生物视觉系统与自然图像统计：理论与应用

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"《自然图像统计》是一本融合了教学与研究价值的书籍，专著于早期计算视觉中的概率方法。该书将自然图像——通常指我们生活环境中的照片——的统计结构作为核心探讨对象，通过一系列统计模型来描述这些图像特征，这些模型的参数则是基于实际图像样本进行估计的。本书的初衷是探索生物视觉系统的计算建模，基于进化适应理论，视觉系统的特性被认为是自然图像统计结构的反映。生物视觉系统的研究不仅限于此，自然图像统计在计算机科学与工程领域也有着重要作用。它推动了更高效、先进的图像处理和计算机视觉技术的发展。通过对自然图像的深入理解，研究人员可以设计出更加符合视觉感知机制的算法，从而提升图像分析、识别和理解的能力。《自然图像统计》系列书籍由Max Viergever等全球顶尖学者主编，涵盖了来自不同国家和研究机构的专家，共同探讨这一跨学科领域的前沿技术和深度研究成果。系列分为四个主要类别：成像系统与图像处理、计算机视觉与图像理解、可视化以及成像技术的应用。只有那些具有独特主题且每个章节都对整个领域有所贡献的多作者专著或研究著作才会被纳入这个系列。《自然图像统计》这本书提供了一个理解自然图像内在规律的窗口，对于理解生物视觉系统的功能、改进计算机视觉技术以及推动成像技术在各个实际应用中的发展具有重要意义。通过阅读这本书，读者不仅能掌握基础理论，还能紧跟最新的科研进展，对于从事计算机视觉、人工智能或者相关领域的专业人士来说，是一本不可多得的参考资料。"

xvi Contents

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

16.6.4IntuitiveExplanationofResults...............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

17.2.1WhatIstheRealLearningPrincipleintheBrain? .....367

17.2.2Naturevs.Nurture......................368

17.2.3 How to Model Whole Images ................369

17.2.4 Are There Clear-Cut Cell Types? ..............369

17.2.5HowFarCanWeGo? ....................371

17.3 Other Mathematical Models of Images ...............371

17.3.1ScalingLaws.........................372

17.3.2 Wavelet Theory .......................372

17.3.3 Physically Inspired Models .................373

17.4FutureWork .............................374

Contents xvii

Part VI Appendix: Supplementary Mathematical Tools

18 Optimization Theory and Algorithms ..................377

18.1LevelsofModeling .........................377

18.2GradientMethod...........................378

18.2.1 Deﬁnition and Meaning of Gradient .............378

18.2.2GradientandOptimization .................380

18.2.3 Optimization of Function of Matrix .............381

18.2.4ConstrainedOptimization..................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

18.6.1Newton’sMethod ......................391

18.6.2 Conjugate Gradient Methods ................393

18.7FastICA,aFixed-PointAlgorithmforICA .............394

18.7.1TheFastICAAlgorithm...................394

18.7.2 Choice of the FastICA Non-linearity . . . .........395

18.7.3MathematicsofFastICA*..................395

19 Crash Course on Linear Algebra ....................399

19.1Vectors................................399

19.2 Linear Transformations .......................400

19.3Matrices ...............................401

19.4 Determinant .............................402

19.5Inverse................................402

19.6BasisRepresentations ........................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

20.2.1Euler’sFormula .......................408

20.2.2 Representation in Complex Exponentials . .........408

20.2.3TheDiscreteFourierTransformandItsInverse.......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

Chapter 1

Introduction

1.1 What this Book Is All About

The purpose of this book is to present a general theory of early vision and image

processing. The theory is normative, i.e. it says what is the optimal way of doing

these things. It is based on construction of statistical models of images combined

with Bayesian inference. Bayesian inference shows how we can use prior infor-

mation on the structure of typical images to greatly improve image analysis, and

statistical models are used for learning and storing that prior information.

The theory predicts what kind of features should be computed from the incoming

visual stimuli in the visual cortex. The predictions on the primary visual cortex

have been largely conﬁrmed by experiments in visual neuroscience. The theory also

predicts something about what should happen in higher areas such as V2, which

gives new hints for people doing neuroscientiﬁc experiments.

Also, the theory can be applied on engineering problems to develop more ef-

ﬁcient methods for de-noising, synthesis, reconstruction, compression, and other

tasks of image analysis, although we do not go into the details of such applications

in this book.

The statistical models presented in this book are quite different from classic sta-

tistical models. In fact, they are so sophisticated that many of them have been devel-

oped only during the last 10 years, so they are interesting in their own right. The key

point in these models is the non-Gaussianity (non-normality) inherent in image data.

The basic model presented is independent component analysis, but that is merely a

starting point for more sophisticated models.

A preview of what kind of properties these models learn is in Fig. 1.1.The

ﬁgure shows a number of linear features learned from natural images by a statis-

tical model. Chapters 5–7 will already consider models which learn such linear

features. In addition to the features themselves, the results in Fig. 1.1 show an-

other visually striking phenomenon, which is their spatial arrangement, or topog-

raphy. The results in the ﬁgure actually come from a model called Topographic

ICA, which is explained in Chap. 11. The spatial arrangement is also related to

computation of non-linear, invariant features, which is the topic of Chap. 10. Thus,

the result in this ﬁgure combines several of the results we develop in this book.

All of these properties are similar to those observed in the visual system of the

brain.

In the rest of this Introduction, we present the basic problem of image analysis,

and an overview of the various ideas discussed in more detail in this book.

A. Hyvärinen, J. Hurri, P.O. Hoyer, Natural Image Statistics,

Computational Imaging and Vision 39,

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