Local Approximation Techniques in Signal and Image Processing

576 Pages·2006·2.57 MB·English

by  Jaakko Astola

Bellingham, Washington USA
Library of Congress Cataloging-in-Publication Data

Katkovnik, V. IA. (Vladimir IAkovlevich).
Local approximation techniques in signal and image processing / Vladimir Katkovnik,
Karen Egiazarian, and Jaakko Astola.
p. cm.
Includes bibliographical references and index.
ISBN: 0-8194-6092-3 (alk. paper)
1. Signal processing--Mathematics. 2. Image processing--Mathematics. 3.
Approximation theory. I. Egiazarian, K. (Karen), 1959- II. Astola, Jaakko. III. Title.

TK5102.9.K38 2006
621.382'2--dc22
2006042318

Published by

SPIE—The International Society for Optical Engineering
P.O. Box 10
Bellingham, Washington 98227-0010 USA
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Copyright © 2006 The Society of Photo-Optical Instrumentation Engineers

All rights reserved. No part of this publication may be reproduced or distributed
in any form or by any means without written permission of the publisher.

The content of this book reflects the work and thought of the author(s).
Every effort has been made to publish reliable and accurate information herein,
but the publisher is not responsible for the validity of the information or for any
outcomes resulting from reliance thereon.

Printed in the United States of America.

The cover image: Spring in Tampere, Finland – May 2006 by Alessandro Foi.

Contents

Preface xi
Notations and Abbreviations xv

1 Introduction 1

1.1 Linear Local Approximation 2

1.1.1 Windowing 2

1.1.2 Nonparametric estimation 3

1.1.3 Scale 5

1.1.4 Ideal scale 8

1.1.5 Adaptive varying scale 9

1.2 Anisotropy 11

1.2.1 Univariate case 11

1.2.2 Multivariate case 12

1.3 Nonlinear Local Approximation 14

1.3.1 Likelihood and quasi-likelihood 14

1.3.2 Robust M-estimation 16

1.3.3 Adaptive varying scale 16

1.4 Multiresolution Analysis 17

1.5 Imaging Applications 17

1.6 Overview of the Book 18

2 Discrete LPA 21

2.1 Introduction 22

2.1.1 Observation modeling 22

2.1.2 Classes of signals 23

2.1.3 Multi-index notation 26

2.2 Basis of LPA 28

2.2.1 Idea of LPA 28

2.2.2 Windowing and scale 34

2.2.3 Estimate calculation 36

2.2.4 Multivariate estimates 37

2.2.5 Examples 38

2.3 Kernel LPA Estimates 47

2.3.1 Estimate of signal 47

2.3.2 Estimate of derivative 48

2.3.3 Reproducing polynomial kernels 48

v
vi Contents
2.4 Nonparametric Regression 50

2.4.1 Regression function 50

2.4.2 Nadaraya-Watson estimate 51

2.5 Nonparametric Interpolation 53

3 Shift-Invariant LPA Kernels 57

3.1 Regular Grid Kernels 57

3.2 Vanishing Moments 60

3.3 Frequency Domain 62

3.3.1 Frequency response 62

3.3.2 Discrete-time Fourier transform 65

3.3.3 Vanishing moments 67

3.3.4 Discrete Fourier transform 68

3.3.5 Examples 71

3.4 Numerical Shift-Invariant Kernels 75

3.4.1 Square uniform window 76

3.4.2 Gaussian window 80

3.5 Numerical Differentiation 83

3.5.1 Univariate differentiation 84

3.5.2 Multivariate differentiation 87

4 Integral LPA 91

4.1 Integral Kernel Estimators 91

4.1.1 Integral LPA 91

4.1.2 Multivariate kernels and estimates 96

4.1.3 Limit LPA estimates 97

4.1.4 Frequency domain 97

4.2 Analytical Kernels 100

4.2.1 1D case 100

4.2.2 2D case 104

4.3 Generalized Singular Functions∗ 107

4.3.1 Gaussian smoothing kernels 107

4.3.2 Univariate Dirac delta function 108

4.3.3 Multivariate Dirac delta function 110

4.3.4 Dirac’s delta sequences 111

4.4 Potential Derivative Estimates 114

5 Discrete LPA Accuracy 121

5.1 Bias andVariance of Estimates 122

5.1.1 Main results 122

5.1.2 Discussion 125

5.2 Ideal Scale 126

5.2.1 Varying scale 126

5.2.2 Invariant scale 131

5.2.3 Example 132

5.3 Accuracy of Potential Differentiators∗ 136
Contents vii
6 Adaptive-Scale Selection 137

6.1 ICI Rule 139

6.1.1 Foundations 139

6.1.2 LPA-ICI algorithm 145

6.1.3 Complexity 147

6.1.4 Convergence 147

6.1.5 Threshold adjustment 148

6.2 Multiple-Window Estimation 151

6.2.1 Multiwindow estimate 152

6.2.2 Combined-window estimate 153

6.3 Denoising Experiments 154

6.3.1 Univariate signals 154

6.3.2 Binary imaging 167

7 Anisotropic LPA 173

7.1 Directional Signal Processing 173

7.1.1 Recent developments 173

7.1.2 Directional LPA-ICI approach 180

7.2 Directional LPA 187

7.2.1 Polynomials 188

7.2.2 Directional windowing 189

7.2.3 Rotation 190

7.2.4 Calculations 193

7.2.5 Shift-invariant kernels 194

7.2.6 Frequency domain 196

7.3 Numerical Directional Kernels 198

7.3.1 Sectorial kernels 198

7.3.2 Line-wise kernels 205

8 Anisotropic LPA-ICI Algorithms 207

8.1 Accuracy Analysis∗ 207

8.1.1 Polynomial smoothness 207

8.1.2 Vanishing moments 211

8.1.3 Taylor series 212

8.1.4 Basic estimates 213

8.1.5 Rotated estimates 217

8.2 Adaptive-Scale Algorithms 219

8.2.1 Calculations 219

8.2.2 Recursive algorithms 221

8.2.3 Algorithm complexity 224

8.3 Directional Image Denoising 224

8.3.1 Criteria and algorithms 224

8.3.2 Experiments 227

8.4 Directional Differentiation 236

8.4.1 Anisotropic gradient 237

8.4.2 Examples 242
viii Contents
8.5 Shading from Depth 253

8.6 Optical Flow Estimation 256

8.6.1 Optical flow equation 257

8.6.2 Velocity estimation 259

8.6.3 Experiments 261

9 Image Reconstruction 263
9.1 Image Deblurring 264
9.1.1 Blur modeling and deblurring 264
9.1.2 LPA deblurring 267
9.2 LPA-ICI Deblurring Algorithms 273
9.2.1 ICI adaptive scale 273
9.2.2 Algorithms 274
9.2.3 Performance study 277
9.3 Motion Deblurring 280
9.4 Super-Resolution Imaging 282
9.5 Inverse Halftoning 285
9.5.1 Error diffusion 286
9.5.2 Linear model of error diffusion 287
9.5.3 Anisotropic deconvolution 289
9.5.4 Simulation results 291
9.6 3D Inverse 294
9.6.1 Sectioning microscopy 294
9.6.2 Observation model 295
9.6.3 3D spatially adaptive inverse 299
9.6.4 Simulation 304

10 Nonlinear Methods 307
10.1 Why Nonlinear Methods? 307
10.1.1 Basic hypotheses 307
10.1.2 M-estimation 309
10.1.3 Statistical approach 311
10.1.4 Some distributions 312
10.2 Robust M-Estimation 319
10.2.1 Median estimates 320
10.2.2 Performance of M-estimates 323
10.2.3 Median in Gaussian noise (simulation) 326
10.2.4 Theory of minimax estimation∗ 330
10.2.5 Median in non-Gaussian noise (simulation) 337
10.3 LPA-ICI Robust M-Estimates 339
10.3.1 Performance of LPA M-estimates 340
10.3.2 ICI algorithm for M-estimates 342
10.4 NonlinearTransform Methods 343
10.4.1 Model simplification 343
10.4.2 Variance stabilization 345
Contents ix
11 Likelihood and Quasi-Likelihood 351

11.1 Local Maximum Likelihood 351

11.1.1 Parametric ML 351

11.1.2 Nonparametric ML 353

11.1.3 Theory for local ML 355

11.1.4 LPA-ICI algorithms 358

11.2 Binary and Counting Observations 361

11.2.1 Bernoulli regression 361

11.2.2 Poisson regression 364

11.3 Local Quasi-Likelihood 366

11.3.1 Parametric quasi-likelihood 367

11.3.2 Nonparametric quasi-likelihood 369

11.4 Quasi-Likelihood LPA-ICI Algorithms 371

11.4.1 Zero-order regression 371

11.4.2 High-order regression 372

12 Photon Imaging 379

12.1 Direct Poisson Observations 381

12.1.1 Anscombe transform 381

12.1.2 Local quasi-likelihood 382

12.1.3 Recursive LPA-ICI(AV) algorithm 382

12.1.4 Numerical experiments 386

12.2 Indirect Poisson Observations 390

12.2.1 ML pure inverse 390

12.2.2 Richardson-Lucy method 391

12.2.3 EM pure inverse 393

12.2.4 Regularized inverse 396

12.2.5 LPA-ICI filtering 397

12.3 Local ML Poisson Inverse 400

12.3.1 Local ML for indirect observations 400

12.3.2 Linear inverse plus LPA-ICI filtering 401

12.3.3 Numerical experiments 405

12.4 ComputerizedTomography 407

12.4.1 Emission tomography 407

12.4.2 Transmission tomography 410

12.4.3 Adaptive LPA-ICI algorithms 422

13 Multiresolution Analysis 427

13.1 MR Analysis:Basic Concepts 427

13.1.1 Orthogonal series 428

13.1.2 Wavelets 430

13.2 Nonparametric LPA Spectrum 440

13.2.1 Finite-difference LPA spectrum 441

13.2.2 Linear independence 443

13.2.3 Gram-Schmidt orthogonalization 444

13.2.4 Orthogonal LPA spectrum analysis 445