Bayesian Modeling of Uncertainty in Low-Level Vision

205 Pages·1989·6.151 MB·English

by  Richard Szeliski (auth.)

BAYESIAN MODELING OF UNCERTAINTY
IN LOW-LEVEL VISION
THE KLUWER INTERNATIONAL SERIES IN
ENGINEERING AND COMPUTER SCIENCE
ROBOTICS: VISION, MANIPULATION AND SENSORS
Consulting Editor
Takeo Kanade
Carnegie Mellon University
Other books in the series:
Robotic Grasping and Fine Manipulation, M. Cutkosky
ISBN 0-89838-200-9
Shadows and Silhouettes in Computer Vision, S. Shafer
ISBN 0-89838-167-3
Perceptual Organization and Visual Recognition, D. Lowe
ISBN 0-89838-172-X
Robot Dynamics Algorithms, R. Featherstone
ISBN 0-89838-230-0
Three Dimensional Machine Vision, T. Kanade (editor)
ISBN 0-89838-188-6
Kinematic Modeling, Identification and Control of Robot Manipulators,
H.W. Stone
ISBN 0-89838-237-8
Object Recognition Using Vision and Touch, P.K. Allen
ISBN 0-89838-245-9
Integration, Coordination and Control of Multi-Sensor Robot Systems,
H.F. Durrant-Whyte
ISBN 0-89838-247-5
Motion Understanding: Robot and Human Vision,
W.N. Martin and 1.K. Aggrawal (Editors)
ISBN 0-89838-258-0
BAYESIAN MODELING OF
UNCERTAINTY IN
LOW-LEVEL VISION
by
Richard Szeliski
Carnegie Mellon University
with a foreword by
Takeo Kanade
1rII...
"
KLUWER ACADEMIC PUBLISHERS
Boston/Dordrecht/London
Distributors for North America:
Kluwer Academic Publishers
tOl Philip Drive
Assinippi Park
Norwell, Massachusetts 02061 USA
Distributors for all other countries:
Kluwer Academic Publishers Group
Distribution Centre
Post Office Box 322
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Library of Congress Cataloging-in-Publication Data
Szeliski, Richard. 1958-
Bayesian modeling of uncertainty in low-level vision I Richard
Szeliski ; foreword by Takeo Kanade.
p. cm. - (The Kluwer international series in engineering and
computer science; #79)
ISBN-13: 978-1-4612-8904-3 e-ISBN-13: 978-1-4613-1637-4
DOl: 10.1007/978-1-4613-1637-4
1. Computer vision-Mathematical models. I. Title. II. Series:
Kluwer international series in engineering and computer science:
SECS 79.
TA1632.S94 1989 89-15632
CIP
Copyright © 1989 by Kluwer Academic Publishers
Softcover reprint of the hardcover 1st edition 1989
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system
or transmitted in any form or by any means, mechanical, photocopying, recording, or other
wise, without the prior written permission of the publisher, Kluwer Academic Publishers, tOl
Philip Drive, Assinippi Park, Norwell, Massachusetts 02061. .
To my parents, Zdzislaw Ludwig Szeliski and Jadwiga Halina Szeliska, who
have always encouraged my academic aspirations, and whose integrity and
love have been a lifelong inspiration.
Contents
Foreword by Takeo Kanade xiii
Preface ..... xv
1 Introduction . 1
1.1 Modeling uncertainty in low-level vision 2
1.2 Previous work . . . 5
1.3 Overview of results 9
1.4 Organization . . . . 12
2 Representations for low-level vision . 15
2.1 Visible surface representations . 15
2.2 Visible surface algorithms . . . . 18
2.2.1 Regularization . . . . . . 18
2.2.2 Finite element discretization . 21
2.2.3 Relaxation . . . . . . . 22
2.3 Multiresolution representations 23
2.3.1 Multigrid algorithms . . 25
2.3.2 Relative representations 27
2.3.3 Hierarchical basis functions 33
2.4 Discontinuities . . . . . . . 45
2.5 Alternative representations . . . . . 46
3 Bayesian models and Markov Random Fields 49
3.1 Bayesian models . . . . . . 49
3.2 Markov Random Fields .. 50
3.3 Using probabilistic models. 56
4 Prior models .......... 59
4.1 Regularization and fractal priors. 60
4.2 Generating constrained fractals 65
4.3 Relative depth representations (reprise) 75
viii Bayesian Modeling of Uncertainty in Low-Level Vision
4.4 Mechanical vs. probabilistic models . 78
5 Sensor models. . . . . . . . . . . . 83
5.1 Sparse data: spring models .. 84
5.2 Sparse data: force field models 87
5.3 Dense data: optical flow ... 93
5.4 Dense data: image intensities 95
6 Posterior estimates . . . . . 99
6.1 MAP estimation . . . . 99
6.2 Uncertainty estimation . 101
6.3 Regularization parameter estimation . 106
6.4 Motion estimation without correspondence 112
7 Incremental algorithms for depth-from-motion 121
7.1 Kalman filtering . . . . . . . . . . . . 122
7.2 Incremental iconic depth-from-motion . 126
7.2.1 Mathematical analysis . . . . 130
7.2.2 Evaluation.......... 133
7.3 Joint modeling of depth and intensity 139
7.3.1 Regularized stereo ..... . 140
7.3.2 Recursive motion estimation 144
7.3.3 Adding discontinuities. 146
8 Conclusions..... 149
8.1 Summary.... 149
8.2 Future research . 151
Bibliography . . . . . . 155
A Finite element implementation. 167
B Fourier analysis. . . . . . . . . . . . . . . . . . 173
B.1 Filtering behavior of regularization . . . . . 173
B.2 Fourier analysis of the posterior distribution 175
B.3 Analysis of gradient descent. . . . . . 176
B.4 Finite element solution. . . . . . . . . 177
B.5 Fourier analysis of multigrid relaxation 179
C Analysis of optical flow computation . . . 183
Contents ix
D Analysis of parameter estimation . . 187
D.1 Computing marginal distributions 187
D.2 Bayesian estimation equations . 188
D.3 Likelihood of observations. 190
Table of symbols 193
Index ..... . 195
List of Figures
1.1 Visual processing hierarchy . . . . . . . . . . . . . . . . 3
1.2 Interpolated surface, typical surface, and uncertainty map 11
2.1 A more complex visual processing hierarchy 17
2.2 Sample data and interpolated surface . . . 20
2.3 Single level relaxation algorithm solutions . 24
2.4 Multiresolution pyramid . . . . . . . . . . . 26
2.5 Coarse-to-fine multiresolution relaxation solution . 28
2.6 Random-dot stereogram showing Comsweet illusion 29
2.7 Disparity profiles for random-dot stereogram 29
2.8 Relative multiresolution decomposition . . . 32
2.9 Hierachical basis pyramid . . . . . . . . . . 34
2.10 Hierarchical basis representation of solution 36
2.11 Conjugate gradient relaxation example ... 38
2.12 Algorithms for nodal and hierarchical conjugate gradient descent 39
2.13 Hierarchical conjugate gradient (L = 4) relaxation example. 41
2.14 Algorithm convergence as a function of L ..... 42
2.15 Algorithm convergence as a function of interpolator . . . . 43
2.16 Dual lattice for representing discontinuities . . . . . . . . . 45
3.1 Simple example of Bayesian modeling: terrain classification. 51
3.2 Conditional probabilities for terrain model . . . . . . . . . . 52
3.3 Restoration of noisy images through Markov Random Fields . 54
4.1 Typical sample from the thin plate prior model . . . 61
4.2 Fractal (random) solution . . . . . . . . . . . . . . 63
4.3 Multiresolution fractal sample from thin plate model 68
4.4 Power spectrum of mixed membrane / thin plate . . 70
4.5 Interpolated surface and fractal surface for mixed membrane /
thin plate . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 70
4.6 Power spectrum of fractal approximation . . . . . . . . . . " 71
4.7 Interpolated surface and fractal surface for fractal approximation 71
4.8 Power spectrum of composite (relative representation) surface. 72
xii Bayesian Modeling of Uncertainty in Low-Level Vision
4.9 Depth constraints for fractal generation example . . . . . . .. 74
4.10 Constrained fractal with spatially varying fractal dimension and
variance . . . . . . . . . . . . . . . . . . . . . 74
4.11 Relative representation for (3 = 3 interpolator . . . . . 77
4.12 Random sample using weak membrane as prior. . . . 80
4.13 Random sample from a three-dimensional elastic net . 81
5.1 Cubic spline fit with different data constraints ... 85
5.2 Constraint energy for contaminated Gaussian . . . . 86
5.3 Depth constraint with three-dimensional uncertainty 90
5.4 Simple camera model showing blur, sampling and noise 95
6.1 Sample covariance and variance fields ..... 102
6.2 Stochastic estimates of covariance and variance. 104
6.3 Cubic spline with confidence interval . . . 105
6.4 Typical solutions for various (>., T) settings 108
6.5 Family of splines of varying smoothness 110
6.6 Maximum likelihood estimate of O"p • • • • 110
6.7 "Blocks world" synthetic range data. . . . 117
6.8 Interpolated surfaces from sparse block data 118
6.9 Motion estimate for blocks worid data 119
7.1 Kalman filter block diagram . . . . . . 124
7.2 Iconic depth estimation block diagram 127
7.3 Illustration of disparity prediction stage 129
7.4 Computation of disparity using least squares fit . 132
7.5 Tiger image and edges . . . . 133
7.6 RMS error in depth estimate. 134
7.7 CIL image ........ . 136
7.8 CIL depth maps ..... . 137
7.9 CIL-2 and SRI depth maps 138
7.10 An Epipolar Plane Image . 141
A.1 Continuity strengths and computational molecules 168
B.1 Filter and noise model . . . . . . . . . . . . . . . 176
B.2 Effective frequency response of multigrid relaxation 180