Image Processing And Analysis

by Chan, Tony F.; Shen, Jianhong

• ISBN13: 9780898715897
• ISBN10: 089871589X
• Format: Paperback
• Copyright: 2005-09-30
• Publisher: Society for Industrial & Applied

Summary

This book develops the mathematical foundation of modern image processing and low-level computer vision, bridging contemporary mathematics with state-of-the-art methodologies in modern image processing, whilst organizing contemporary literature into a coherent and logical structure. The authors have integrated the diversity of modern image processing approaches by revealing the few common threads that connect them to Fourier and spectral analysis, the machinery that image processing has been traditionally built on. The text is systematic and well organized: the geometric, functional, and atomic structures of images are investigated, before moving to a rigorous development and analysis of several image processors. The book is comprehensive and integrative, covering the four most powerful classes of mathematical tools in contemporary image analysis and processing while exploring their intrinsic connections and integration. The material is balanced in theory and computation, following a solid theoretical analysis of model building and performance with computational implementation and numerical examples.

Author Biography

Tony F. Chan is Professor of Mathematics and currently also Dean of the Division of Physical Sciences at the University of California, Los Angeles. His research interests include mathematical and computational methods in image processing and computer vision, brain mapping, and VLSI physical design. URL: www.math.ucla.edu/~imagers.Jianhong (Jackie) Shen is Assistant Professor of Mathematics at the University of Minnesota. In addition to doing extensive research in imaging and vision sciences, he is interested in multiscale structures and patterns in scientific data analysis as well as modeling, analysis, and computation in biological and medical sciences. URL: www.math.umn.edu/~jhshen.

Table of Contents

List of Figures	xiii
Preface	xix
Introduction	1	(30)
Dawning of the Era of Imaging Sciences	1	(5)
Image Acquisition	1	(4)
Image Processing	5	(1)
Image Interpretation and Visual Intelligence	6	(1)
Image Processing by Examples	6	(6)
Image Contrast Enhancement	6	(2)
Image Denoising	8	(1)
Image Deblurring	9	(1)
Image Inpainting	9	(2)
Image Segmentation	11	(1)
An Overview of Methodologies in Image Processing	12	(12)
Morphological Approach	12	(2)
Fourier and Spectral Analysis	14	(1)
Wavelet and Space-Scale Analysis	15	(1)
Stochastic Modeling	16	(1)
Variational Methods	17	(2)
Partial Differential Equations (PDEs)	19	(2)
Different Approaches Are Intrinsically Interconnected	21	(3)
Organization of the Book	24	(2)
How to Read the Book	26	(5)
Some Modern Image Analysis Tools	31	(60)
Geometry of Curves and Surfaces	31	(14)
Geometry of Curves	31	(5)
Geometry of Surfaces in Three Dimensions	36	(8)
Hausdorff Measures and Dimensions	44	(1)
Functions with Bounded Variations	45	(9)
Total Variation as a Radon Measure	46	(3)
Basic Properties of BV Functions	49	(3)
The Co-Area Formula	52	(2)
Elements of Thermodynamics and Statistical Mechanics	54	(7)
Essentials of Thermodynamics	55	(1)
Entropy and Potentials	56	(2)
Statistical Mechanics of Ensembles	58	(3)
Bayesian Statistical Inference	61	(4)
Image Processing or Visual Perception as Inference	61	(1)
Bayesian Inference: Bias Due to Prior Knowledge	62	(2)
Bayesian Method in Image Processing	64	(1)
Linear and Nonlinear Filtering and Diffusion	65	(8)
Point Spreading and Markov Transition	65	(2)
Linear Filtering and Diffusion	67	(3)
Nonlinear Filtering and Diffusion	70	(3)
Wavelets and Multiresolution Analysis	73	(18)
Quest for New Image Analysis Tools	73	(2)
Early Edge Theory and Marr's Wavelets	75	(1)
Windowed Frequency Analysis and Gabor Wavelets	76	(1)
Frequency-Window Coupling: Malvar-Wilson Wavelets	77	(3)
The Framework of Multiresolution Analysis (MRA)	80	(6)
Fast Image Analysis and Synthesis via Filter Banks	86	(5)
Image Modeling and Representation	91	(54)
Modeling and Representation: What, Why, and How	91	(2)
Deterministic Image Models	93	(6)
Images as Distributions (Generalized Functions)	93	(3)
Lp Images	96	(2)
Sobolev Images Hn(Ω)	98	(1)
BV Images	98	(1)
Wavelets and Multiscale Representation	99	(16)
Construction of 2-D Wavelets	99	(5)
Wavelet Responses to Typical Image Features	104	(3)
Besov Images and Sparse Wavelet Representation	107	(8)
Lattice and Random Field Representation	115	(11)
Natural Images of Mother Nature	115	(1)
Images as Ensembles and Distributions	116	(1)
Images as Gibbs' Ensembles	117	(2)
Images as Markov Random Fields	119	(3)
Visual Filters and Filter Banks	122	(2)
Entropy-Based Learning of Image Patterns	124	(2)
Level-Set Representation	126	(6)
Classical Level Sets	127	(1)
Cumulative Level Sets	127	(2)
Level-Set Synthesis	129	(1)
An Example: Level Sets of Piecewise Constant Images	129	(1)
High Order Regularity of Level Sets	130	(1)
Statistics of Level Sets of Natural Images	131	(1)
The Mumford-Shah Free Boundary Image Model	132	(13)
Piecewise Constant 1-D Images: Analysis and Synthesis	132	(2)
Piecewise Smooth 1-D Images: First Order Representation	134	(1)
Piecewise Smooth 1-D Images: Poisson Representation	135	(1)
Piecewise Smooth 2-D Images	136	(2)
The Mumford-Shah Model	138	(2)
The Role of Special BV Images	140	(5)
Image Denoising	145	(62)
Noise: Origins, Physics, and Models	145	(11)
Origins and Physics of Noise	145	(2)
A Brief Overview of 1-D Stochastic Signals	147	(3)
Stochastic Models of Noises	150	(1)
Analog White Noises as Random Generalized Functions	151	(2)
Random Signals from Stochastic Differential Equations	153	(2)
2-D Stochastic Spatial Signals: Random Fields	155	(1)
Linear Denoising: Lowpass Filtering	156	(3)
Signal vs. Noise	156	(1)
Denoising via Linear Filters and Diffusion	157	(2)
Data-Driven Optimal Filtering: Wiener Filters	159	(1)
Wavelet Shrinkage Denoising	160	(14)
Shrinkage: Quasi-statistical Estimation of Singletons	160	(3)
Shrinkage: Variational Estimation of Singletons	163	(2)
Denoising via Shrinking Noisy Wavelet Components	165	(6)
Variational Denoising of Noisy Besov Images	171	(3)
Variational Denoising Based on BV Image Model	174	(17)
TV, Robust Statistics, and Median	174	(1)
The Role of TV and BV Image Model	175	(1)
Biased Iterated Median Filtering	175	(2)
Rudin, Osher, and Fatemi's TV Denoising Model	177	(1)
Computational Approaches to TV Denoising	178	(5)
Duality for the TV Denoising Model	183	(2)
Solution Structures of the TV Denoising Model	185	(6)
Denoising via Nonlinear Diffusion and Scale-Space Theory	191	(7)
Perona and Malik's Nonlinear Diffusion Model	191	(3)
Axiomatic Scale-Space Theory	194	(4)
Denoising Salt-and-Pepper Noise	198	(5)
Multichannel TV Denoising	203	(4)
Variational TV Denoising of Multichannel Images	203	(1)
Three Versions of TV[u]	204	(3)
Image Deblurring	207	(38)
Blur: Physical Origins and Mathematical Models	207	(9)
Physical Origins	207	(1)
Mathematical Models of Blurs	208	(6)
Linear vs. Nonlinear Blurs	214	(2)
Ill-posedness and Regularization	216	(1)
Deblurring with Wiener Filters	217	(3)
Intuition on Filter-Based Deblurring	217	(1)
Wiener Filtering	218	(2)
Deblurring of BV Images with Known PSF	220	(6)
The Variational Model	220	(2)
Existence and Uniqueness	222	(2)
Computation	224	(2)
Variational Blind Deblurring with Unknown PSF	226	(19)
Parametric Blind Deblurring	226	(4)
Parametric-Field-Based Blind Deblurring	230	(3)
Nonparametric Blind Deblurring	233	(12)
Image Inpainting	245	(64)
A Brief Review on Classical Interpolation Schemes	246	(10)
Polynomial Interpolation	246	(2)
Trigonometric Polynomial Interpolation	248	(1)
Spline Interpolation	249	(2)
Shannon's Sampling Theorem	251	(2)
Radial Basis Functions and Thin-Plate Splines	253	(3)
Challenges and Guidelines for 2-D Image Inpainting	256	(2)
Main Challenges for Image Inpainting	256	(1)
General Guidelines for Image Inpainting	257	(1)
Inpainting of Sobolev Images: Green's Formulae	258	(5)
Geometric Modeling of Curves and Images	263	(7)
Geometric Curve Models	264	(1)
2-, 3-Point Accumulative Energies, Length, and Curvature	265	(3)
Image Models via Functionalizing Curve Models	268	(1)
Image Models with Embedded Edge Models	269	(1)
Inpainting BV Images (via the TV Radon Measure)	270	(9)
Formulation of the TV Inpainting Model	270	(2)
Justification of TV Inpainting by Visual Perception	272	(2)
Computation of TV Inpainting	274	(1)
Digital Zooming Based on TV Inpainting	274	(2)
Edge-Based Image Coding via Inpainting	276	(1)
More Examples and Applications of TV Inpainting	277	(2)
Error Analysis for Image Inpainting	279	(3)
Inpainting Piecewise Smooth Images via Mumford and Shah	282	(2)
Image Inpainting via Euler's Elasticas and Curvatures	284	(5)
Inpainting Based on the Elastica Image Model	284	(3)
Inpainting via Mumford--Shah--Euler Image Model	287	(2)
Inpainting of Meyer's Texture	289	(2)
Image Inpainting with Missing Wavelet Coefficients	291	(4)
PDE Inpainting: Transport, Diffusion, and Navier--Stokes	295	(12)
Second Order Interpolation Models	295	(4)
A Third Order PDE Inpainting Model and Navier--Stokes	299	(2)
TV Inpainting Revisited: Anisotropic Diffusion	301	(1)
CDD Inpainting: Curvature Driven Diffusion	302	(1)
A Quasi-axiomatic Approach to Third Order Inpainting	303	(4)
Inpainting of Gibbs/Markov Random Fields	307	(2)
Image Segmentation	309	(64)
Synthetic Images: Monoids of Occlusive Preimages	309	(9)
Introduction and Motivation	309	(1)
Monoids of Occlusive Preimages	310	(5)
Mimimal and Prime (or Atomic) Generators	315	(3)
Edges and Active Contours	318	(20)
Pixelwise Characterization of Edges: David Marr's Edges	318	(2)
Edge-Regulated Data Models for Image Gray Values	320	(2)
Geometry-Regulated Prior Models for Edges	322	(3)
Active Contours: Combining Both Prior and Data Models	325	(2)
Curve Evolutions via Gradient Descent	327	(2)
Γ-Convergence Approximation of Active Contours	329	(2)
Region-Based Active Contours Driven by Gradients	331	(1)
Region-Based Active Contours Driven by Stochastic Features	332	(6)
Geman and Geman's Intensity-Edge Mixture Model	338	(6)
Topological Pixel Domains, Graphs, and Cliques	338	(1)
Edges as Hidden Markov Random Fields	339	(3)
Intensities as Edge-Regulated Markov Random Fields	342	(1)
Gibbs' Fields for Joint Bayesian Estimation of u and Γ	343	(1)
The Mumford--Shah Free-Boundary Segmentation Model	344	(25)
The Mumford--Shah Segmentation Model	344	(1)
Asymptotic M.--S. Model I: Sobolev Smoothing	345	(2)
Asymptotic M.--S. Model II: Piecewise Constant	347	(4)
Asymptotic M.--S. Model III: Geodesic Active Contours	351	(4)
Nonuniqueness of M.--S. Segmentation: A 1-D Example	355	(1)
Existence of M.--S. Segmentation	355	(4)
How to Segment Sierpinski Islands	359	(3)
Hidden Symmetries of M.--S. Segmentation	362	(2)
Computational Method I: Γ-Convergence Approximation	364	(2)
Computational Method II: Level-Set Method	366	(3)
Multichannel Logical Segmentation	369	(4)
Bibliography	373	(20)
Index	393