u n : possible) ~ m b c i cf storage wi ts when the input is available row

by row. You may assume that the input x(n,, n,) is zero outside 0 5 n, 5 N - 1,

0 s n, 5 N - 1 where N >> 1, and that we wish to compute y(n,, n,) for 0 5 n, 5

N - 1 , 0 s n 2 s N - 1 .

Figure P5.27

(a) Write a series of difference equations that will realize the above signal flowgraph.

(b) Determine the system function of the above signal flowgraph.

5.28. Consider a first-quadrant support IIR filter whose system function is given by

The filter may or may not be stable, depending on the coefficients, a, b, and c.

(a) If a = 2, b = 4, and c = 8, the filter is unstable. Determine H'(zl , z,) which

is stable and is a first-quadrant support system, and which has the same magnitude

response as H(zl, 2,). Note that the denominator polynomial is factorable for

this choice of a, b, and c.

(b) We wish to implement H(z,, z,), even though it may be unstable. Sketch a signal

flowgraph of H(zl, z,), which minimizes (within a few storage elements from the

minimum possible) the number of storage elements needed when the input is

available column by column. Find the total number of storage elements needed

when the input is zero outside 0 s n, 5 N - 1, 0 5 n2 5 N - 1 and we wish to

compute the output for 0 s n, s N - 1 , 0 n, 5 N - 1.

5.29. Consider the following signal flowgraph, which implements an IIR filter:

v, In,, n,)

t v(nl, nz)

x(nl, nz) o

Figure P5.29

Chap. 5 Problems

Infinite Impulse Response Filters Chap. 5

; 1

Page 357

Spatial domain design (cont.)

linear closed-form algorithms,

270-76

zero-phase filter design, 283-85

Spatial frequency response, mach

band effect and, 432-34

Spatial interpolation, 495-97

motion estimation methods and,

509- 11

Spatial masking, 434-35

Spatial resolution, 440, 441

Spatio-temporal constraint equation,

499

Spatio-temporal constraint methods,

503-7

Special support systems, 20-22

Spectral estimation, random

processes and, 347 -58

Spectral estimation methods:

application of, 391 -97

based on autoregressive signal

modeling, 369, 371-73, 375,

377

conventional methods, 361-63

data or correlation extension, 377

dimension-dependent processing,

363, 365

estimating correlation, 388-89, 391

maximum entropy method, 377-81

maximum likelihood method,

365-67, 369

modern or high resolution, 363

performance comparisons of,

384-88

Spectral subtraction, short-space,

545 -46

Stability:

algorithms for testing, 117- 19

one-dimensional tests, 119-24

problem of testing, 102-5

theorems, 105-17

Stabilization of unstable filters, 304

by the complex cepstrum method,

306-8

by the planar least squares inverse

method, 308-9

Stable systems, 20 I

State-space representation, 325-28

Stationary random process, 349-50

Statistical coding, 613

Statistical parameter estimation,

356-58

Steepest descent method, 279 -80

with accelerated convergence

method, 281

Stopband region, 201

Stopband tolerance, 202

Subband signal coding, 632, 639

Subimage-by-subimage coding, 647

Subimage-by-subimage processing,

534

Subtractive color systems, 419-20

Symmetry properties:

discrete cosine transform and, 158

discrete Fourier series and, 138

discrete Fourier transform and, 142

discrete-space cosine transform

and, 162

Fourier transform and, 25

z-transform and, 76

Systems:

convolution, 14-20

linear, 12-14

purpose of, 1-2

shift-invariant , 12- 14

special support, 20-22

stable, 20

See also Linear shift-invariant

systems

Television images, improving,

410-11

Temporal filtering for image

restoration:

frame averaging, 568-70

motion-compensated, 570, 573-75

Threshold coding, 647-48

Tolerance scheme? 2.01

Transform image coding:

adaptive coding and vector quan-

tization, 655-56

description of, 42, 590, 642

hybrid, 654-55

implementation considerations and

examples, 647-52

properties of, 642-44

reduction of blocking effect,

653-54

type of coders, 644-47

Transition band, 202

Tree codebook and binary search,

609- 11

Tristimulus values, 420-21

Tube sensors, 440

Two-channel coders, 466, 468, 538,

630, 632

Two-dimensional:

comparison of one- and two-

dimensional linear constant

coefficient difference equa-

tions, 79

comparison of one- and two-

dimensional optimal filter

design, 240 - 43

complex cepstrum, 301 -4

discrete cosine transform, 154-57

discrete-space cosine transform,

160-62

See also Discrete-space signals,

two-dimensional

Ultraviolet radiation, 414

Uniform convergence, Fourier trans-

form and, 25

Uniform-length codeword

assignment, 612- 13

Uniquely decodable, 612

Unit sample sequence, 3-5

Unit step sequence, 5-6

lndex

Unit surface, 66

Unsharp masking, 459, 462-63

Useful relations, z-transform and, 76

Variable-length codeword

assignment, 613- 16

Vector quantization, 598-611

adaptive coding and, 640-42,

655-56

Vector radix fast Fourier transforms,

172-77

Vertical state variables, 327

Video communications and

conferencing, 411

Vidicon, 438-40

Visual system, human:

adaptation, 431-32

the eye, 423-28

intensity discrimination, 429-31

mach band effect and spatial

frequency response, 432-34

model for peripheral level of,

428-29

other visual phenomena, 435-37

spatial masking, 434-35

Waveform coding:

advantages of, 617

delta modulation, 622, 624-27

description of, 590

differential pulse code modulation,

627-30

pulse code modulation, 618-21

pyramid coding, 632-34, 636-40

subband coding, 632, 639

two-channel coders, 630, 632

lndex I

Page 358

vector quantization and adaptive,

640-42

Weber's law, 430

Wedge support output mask, 95

Wedge support sequence, 21 -22

Weighted Chebyshev approximation

problem, 236, 239, 268

White noise process, 351

Wiener filtering:

adaptive, 536-39

noncausal, 354-56

reducing additive random noise

and, 527-31

variations of, 531 -33

Window method of design for finite

impulse response filters,

202- 13

Winograd Fourier transform

algorithm (WFTA), 178-82

Zero-mean random process, 349

Zero-order interpolation, 496

Zero-phase filter design:

frequency domain design, 313- 15

spatial domain design, 283-85

Zero-phase filters, 196-99

Zonal coding, 647-48

z-transform:

definition of, 65-66

examples of, 67-72

inverse, 76-78

linear constant coefficient

difference equations and,

78- 102

properties of, 74, 76

rational, 102

stability and, 102-24

Index

PRENTICE HALL SIGNAL PROCESSING SERIES

Alan V. Oppenheim, Editor

ANDREWS AND HUNT Digital lmage Restoration

BRIGHAM The Fast Fourier Transform

BRIGHAM The Fast Fourier Transform and Its Applications

BURDIC Underwater Acoustic System Analysis

CASTLEMAN Digital Image Processing

COWAN AND GRANT Adaptive Filters

CROCHIERE AND RABINER Multirate Digital Signal Processing

DUDGEON AND MERSEREAU Multidimensional Digital Signal Processing

HAMMING Digital Filters, 3IE

HAYKIN, ED. Array Signal Processing

JAYANT AND NOLL Digital Coding of W a v e f o m

KAY Modern Spectral Estimation

KINO Acoustic Waves: Devices, Imaging, and Analog Signal Processing

LEA, ED. Trends in Speech Recognition

LIM Two-Dimensional Signal and Image Processing

LIM, ED. Speech Enhancement

LIM AND OPPENHEIM, EDS. Advanced Topics in Signal Processing

MARPLE Digital Spectral Analysis with Applications

MCCLELLAN AND RADER Number Theory in Digital Signal Processing

MENDEL Lessons in Digital Estimation Theory

OPPENHEIM, ED. Applicatiom of Digital Signal Processing

OPPENHEIM, WILLSKY, WITH YOUNG Signals and Systems

OPPENHEIM AND SCHAFER Digital Signal Processing

OPPENHEIM AND SCHAFER Discrete-Time Signal Processing

QUACKENBUSH ET AL. Objective Measures of Speech Quality

RABINER AND GOLD Theory and Applications of Digital Signal Processing

RABINER AND SCHAFER Digital Processing of Speech Signals

ROBINSON AND TREITEL Geophysical Signal Analysis

STEARNS AND DAVID Signal Processing Algor i thm

TRIBOLET Seismic Applications of Homomorphic Signal Processing

WIDROW AND STEARNS Adaptive Signal Processing

PROCESSING

JAE S. LIM

Department of Electrical Engineering

and Computer Science

Massachusetts Institute of Technology

PRENTICE HALL PTR, Upper Saddle River, New Jersey 07458

Page 2

Library of Congress Cataloging-in-Publication Data

Lim, Jae S.

Two-dimensional signal and image processing 1 Jae S. Lim

p. cm.- rentic ice Hall signal processing series)

~ i b l i o ~ r a ~ h ~ : p.

Includes index.

ISBN 0-13-935322-4

1. Signal processing-Digital techniques. 2. Image processing-

Digital techniques. I. Title. 11. Series.

TK5102.5.L54 1990

621.382'2-dc20 89-33088

CIP

EditoriaYproduction supervision: Raeia Maes

Cover design: Ben Santora

Manufacturing buyer: Mary Ann Gloriande

O 1990 Prentice Hall PTR

Prentice-Hall, Inc.

Simon & Schuster I A Viacom Company

Upper Saddle River, New Jersey 07458

All rights reserved. No part of this book may be

reproduced, in any form or by any means,

without permission in writing from the publisher.

Printed in the United States of America

ISBN 0-13-735322-q

Prentice-Hall International (UK) Limited, London

Prentice-Hall of Australia Pty. Limited, Sydney

Prentice-Hall Canada Inc., Toronto

Prentice-Hall Hispanoamericana, S. A , , Mexico

Prentice-Hall of India Private Limited, New Delhi

Prentice-Hall of Japan, Inc., Tokyo

Simon & Schuster Asia Pte. Ltd., Singapore

Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro

TO

KYUHO and TAEHO

Page 179

Figure P5.19

(a) Let a sequence c(n, , n,) denote the inverse of b(n , , n,); that is,

Is c(n, , n,) a well-defined sequence; that is, is C ( w , , w,) a valid Fourier transform?

Explain your answer.

(b) Can you recover a(n, , n,) from b(n , , n,)? If so, express a(n , , n,) in terms of

b(n , , n,). If not, explain why a(n , , n,) cannot be recovered from b(n, , n,).

5.20. Suppose an image x(n , , n,) is degraded by blurring in such a way that the blurred

image y(n , , n,) can be represented by

where b(n, , n,) represents the impulse response of the blurring system. Using the

complex cepstrum, develop a method to reduce the effect of b(n , , n,) by linear filtering

in the complex cepstrum domain. This type of system is referred to as a homomorphic

system for convolution. Determine the conditions under which the effect of b (n , , n,)

can be completely eliminated by linear filtering in the complex cepstrum domain.

5.21. Consider a 2 x 2-point first-quadrant support sequence a(n , , n,) sketched in the

following figure.

Infinite Impulse Response Filters Chap. 5

Figure P5.21

Let b(n, , n,) be a 2 x I-point sequence that is zero outside 0 5 n , 5 1 , n, = 0.

Determine b(n , , n,), the planar least squares inverse of a (n , , n,).

5.22. In this problem, we study one significant difference between spatial and frequency

domain methods. For simplicity, we consider the I-D case. The result extends

straightforwardly to the 2-D case. Let the desired impulse response hd(n) be given

by

hd(n) = ( f )"u(n) + 8(4)"u(-n - 1). (1)

Let the desired frequency response Hd(w) be the Fourier transform of hd(n). We

wish to design an TIR filter with the system function given by

Let h(n) denote the impulse response of the filter.

(a) In a spatial domain design method, we assume that the computational procedure

is given by

We estimate the filter coefficients a , b , and c by minimizing

Error = 2 (hd(n) - h(n) )2 .

" = -r

Show that a = -4, b = 0, c = 1 , and h(n) is given by h(n) = (4lnu(n).

(b) Is the system in (a) stable?

(c) In a frequency domain design method, we assume that

We estimate a , b , and c by minimizing

' j H a w ) - H(w)12 dw. Error = - 27T w = - =

Chap. 5 Problems

Page 180

Show that a = - 912, b = 2, c = - 7, and h(n) obtained by inverse Fourier

transforming H(w) is given by

Note that the filter is stable, but it does not have the same region of support as

h(n) in (a), and it is not even recursively computable.

(d) Suppose we use the a, b, and c obtained in (c) but use the computational procedure

in (3). Determine h(n) and show that the system is unstable.

(e) Suppose hd(n) = (f)"u(n). Show that the impulse responses obtained from the

spatial domain design method in (a) and the frequency domain design method in

(c) are the same.

Even though Parseval's theorem states that the two error criteria in (4) and (6) are

the same, they can lead to different results in practice, depending on how they are

used.

5.23. One error criterion that penalizes an unstable filter is given by (5.127). If we let the

parameter a in (5.127) approach a , is the filter designed guaranteed to be stable? Is

I

minimizing the Error in (5.127) while letting a approach a good way to design an

IIR filter?

5.24. In one design method by spectral transformation, a 2-D IIR filter is designed from a

I-D IIR filter. Let H(z) represent a 1-D causal and stable IIR filter. The filter is

a lowpass filter with cutoff frequency of 312. We design two 2-D filters H,(z,, z,)

and H2(z1, 2,) by

(a) What is the approximate magnitude response of H,(z,, z,)?

(b) Is H,(zI, 2,) stable and recursively computable?

(c) What is the approximate magnitude response of [email protected],, z,)?

(d) Is H2(z1, 2,) stable and recursively computable?

(e) We design another 2-D filter HT(zI, 2,) by

What is the approximate magnitude response of HT(zI, z2)? : I

5.25. Let H(z,, z,) denote a system function given by 1

(a) Sketch a signal flowgraph that realizes the above system function using a direct

form.

(b) Sketch a signal flowgraph that realizes the above system function using a cascade

form.

5.26. Consider the following computational procedure:

Draw a signal flowgraph that requires the smallest (within a few storage elements

from the ~ i n i m u n : possible) ~ m b c i cf storage wi ts when the input is available row

by row. You may assume that the input x(n,, n,) is zero outside 0 5 n, 5 N - 1,

0 s n, 5 N - 1 where N >> 1, and that we wish to compute y(n,, n,) for 0 5 n, 5

N - 1 , 0 s n 2 s N - 1 .

Figure P5.27

(a) Write a series of difference equations that will realize the above signal flowgraph.

(b) Determine the system function of the above signal flowgraph.

5.28. Consider a first-quadrant support IIR filter whose system function is given by

The filter may or may not be stable, depending on the coefficients, a, b, and c.

(a) If a = 2, b = 4, and c = 8, the filter is unstable. Determine H'(zl , z,) which

is stable and is a first-quadrant support system, and which has the same magnitude

response as H(zl, 2,). Note that the denominator polynomial is factorable for

this choice of a, b, and c.

(b) We wish to implement H(z,, z,), even though it may be unstable. Sketch a signal

flowgraph of H(zl, z,), which minimizes (within a few storage elements from the

minimum possible) the number of storage elements needed when the input is

available column by column. Find the total number of storage elements needed

when the input is zero outside 0 s n, 5 N - 1, 0 5 n2 5 N - 1 and we wish to

compute the output for 0 s n, s N - 1 , 0 n, 5 N - 1.

5.29. Consider the following signal flowgraph, which implements an IIR filter:

v, In,, n,)

t v(nl, nz)

x(nl, nz) o

Figure P5.29

Chap. 5 Problems

Infinite Impulse Response Filters Chap. 5

; 1

Page 357

Spatial domain design (cont.)

linear closed-form algorithms,

270-76

zero-phase filter design, 283-85

Spatial frequency response, mach

band effect and, 432-34

Spatial interpolation, 495-97

motion estimation methods and,

509- 11

Spatial masking, 434-35

Spatial resolution, 440, 441

Spatio-temporal constraint equation,

499

Spatio-temporal constraint methods,

503-7

Special support systems, 20-22

Spectral estimation, random

processes and, 347 -58

Spectral estimation methods:

application of, 391 -97

based on autoregressive signal

modeling, 369, 371-73, 375,

377

conventional methods, 361-63

data or correlation extension, 377

dimension-dependent processing,

363, 365

estimating correlation, 388-89, 391

maximum entropy method, 377-81

maximum likelihood method,

365-67, 369

modern or high resolution, 363

performance comparisons of,

384-88

Spectral subtraction, short-space,

545 -46

Stability:

algorithms for testing, 117- 19

one-dimensional tests, 119-24

problem of testing, 102-5

theorems, 105-17

Stabilization of unstable filters, 304

by the complex cepstrum method,

306-8

by the planar least squares inverse

method, 308-9

Stable systems, 20 I

State-space representation, 325-28

Stationary random process, 349-50

Statistical coding, 613

Statistical parameter estimation,

356-58

Steepest descent method, 279 -80

with accelerated convergence

method, 281

Stopband region, 201

Stopband tolerance, 202

Subband signal coding, 632, 639

Subimage-by-subimage coding, 647

Subimage-by-subimage processing,

534

Subtractive color systems, 419-20

Symmetry properties:

discrete cosine transform and, 158

discrete Fourier series and, 138

discrete Fourier transform and, 142

discrete-space cosine transform

and, 162

Fourier transform and, 25

z-transform and, 76

Systems:

convolution, 14-20

linear, 12-14

purpose of, 1-2

shift-invariant , 12- 14

special support, 20-22

stable, 20

See also Linear shift-invariant

systems

Television images, improving,

410-11

Temporal filtering for image

restoration:

frame averaging, 568-70

motion-compensated, 570, 573-75

Threshold coding, 647-48

Tolerance scheme? 2.01

Transform image coding:

adaptive coding and vector quan-

tization, 655-56

description of, 42, 590, 642

hybrid, 654-55

implementation considerations and

examples, 647-52

properties of, 642-44

reduction of blocking effect,

653-54

type of coders, 644-47

Transition band, 202

Tree codebook and binary search,

609- 11

Tristimulus values, 420-21

Tube sensors, 440

Two-channel coders, 466, 468, 538,

630, 632

Two-dimensional:

comparison of one- and two-

dimensional linear constant

coefficient difference equa-

tions, 79

comparison of one- and two-

dimensional optimal filter

design, 240 - 43

complex cepstrum, 301 -4

discrete cosine transform, 154-57

discrete-space cosine transform,

160-62

See also Discrete-space signals,

two-dimensional

Ultraviolet radiation, 414

Uniform convergence, Fourier trans-

form and, 25

Uniform-length codeword

assignment, 612- 13

Uniquely decodable, 612

Unit sample sequence, 3-5

Unit step sequence, 5-6

lndex

Unit surface, 66

Unsharp masking, 459, 462-63

Useful relations, z-transform and, 76

Variable-length codeword

assignment, 613- 16

Vector quantization, 598-611

adaptive coding and, 640-42,

655-56

Vector radix fast Fourier transforms,

172-77

Vertical state variables, 327

Video communications and

conferencing, 411

Vidicon, 438-40

Visual system, human:

adaptation, 431-32

the eye, 423-28

intensity discrimination, 429-31

mach band effect and spatial

frequency response, 432-34

model for peripheral level of,

428-29

other visual phenomena, 435-37

spatial masking, 434-35

Waveform coding:

advantages of, 617

delta modulation, 622, 624-27

description of, 590

differential pulse code modulation,

627-30

pulse code modulation, 618-21

pyramid coding, 632-34, 636-40

subband coding, 632, 639

two-channel coders, 630, 632

lndex I

Page 358

vector quantization and adaptive,

640-42

Weber's law, 430

Wedge support output mask, 95

Wedge support sequence, 21 -22

Weighted Chebyshev approximation

problem, 236, 239, 268

White noise process, 351

Wiener filtering:

adaptive, 536-39

noncausal, 354-56

reducing additive random noise

and, 527-31

variations of, 531 -33

Window method of design for finite

impulse response filters,

202- 13

Winograd Fourier transform

algorithm (WFTA), 178-82

Zero-mean random process, 349

Zero-order interpolation, 496

Zero-phase filter design:

frequency domain design, 313- 15

spatial domain design, 283-85

Zero-phase filters, 196-99

Zonal coding, 647-48

z-transform:

definition of, 65-66

examples of, 67-72

inverse, 76-78

linear constant coefficient

difference equations and,

78- 102

properties of, 74, 76

rational, 102

stability and, 102-24

Index