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E-raamat: Natural Image Statistics: A Probabilistic Approach to Early Computational Vision.

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Natural Image Statistics: A Probabilistic Approach to Early Computational Vision.Suurem pilt
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This is the first comprehensive introduction to the multidisciplinary field of natural image statistics. It offers a clear overview of all the basic theory plus the most recent developments and research, and it includes exercises and computer assignments.



Aims and Scope This book is both an introductory textbook and a research monograph on modeling the statistical structure of natural images. In very simple terms, “natural images” are photographs of the typical environment where we live. In this book, their statistical structure is described using a number of statistical models whose parameters are estimated from image samples. Our main motivation for exploring natural image statistics is computational m- eling of biological visual systems. A theoretical framework which is gaining more and more support considers the properties of the visual system to be re ections of the statistical structure of natural images because of evolutionary adaptation processes. Another motivation for natural image statistics research is in computer science and engineering, where it helps in development of better image processing and computer vision methods. While research on natural image statistics has been growing rapidly since the mid-1990s, no attempt has been made to cover the ?eld in a single book, providing a uni ed view of the different models and approaches. This book attempts to do just that. Furthermore, our aim is to provide an accessible introduction to the ?eld for students in related disciplines.

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From the reviews:

The authors did a wonderful job of introducing the field of natural image statistics, comprehensively. The book provides the underlying fundamental mathematics accessible to a wide audience. provides exercises and computer assignments at the end of the chapters. the advanced topics are treated in a similar manner to basic theory, makes the book suitable to be used as a textbook for advanced students and by researchers in any discipline related to computer vision. (Michael Goldberg and R. Goldberg, ACM Computing Reviews, October, 2010)

1 Introduction 1
1.1 What this Book Is All About
1
1.2 What Is Vision?
2
1.3 The Magic of Your Visual System
3
1.4 Importance of Prior Information
7
1.4.1 Ecological Adaptation Provides Prior Information
7
1.4.2 Generative Models and Latent Quantities
8
1.4.3 Projection onto the Retina Loses Information
9
1.4.4 Bayesian Inference and Priors
9
1.5 Natural Images
10
1.5.1 The Image Space
10
1.5.2 Definition of Natural Images
11
1.6 Redundancy and Information
13
1.6.1 Information Theory and Image Coding
13
1.6.2 Redundancy Reduction and Neural Coding
14
1.7 Statistical Modeling of the Visual System
15
1.7.1 Connecting Information Theory and Bayesian Inference
15
1.7.2 Normative vs. Descriptive Modeling of Visual System
15
1.7.3 Toward Predictive Theoretical Neuroscience
16
1.8 Features and Statistical Models of Natural Images
17
1.8.1 Image Representations and Features
17
1.8.2 Statistics of Features
18
1.8.3 From Features to Statistical Models
19
1.9 The Statistical–Ecological Approach Recapitulated
20
1.10 References
21
Part I Background
2 Linear Filters and Frequency Analysis
25
2.1 Linear Filtering
25
2.1.1 Definition
25
2.1.2 Impulse Response and Convolution
28
2.2 Frequency-Based Representation
29
2.2.1 Motivation
29
2.2.2 Representation in One and Two Dimensions
29
2.2.3 Frequency-Based Representation and Linear Filtering
34
2.2.4 Computation and Mathematical Details
37
2.3 Representation Using Linear Basis
38
2.3.1 Basic Idea
38
2.3.2 Frequency-Based Representation as a Basis
40
2.4 Space-Frequency Analysis
41
2.4.1 Introduction
41
2.4.2 Space-Frequency Analysis and Gabor Filters
43
2.4.3 Spatial Localization vs. Spectral Accuracy
46
2.5 References
48
2.6 Exercises
48
3 Outline of the Visual System
51
3.1 Neurons and Firing Rates
51
3.2 From the Eye to the Cortex
53
3.3 Linear Models of Visual Neurons
54
3.3.1 Responses to Visual Stimulation
54
3.3.2 Simple Cells and Linear Models
56
3.3.3 Gabor Models and Selectivities of Simple Cells
57
3.3.4 Frequency Channels
58
3.4 Non-linear Models of Visual Neurons
59
3.4.1 Non-linearities in Simple-Cell Responses
59
3.4.2 Complex Cells and Energy Models
61
3.5 Interactions between Visual Neurons
62
3.6 Topographic Organization
64
3.7 Processing after the Primary Visual Cortex
64
3.8 References
65
3.9 Exercises
65
4 Multivariate Probability and Statistics
67
4.1 Natural Images Patches as Random Vectors
67
4.2 Multivariate Probability Distributions
68
4.2.1 Notation and Motivation
68
4.2.2 Probability Density Function
69
4.3 Marginal and Joint Probabilities
70
4.4 Conditional Probabilities
73
4.5 Independence
75
4.6 Expectation and Covariance
77
4.6.1 Expectation
77
4.6.2 Variance and Covariance in One Dimension
78
4.6.3 Covariance Matrix
78
4.6.4 Independence and Covariances
79
4.7 Bayesian Inference
81
4.7.1 Motivating Example
81
4.7.2 Bayes' Rule
83
4.7.3 Non-informative Priors
83
4.7.4 Bayesian Inference as an Incremental Learning Process
84
4.8 Parameter Estimation and Likelihood
86
4.8.1 Models, Estimation, and Samples
86
4.8.2 Maximum Likelihood and Maximum a Posteriori
87
4.8.3 Prior and Large Samples
89
4.9 References
89
4.10 Exercises
89
Part II Statistics of Linear Features
5 Principal Components and Whitening
93
5.1 DC Component or Mean Grey-Scale Value
93
5.2 Principal Component Analysis
94
5.2.1 A Basic Dependency of Pixels in Natural Images
94
5.2.2 Learning One Feature by Maximization of Variance
96
5.2.3 Learning Many Features by PCA
98
5.2.4 Computational Implementation of PCA
101
5.2.5 The Implications of Translation-Invariance
102
5.3 PCA as a Preprocessing Tool
103
5.3.1 Dimension Reduction by PCA
103
5.3.2 Whitening by PCA
104
5.3.3 Anti-aliasing by PCA
106
5.4 Canonical Preprocessing Used in This Book
109
5.5 Gaussianity as the Basis for PCA
109
5.5.1 The Probability Model Related to PCA
109
5.5.2 PCA as a Generative Model
110
5.5.3 Image Synthesis Results
111
5.6 Power Spectrum of Natural Images
111
5.6.1 The 1 /f Fourier Amplitude or 1/1'2 Power Spectrum
111
5.6.2 Connection between Power Spectrum and Covariances
113
5.6.3 Relative Importance of Amplitude and Phase
114
5.7 Anisotropy in Natural Images
115
5.8 Mathematics of Principal Component Analysis*
116
5.8.1 Eigenvalue Decomposition of the Covariance Matrix
117
5.8.2 Eigenvectors and Translation-Invariance
119
5.9 Decorrelation Models of Retina and LGN
120
5.9.1 Whitening and Redundancy Reduction
120
5.9.2 Patch-Based Decorrelation
121
5.9.3 Filter-Based Decorrelation
124
5.10 Concluding Remarks and References
128
5.11 Exercises
129
6 Sparse Coding and Simple Cells
131
6.1 Definition of Sparseness
131
6.2 Learning One Feature by Maximization of Sparseness
132
6.2.1 Measuring Sparseness: General Framework
133
6.2.2 Measuring Sparseness Using Kurtosis
133
6.2.3 Measuring Sparseness Using Convex Functions of Square
134
6.2.4 The Case of Canonically Preprocessed Data
138
6.2.5 One Feature Learned from Natural Images
138
6.3 Learning Many Features by Maximization of Sparseness
139
6.3.1 Deflationary Decorrelation
140
6.3.2 Symmetric Decorrelation
141
6.3.3 Sparseness of Feature vs. Sparseness of Representation
141
6.4 Sparse Coding Features for Natural Images
143
6.4.1 Full Set of Features
143
6.4.2 Analysis of Tuning Properties
144
6.5 How Is Sparseness Useful?
147
6.5.1 Bayesian Modeling
147
6.5.2 Neural Modeling
148
6.5.3 Metabolic Economy
148
6.6 Concluding Remarks and References
148
6.7 Exercises
149
7 Independent Component Analysis
151
7.1 Limitations of the Sparse Coding Approach
151
7.2 Definition of ICA
152
7.2.1 Independence
152
7.2.2 Generative Model
152
7.2.3 Model for Preprocessed Data
154
7.3 Insufficiency of Second-Order Information
154
7.3.1 Why Whitening Does Not Find Independent Components
154
7.3.2 Why Components Have to Be Non-Gaussian
156
7.4 The Probability Density Defined by ICA
158
7.5 Maximum Likelihood Estimation in ICA
159
7.6 Results on Natural Images
160
7.6.1 Estimation of Features
160
7.6.2 Image Synthesis Using ICA
160
7.7 Connection to Maximization of Sparseness
161
7.7.1 Likelihood as a Measure of Sparseness
161
7.7.2 Optimal Sparseness Measures
163
7.8 Why Are Independent Components Sparse
166
7.8.1 Different Forms of Non-Gaussianity
167
7.8.2 Non-Gaussianity in Natural Images
167
7.8.3 Why Is Sparseness Dominant?
168
7.9 General ICA as Maximization of Non-Gaussianity
168
7.9.1 Central Limit Theorem
169
7.9.2 "Non-Gaussian Is Independent"
169
7.9.3 Sparse Coding as a Special Case of ICA
170
7.10 Receptive Fields vs. Feature Vectors
171
7.11 Problem of Inversion of Preprocessing
172
7.12 Frequency Channels and ICA
173
7.13 Concluding Remarks and References
173
7.14 Exercises
174
8 Information-Theoretic Interpretations
177
8.1 Basic Motivation for Information Theory
177
8.1.1 Compression
177
8.1.2 Transmission
178
8.2 Entropy as a Measure of Uncertainty
179
8.2.1 Definition of Entropy
179
8.2.2 Entropy as Minimum Coding Length
180
8.2.3 Redundancy
181
8.2.4 Differential Entropy
182
8.2.5 Maximum Entropy
183
8.3 Mutual Information
184
8.4 Minimum Entropy Coding of Natural Images
185
8.4.1 Image Compression and Sparse Coding
185
8.4.2 Mutual Information and Sparse Coding
187
8.4.3 Minimum Entropy Coding in the Cortex
187
8.5 Information Transmission in the Nervous System
188
8.5.1 Definition of Information Flow and Infomax
188
8.5.2 Basic Infomax with Linear Neurons
188
8.5.3 Infomax with Non-linear Neurons
189
8.5.4 Infomax with Non-constant Noise Variance
190
8.6 Caveats in Application of Information Theory
193
8.7 Concluding Remarks and References
195
8.8 Exercises
195
Part III Nonlinear Features and Dependency of Linear Features
9 Energy Correlation of Linear Features and Normalization
199
9.1 Why Estimated Independent Components Are Not Independent
199
9.1.1 Estimates vs. Theoretical Components
199
9.1.2 Counting the Number of Free Parameters
200
9.2 Correlations of Squares of Components in Natural Images
201
9.3 Modeling Using a Variance Variable
201
9.4 Normalization of Variance and Contrast Gain Control
203
9.5 Physical and Neurophysiological Interpretations
205
9.5.1 Canceling the Effect of Changing Lighting Conditions
205
9.5.2 Uniform Surfaces
206
9.5.3 Saturation of Cell Responses
206
9.6 Effect of Normalization on ICA
207
9.7 Concluding Remarks and References
210
9.8 Exercises
211
10 Energy Detectors and Complex Cells
213
10.1 Subspace Model of Invariant Features
213
10.1.1 Why Linear Features Are Insufficient
213
10.1.2 Subspaces or Groups of Linear Features
213
10.1.3 Energy Model of Feature Detection
214
10.2 Maximizing Sparseness in the Energy Model
216
10.2.1 Definition of Sparseness of Output
216
10.2.2 One Feature Learned from Natural Images
217
10.3 Model of Independent Subspace Analysis
219
10.4 Dependency as Energy Correlation
220
10.4.1 Why Energy Correlations Are Related to Sparseness
220
10.4.2 Spherical Symmetry and Changing Variance
221
10.4.3 Correlation of Squares and Convexity of Non-linearity
222
10.5 Connection to Contrast Gain Control
223
10.6 ISA as a Non-linear Version of ICA
224
10.7 Results on Natural Images
225
10.7.1 Emergence of Invariance to Phase
225
10.7.2 The Importance of Being Invariant
230
10.7.3 Grouping of Dependencies
232
10.7.4 Superiority of the Model over ICA
232
10.8 Analysis of Convexity and Energy Correlations*
234
10.8.1 Variance Variable Model Gives Convex h
234
10.8.2 Convex h Typically Implies Positive Energy Correlations
235
10.9 Concluding Remarks and References
236
10.10 Exercises
236
11 Energy Correlations and Topographic Organization
239
11.1 Topography in the Cortex
239
11.2 Modeling Topography by Statistical Dependence
240
11.2.1 Topographic Grid
240
11.2.2 Defining Topography by Statistical Dependencies
240
11.3 Definition of Topographic ICA
242
11.4 Connection to Independent Subspaces and Invariant Features
243
11.5 Utility of Topography
244
11.6 Estimation of Topographic ICA
245
11.7 Topographic ICA of Natural Images
246
11.7.1 Emergence of V1-like Topography
246
11.7.2 Comparison with Other Models
253
11.8 Learning Both Layers in a Two-Layer Model "I'
253
11.8.1 Generative vs. Energy-Based Approach
253
11.8.2 Definition of the Generative Model
254
11.8.3 Basic Properties of the Generative Model
255
11.8.4 Estimation of the Generative Model
256
11.8.5 Energy-Based Two-Layer Models
259
11.9 Concluding Remarks and References
260
12 Dependencies of Energy Detectors: Beyond V1
263
12.1 Predictive Modeling of Extrastriate Cortex
263
12.2 Simulation of V 1 by a Fixed Two-Layer Model
263
12.3 Learning the Third Layer by Another ICA Model
265
12.4 Methods for Analyzing Higher-Order Components
266
12.5 Results on Natural Images
268
12.5.1 Emergence of Collinear Contour Units
268
12.5.2 Emergence of Pooling over Frequencies
269
12.6 Discussion of Results
273
12.6.1 Why Coding of Contours9
273
12.6.2 Frequency Channels and Edges
274
12.6.3 Toward Predictive Modeling
274
12.6.4 References and Related Work
275
12.7 Conclusion
276
13 Overcomplete and Non-negative Models
277
13.1 Overcomplete Bases
277
13.1.1 Motivation
277
13.1.2 Definition of Generative Model
278
13.1.3 Nonlinear Computation of the Basis Coefficients
279
13.1.4 Estimation of the Basis
281
13.1.5 Approach Using Energy-Based Models
282
13.1.6 Results on Natural Images
285
13.1.7 Markov Random Field Models
285
13.2 Non-negative Models
288
13.2.1 Motivation
288
13.2.2 Definition
288
13.2.3 Adding Sparseness Constraints
290
13.3 Conclusion
293
14 Lateral Interactions and Feedback
295
14.1 Feedback as Bayesian Inference
295
14.1.1 Example: Contour Integrator Units
296
14.1.2 Thresholding (Shrinkage) of a Sparse Code
298
14.1.3 Categorization and Top-Down Feedback
302
14.2 Overcomplete Basis and End-stopping
302
14.3 Predictive Coding
304
14.4 Conclusion
305
Part IV Time, Color, and Stereo
15 Color and Stereo Images
309
15.1 Color Image Experiments
309
15.1.1 Choice of Data
309
15.1.2 Preprocessing and PCA
310
15.1.3 ICA Results and Discussion
313
15.2 Stereo Image Experiments
315
15.2.1 Choice of Data
315
15.2.2 Preprocessing and PCA
316
15.2.3 ICA Results and Discussion
317
15.3 Further References
322
15.3.1 Color and Stereo Images
322
15.3.2 Other Modalities, Including Audition
323
15.4 Conclusion
323
16 Temporal Sequences of Natural Images
325
16.1 Natural Image Sequences and Spatiotemporal Filtering
325
16.2 Temporal and Spatiotemporal Receptive Fields
326
16.3 Second-Order Statistics
328
16.3.1 Average Spatiotemporal Power Spectrum
328
16.3.2 The Temporally Decorrelating Filter
332
16.4 Sparse Coding and ICA of Natural Image Sequences
333
16.5 Temporal Coherence in Spatial Features
336
16.5.1 Temporal Coherence and Invariant Representation
336
16.5.2 Quantifying Temporal Coherence
337
16.5.3 Interpretation as Generative Model
338
16.5.4 Experiments on Natural Image Sequences
339
16.5.5 Why Gabor-Like Features Maximize Temporal Coherence
341
16.5.6 Control Experiments
344
16.6 Spatiotemporal Energy Correlations in Linear Features
345
16.6.1 Definition of the Model
345
16.6.2 Estimation of the Model
347
16.6.3 Experiments on Natural Images
348
16.6.4 Intuitive Explanation of Results
350
16.7 Unifying Model of Spatiotemporal Dependencies
352
16.8 Features with Minimal Average Temporal Change
354
16.8.1 Slow Feature Analysis
354
16.8.2 Quadratic Slow Feature Analysis
357
16.8.3 Sparse Slow Feature Analysis
359
16.9 Conclusion
361
Part V Conclusion
17 Conclusion and Future Prospects
365
17.1 Short Overview
365
17.2 Open, or Frequently Asked, Questions
367
17.2.1 What Is the Real Learning Principle in the Brain?
367
17.2.2 Nature vs. Nurture
368
17.2.3 How to Model Whole Images
369
17.2.4 Arc There Clear-Cut Cell Types?
369
17.2.5 How Far Can We Go?
371
17.3 Other Mathematical Models of Images
371
17.3.1 Scaling Laws
372
17.3.2 Wavelet Theory
372
17.3.3 Physically Inspired Models
373
17.4 Future Work
374
Part VI Appendix: Supplementary Mathematical Tools
18 Optimization Theory and Algorithms
377
18.1 Levels of Modeling
377
18.2 Gradient Method
378
18.2.1 Definition and Meaning of Gradient
378
18.2.2 Gradient and Optimization
380
18.2.3 Optimization of Function of Matrix
381
18.2.4 Constrained Optimization
381
18.3 Global and Local Maxima
383
18.4 Hebb's Rule and Gradient Methods
384
18.4.1 Hebb's Rule
384
18.4.2 Hebb's Rule and Optimization
385
18.4.3 Stochastic Gradient Methods
386
18.4.4 Role of the Hebbian Non-linearity
387
18.4.5 Receptive Fields vs. Synaptic Strengths
388
18.4.6 The Problem of Feedback
388
18.5 Optimization in Topographic ICA *
389
18.6 Beyond Basic Gradient Methods*
390
18.6.1 Newton's Method
391
18.6.2 Conjugate Gradient Methods
393
18.7 FastICA, a Fixed-Point Algorithm for ICA
394
18.7.1 The FastICA Algorithm
394
18.7.2 Choice of the FastICA Non-linearity
395
18.7.3 Mathematics of FastICA*
395
19 Crash Course on Linear Algebra
399
19.1 Vectors
399
19.2 Linear Transformations
400
19.3 Matrices
401
19.4 Determinant
402
19.5 Inverse
402
19.6 Basis Representations
403
19.7 Orthogonality
404
19.8 Pseudo-Inverse*
405
20 The Discrete Fourier Transform
407
20.1 Linear Shift-Invariant Systems
407
20.2 One-Dimensional Discrete Fourier Transform
408
20.2.1 Euler's Formula
408
20.2.2 Representation in Complex Exponentials
408
20.2.3 The Discrete Fourier Transform and Its Inverse
411
20.3 Two- and Three-Dimensional Discrete Fourier Transforms
417
21 Estimation of Non-normalized Statistical Models
419
21.1 Non-normalized Statistical Models
419
21.2 Estimation by Score Matching
420
21.3 Example 1: Multivariate Gaussian Density
422
21.4 Example 2: Estimation of Basic ICA Model
424
21.5 Example 3: Estimation of an Overcomplete ICA Model
425
21.6 Conclusion
425
References 427
Index 441
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