NaturalImageStatistics

A Probabilistic Approach to Early Computational Vision一本介绍计算机视觉的好书！ 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 thiComputational Imaging and visionManaging editorMAⅩ VIERGEVERUtrecht University, The NetherlandsSeries editorsGUNILLA BORGEFORSRACHID DERICHECentre for Image Analysis, SLU, Uppsala, SwedenIA. franceTHOMAS S HUANG, University of lllinois, Urbana, U.S.AKATSUSHI IKEUCHI, Tokyo University, JapanTIANZI JIANG, Institute of Automation, CAS, BeijingREINHARD KLETTE, University of auckland, New ZealandALES LEONARDIS, ViCoS, University of ljubljana, SloveniaHEINZ-OTTO PEITGEN, Ce vis, Bremen, germanyJOHN K. TSOTSOS, York University, CanadaThis comprehensive book series embraces state-of-the-art expository works and advancedresearch monographs on any aspect of this interdisciplinary fieleTopics covered by the series fall in the following four main categoriesImaging Systems and Image ProcessingComputer Vision and Image UnderstandingⅤ isualizationApplications of Imaging TechnologiesOnly monographs or multi-authored books that have a distinct subject area, that is whereeach chapter has been invited in order to fulfill this purpose, will be considered for theriesVolume 39For other titles published in this series, go toww.springer. com/series/5754Natural Image statisticsA Probabilistic Approachto Early Computational VisionWritten byAapo hyvarinenUniversity of helsinki, FinlandJarmo hurriUniversity of Helsinki, FinlandndPatrik o. hoyerUniversity of helsinki, FinlandSringerAapo hyvarinenPatrik O. HoyerUniversity of HelsinkiUniversity of helsinkiDept Mathematics Statistics andDept. Computer ScienceDept. Computer ScienceFinlandFinlandJarmo hurriniversityof helsinkiDept. Computer ScienceFinlandISSN1381-6446Computational Imaging and visionISBN978-1-84882-490-4e-ISBN978-1-84882-491-1Springer Dordrecht Heidelberg London New YorkBritish Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryLibrary of Congress Control Number: 2009923579AMS Codes:62H35,92B20,91E30,68U10,68T05,94A08,92C20OSpringer-Verlag London Limited 2009part from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproducedstored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by theCopyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent tothe publishersThe use of registered names, trademarks, etc, in this publication does not imply, even in the absence of aspecific statement, that such names are exempt from the relevant laws and regulations and therefore freefor general useThe publisher makes no representation, express or implied, with regard to the accuracy of the informationcontained in this book and cannot accept any legal responsibility or liability for any errors or omissionsthat may be madeCover design: WMHDesign GmbhPrinted on acid-free paperSpringerispartofSpringerScience+businessMedia(www.springer.com)PrefaceAims and ScopeThis book is both an introductory textbook and a research monograph on modelingthe statistical structure of natural images. In very simple terms, natural images" arehotographs of the typical environment where we live. In this book their statisticalstructure is described using a number of statistical models whose parameters areestimated from image samplesOur main motivation for exploring natural image statistics is computational modeling of biological visual systems. A theoretical framework which is gaining moreand more support considers the properties of the visual system to be reflectionsof the statistical structure of natural images because of evolutionary adaptationprocesses. Another motivation for natural image statistics research is in computerscience and engineering, where it helps in development of better image processingand computer viSion methodsWhile research on natural image statistics has been growing rapidly since themid-1990s. no attempt has been made to cover the field in a single book providinga unified view of the different models and approaches. This book attempts to do justthat. Furthermore, our aim is to provide an accessible introduction to the field forstudents in related disciplinesHowever, not all aspects of such a large field of study can be completely coveredin a single book. so we have had to make some choices basically we concentrateon the neural modeling approaches at the expense of engineering applications. Fur-thermore those topics on which the authors themselves have been doing researchare inevitably given more emphasisTargeted Audience and PrerequisitesThe book is targeted for advanced undergraduate students, graduate students andresearchers in vision science, computational neuroscience, computer vision, andimage processing. It can also be read as an introduction to the area by peoplewith a background in mathematical disciplines(mathematics, statistics, theoreticalphySics)Due to the multi-disciplinary nature of the subject, the book has been written soas to be accessible to an audience coming from very different backgrounds such aspsychology, computer science, electrical engineering, neurobiology, mathematicsstatistics, and physics. Therefore, we have attempted to reduce the prerequisites toa minimum. The main thing needed are basic mathematical skills as taught in introductory university-level mathematics courses. In particular, the reader is assumed toPrefaceknow the basics ofunivariate calculus(e. g. one-dimensional derivatives and integrals)linear algebra(e. g inverse matrix, orthogonality)probability and statistics(e. g expectation, probability density function, variance,covariance)To help readers with a modest mathematical background, a crash course on linearalgebra is offered at Chap. 19, and Chap 4 reviews probability theory and statisticson a rather elementary levelNo previous knowledge of neuroscience or vision science is necessary for readingthis book. All the necessary background on the visual system is given in Chap. 3and an introduction to some basic image processing methods is given in Chap 2Structure of the book and its use as a TextbookThis book is a hybrid of a monograph and an advanced graduate textbook. It startswith background material which is rather classic, whereas the latter parts of the bookconsider very recent work with many open problems. The material in the middle isquite recent but relatively establishedThe book is divided into the following partsIntroduction which explains the basic setting and motivationPart I which consists of background chapters. This is mainly classic material foundin many textbooks in statistics, neuroscience, and signal processing. However, hereit has been carefully selected to ensure that the reader has the right back ground forthe main part of the bookPart II starts the main topic, considering the most basic models for natural imagestatistics. These models are based on the statistics of linear features. i.e. linearcombinations of image pixel valuesPart III considers more sophisticated models of natural image statistics, in whichdependencies (interactions) of linear features are considered, which is related tocomputing non-linear featuresPart Iv applies the models already introduced to new kinds of data: color imagesstereo images, and image sequences(video). Some new models on the temporalstructure of sequences are also introducedPart v consists of a concluding chapter. It provides a short overview of the bookand discusses open questions as well as alternative approaches to image modelinPart VI consists of mathematical chapters which are provided as a kind of an appendix Chapter 18 is a rather independent chapter on optimization theory. Chapter 19 is background material which the reader is actually supposed to know: it isprovided here as a reminder. Chapters 20 and 2l provide sophisticated supplementary mathematical material for readers with such interestsDependencies of the parts are rather simple. When the book is used as a textbook,all readers should start by reading the first seven chapters in the order they arePrefacegiven (i.e. Introduction, Part I, and Part II except for the last chapter), unless thereader is already familiar with some of the material. After that, it is possible to jumpto later chapters in almost any order, except for the followingChapter 10 requires Chap. 9, and Chap. ll requires Chaps. 9 and 10Chapter 14 requires sect. 13.1Some of the sections are marked with an asterisk s, which means that they are moresophisticated material which can be skipped without interrupting the flow of ideasAn introductory course on natural image statistics can be simply constructed bygoing through the first n chapters of the book, where n would typically be between7 and 17, depending on the amount of time availableReferencing and exercisesTo keep the text readable and suitable for a textbook, the first ll chapters do notinclude references in the main text. References are given in a separate section atthe end of the chapter. In the latter chapters, the nature of the material requiresthat references are given in the text, so the style changes to a more scholarly oneLikewise, mathematical exercises and computer assignments are given for the first10 chaptersCode for Reproducing experimentsFor pedagogical purposes as well as to ensure the reproducibility of the experimentsthe Matlab Tm code for producing most of the experiments in the first 11 chaptersand some in Chap. 13, is distributed on the Internet atwww.naturalimagestatistics.netThis web site will also include other related materialAcknowledgementsWe would like to thank Michael Gutmann, Asun Vicente, and Jussi Lindgren fordetailed comments on the manuscript. We have also greatly benefited from discussions with Bruno Olshausen, Eero Simoncelli, Geoffrey Hinton, David Field, PeterDayan, David Donoho, Pentti Laurinen, Jussi saarinen, Simo Vanni, and many oth-ers. We are also very grateful to Dario Ringach for providing the reverse correla-tion results in Fig. 3.7. During the writing process, the authors were funded by theUniversity of Helsinki (Department of Computer Science and Department of mathematics and Statistics), the Helsinki Institute for Information Technology, and theAcademy of finlandHelsinkiAapo hyvarinenJarmo hurriPatrik HoyerContents1 Introduction1.1 What this book is all about12 What IsⅤ ISion?1.3 The Magic of Your Visual System1. 4 Importance of Prior Information1. 4.1 Ecological Adaptation Provides Prior Information1.4.2 Generative Models and Latent Quantities1.4.3 Projection onto the Retina Loses Information23778991.4.4 Bayesian Inference and priors1.5 Natural images101.5.1 The Image space1.5.2 Definition of Natural Images1.6 Redundancy and Information131.6.1 Information Theory and Image Codin1.6.2 Redundancy Reduction and Neural Coding141.7 Statistical Modeling of the Visual System1.7.1 Connecting Information Theory and Bayesian Inference151.7.2 Normative vs Descriptive Modeling of visual system151.7. 3 Toward Predictive Theoretical Neuroscience161. 8 Features and Statistical Models of Natural Images171. 8.1 Image Representations and Features171. 8.2 Statistics of features181. 8. 3 From Features to Statistical Models191. 9 The Statistical-Ecological Approach Recapitulated201.10 ReferencesPartI Background2 Linear Filters and Frequency analysis22. 1 Linear Filtering252.1.1 Definition2.1.2 Impulse response and Convolution2.2 Frequency-Based Representation92.2.1 Motivation292.2.2 Representation in One and Two dimensions292.2.3 Frequency-Based Representation and linear Filtering342.2.4 Computation and Mathematical Details.372.3 Representation Using Linear Basis2.3.1 Basic Idea882.3.2 Frequency-Based Representation as a BasisContents2.4 Space-Frequency Analysis2.4.1 Introduction2.4.2 Space-Frequency Analysis and gabor Filters432.4.3 Spatial Localization vS Spectral Accuracy462.5 References482. 6 Exercises483 Outline of the visual System513.1 Neurons and firing rates513.2 From the eye to the cortex533.3 Linear models of visual neurons.543.3.1 Responses to Visual stimulation543.3.2 Simple Cells and Linear models563.3.3 Gabor Models and Selectivities of Simple Cells573.3.4 Frequency Channels583.4 Non-linear Models of Visual Neurons3.4.1 Non-linearities in Simple-Cell Responses593.4.2 Complex Cells and Energy models3.5 Interactions between Visual Neurons.623.6 Topographic Organization3.7 Processing after the Primary Visual Cortex643. 8 References.653.9 Exercises654 Multivariate Probability and Statistics4.1 Natural images patches as random vectors674. 2 Multivariate Probability Distributions684.2.1 Notation and motivation4.2.2 Probability Density Function4.3 Marginal and Joint probabilities704. 4 Conditional probabilities4.5 Independence754.6 Expectation and Covariance4.6.1 Expectation774.6.2 Variance and Covariance in One Dimension.784.6. 3 Covariance matrix784.6.4 Independence and covariances794.7 Bayesian Inference814.7.1 Motivating EXample814.7.2 Bayes Rule834.7.3 Non-informative priors834.7.4 Bayesian Inference as an Incremental learning process .. 844. 8 Parameter Estimation and likelihood.864.8.1 Models, Estimation, and Samples.864.8.2 Maximum Likelihood and maximum a posteriori4.8.3 Prior and Large Samples.89