Statistical Principles in Image Modeling - UCLA Statistics

Statistical Principles in Image ModelingYing Nian WU Jinhui LI Ziqiang LIU and Song-Chun ZHU

Department of StatisticsUniversity of California Los Angeles

Los Angeles CA 90024(ywustatuclaedu)

Images of natural scenes contain a rich variety of visual patterns To learn and recognize these patternsfrom natural images it is necessary to construct statistical models for these patterns In this review arti-cle we describe three statistical principles for modeling image patterns the sparse coding principle theminimax entropy principle and the meaningful alignment principle We explain these three principlesand their relationships in the context of modeling images as compositions of Gabor wavelets These threeprinciples correspond to three regimes of composition patterns of Gabor wavelets and these three regimesare connected by changes in scale or resolution

KEY WORDS Gabor wavelet Meaningful alignment Minimax entropy Scale Sparse coding

1 INTRODUCTION

11 Three Regimes of Image Patterns andScaling Connection

Images of natural scenes are characterized by a bewilderingrichness of visual patterns The goal of computer vision is tolearn and recognize these patterns from natural images To ac-complish this goal it is necessary to construct statistical mod-els for these patterns The parameters of these models can belearned from training image data The learned models then canbe used to interpret future image data to recognize the objectsand patterns Thus vision is essentially a statistical modelingand inference problem

But vision also proves to be a highly specialized modelingand inference problem A special property of vision that distin-guishes it from other recognition tasks such as speech recog-nition is that visual objects in natural scenes can appear at awide range of distance from the camera At different viewingdistances the objects project different sizes (sometimes evensmaller than a single pixel) on the resulting image and thus cre-ate different image patterns To illustrate this concept Figure 1displays three images of maple leaves at three different view-ing distances (with the scope of the camera remaining fixed)Image (a) is observed at a far viewing distance In this imagemaple leaves are so small and densely packed that the individ-ual leaf shapes cannot be recognized The leaves collectivelyproduce a foliage texture pattern Image (b) is observed at amedium distance The leaf sizes are larger than those in im-age (a) and their shapes can be individually recognized Im-age (c) is observed at a very close distance The overall shapeof a single maple leaf is larger than the scope of the cameraso that only an edge pattern is recognized within this imageThese three images have different statistical properties Image(a) is very random image (c) is very simple and image (b) isin between The three images are instances of three regimes ofimage patterns Image (a) is from the texture regime where noclear shapes exist Image (b) is from the object regime wherethe overall shapes of the objects can be recognized Image (c) isfrom the geometry regime which comprises lines regions cor-ners junctions and so on From Figure 1 it is clear that thesethree regimes of image patterns are connected by the variabilityof viewing distance This variability is a primary cause for the

richness of visual patterns The effect of varying viewing dis-tance can be equally achieved by varying the camera resolutionwith zooming-in and zooming-out operations

12 Multiscale Representations for Both Image Dataand Their Patterns

The variability of viewing distance requires that the visualsystem be able to recognize patterns at different scales or res-olutions For a fixed image it is natural to use this ability tosimultaneously analyze the image at multiple scales or resolu-tions

Figure 2(a) displays an image at multiple resolutions in apyramidal structure The original image is at the bottom layerOn top of that the image at each layer is a zoomed-out versionof the image at the layer beneath The zooming-out operationcan be accomplished by smoothing the current image by con-volving it with a Gaussian kernel of a certain standard devia-tion and then subsampling the pixels by a certain factor to makethe image smaller This pyramidal structure called a Gaussianpyramid (Burt and Adelson 1983) can have many layers Eachlayer of the Gaussian pyramid is said to be a low-pass version ofthe original image because only low-frequency content of theimage is preserved A companion structure is called a Laplacianpyramid The image at each layer of the Laplacian pyramid isthe difference between the images at two adjacent layers of theGaussian pyramid or more precisely the difference betweenthe image at the corresponding layer of the Gaussian pyramidand its smoothed version obtained by convolving it with theGaussian kernel Each layer of the Laplacian pyramid is saidto be a bandpass version of the original image because onlythe content within a certain frequency band is preserved TheLaplacian pyramid is a decomposition of the original imageinto different frequency bands (See Sec 22 for explanationsof low-pass and bandpass images)

If we move a window of a certain size (say 40 times 40) over theimages at multiple layers in the Gaussian or Laplacian pyramid

copy 2007 American Statistical Association andthe American Society for Quality

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249

250 YING NIAN WU ET AL

(a) (b) (c)

Figure 1 Examples from three regimes of image patterns (a) texture regime (b) object regime and (c) geometry regime The three regimesare connected by the change in viewing distance or equivalently the change in camera resolution

then we can recognize different image patterns within the win-dow and these patterns belong to different regimes describedin the previous section We recognize lines and regions at highresolution or small scale shapes and objects at medium reso-lution or scale and textures and clutters at low resolution orlarge scale In addition these patterns are highly organizedThe large-scale texture patterns are composed of medium-scaleshape patterns which in turn are composed of small-scale edgeand corner patterns Figure 2(b) provides an illustration wherethe image patches at different resolutions are labeled as variouspatterns such as foliage leaves and edges or corners The ver-tical and diagonal arrows indicate the compositional relation-ships and the horizontal arrows indicate the spatial arrange-ments Thus a complete interpretation of an image must con-sist of the labels of image patches at all of these layers It alsoshould include the compositional and spatial organizations ofthe labels

Figure 2 illustrates the multiscale structures of both the im-age data and the recognized patterns The multiscale structureof the image data has been extensively studied in wavelet the-ory (Mallat 1989 Simoncelli Freeman Adelson and Heeger1992) and scale-space theory (Witkin 1983 Lindberg 1994)

however the multiscale structure of the recognized patterns hasnot received as much treatment Recent attempts to understandthis issue have been made by Wu Zhu and Guo (2007) andWang Bahrami and Zhu (2005) The compositional relation-ships have been studied by Geman Potter and Chi (2002) TheANDndashOR grammars for the multilayer organizations of the im-age patterns have been studied by Zhu and Mumford (2007)

13 Bayesian Generative Modeling andPosterior Inference

An elegant approach to recognizing the multiscale patternsfrom the multiscale image data is to construct a Bayesian gen-erative model and use the posterior distribution to guide the in-ferential process (Grenander 1993 Grenander and Miller 1994Geman and Geman 1984 Mumford 1994) Given the multilayerstructures of both the image data and the labels of the recog-nized patterns as shown in Figure 2(b) a natural form of thismodel is a recursive coarse-to-fine generative process consist-ing of the following two schemes

(a) (b)

Figure 2 Multiscale representations of data and patterns (a) A Gaussian pyramid The image at each layer is a zoomed-out version of theimage at the layer beneath The Laplacian pyramid can be obtained by computing the differences between consecutive layers of the Gaussianpyramid (b) For the same image different patterns are recognized at different scales These patterns are organized in terms of compositionaland spatial relationships The vertical and diagonal arrows represent compositional relationships the horizontal arrows spatial relationships

TECHNOMETRICS AUGUST 2007 VOL 49 NO 3

STATISTICAL PRINCIPLES IN IMAGE MODELING 251

(a) (b) (c)

Figure 3 Gabor wavelets (a) A sample of Gabor wavelets at different locations scales and orientations (b) An example of a Gabor sinewavelet (c) An example of a Gabor cosine wavelet

Scheme 1 The labels generate their image data at the cor-responding scale or frequency band for example a leafpattern generates a leaf image patch at the same layer

Scheme 2 The labels generate those constituent labels at thelower scale for example a leaf pattern is decomposedinto edge and corner patterns at the lower layer

These two schemes are closely related and lead to a jointdistribution of the labels of the recognized patterns and thebandpass image patches over all of the scales Pr(labels imagepatches) In principle this model can be learned from trainingimages that have already been labeled A new image can beinterpreted by sampling from or maximizing the posterior dis-tribution Pr(labels|image patches)

In the posterior distribution Pr(labels|image patches) the la-bel of a particular image patch at a certain layer is determinednot only by this image patch itself but also by the labels ofother image patches For instance the leaf pattern in the centerof Figure 2(b) can be recognized by combining the followingfour sources of information

Source 1 The image patch of this leaf at the current layerSource 2 The bottom-up information from the edge and cor-

ner patterns of the image patches at the layers belowSource 3 The top-down information from the foliage pat-

terns of the image patches at the layers aboveSource 4 The contextual information from the leaf patterns

of the adjacent image patches at the same layer

Although source 1 is the most direct the other sources alsocan be important especially when source 1 contains too muchambiguity which can often be the case in natural images Thecombination of information can be accomplished by Bayesianposterior computation

A realistic generative model Pr(labels image patches)based on the two schemes mentioned earlier remains beyondour reach The posterior computation for propagating and com-bining information also is not well understood In this articlewe study statistical principles that may eventually lead to sucha generative model The focus of this article is on Scheme 1mdashthat is how the patterns generate their image data at the samelayer A recent article by Zhu and Mumford (2007) on the sto-chastic ANDndashOR grammars for Scheme 2 explores how thepatterns are decomposed into the constituent patterns at lowerlayers in the compositional hierarchy

14 Hint From Biological Vision

Given the fact that biological vision is far superior to com-puter vision in terms of learning and recognizing natural imagepatterns it is useful to take some clues from biological visionwhen constructing image models to be used for computer vi-sion In neuroscience neuron recording data (Hubel and Wiesel1962) indicate that visual cells in the primary visual cortex orV1 area (the part of the brain responsible for the initial stageof visual processing) respond to local image structures such asbars edges and gratings at different positions scales and ori-entations

Specifically the V1 cells are classified into simple cellsand complex cells The simple cells are modeled by Gaborwavelets (Daugman 1985) Figure 3(a) displays a sample ofGabor wavelets (Lee 1996) which are localized functions atdifferent locations with different orientations and scales Thesefunctions are in the form of sinusoidal waves multiplied byGaussian functions Figures 3(b) and 3(c) display the perspec-tive plots of two such functions A more detailed explanationis provided in Section 2 Responses of simple cells are mod-eled as projection coefficients of the image onto these func-tions These projection coefficients capture image structuresat different locations orientations and scales The complexcells are also sensitive to similar local image structures how-ever they are more tolerant or invariant of the shifting of loca-tions and scales of the image structures (Lampl Ferster Pog-gis and Riesenhuber 2004) Such nonlinear behavior can bemodeled by combining or pooling the outputs from the sim-ple cells (Adelson and Bergen 1985 Riesenhuber and Poggio1999)

The set of Gabor wavelets is self-similar and has a multiscalestructure that is consistent with the multiscale structures of theimage data described in the previous section Specifically pro-jecting an image onto a large-scale Gabor wavelet is equivalentto projecting a zoomed-out version of this image onto a smaller-scale Gabor wavelet at the same location and orientation TheseGabor wavelets can be the link between the image data and thecorresponding patterns Specifically these Gabor wavelets mayserve as the building elements in the generative model Pr(imagepatches|labels) as well as the intermediate step in computingPr(labels|image patches)



15 Three Statistical Principles for ComposingGabor Wavelets

In this article we examine three existing statistical principlesfor image modeling These principles shed light on the roles ofGabor wavelets in pattern recognition and provide useful guid-ance for constructing image models as compositions or pool-ings of Gabor wavelets

151 Sparse Coding Principle Olshausen and Field(1996) proposed that sparse coding is a strategy used by theprimary visual cortex to represent image data This principleseeks to find a dictionary of linear basis elements so that anytypical natural image can be expressed as a linear combinationof a small number of basis elements chosen from the dictionaryThe basis elements learned by Olshausen and Field from naturalimages exhibit close resemblance to the Gabor wavelets

152 Minimax Entropy Principle Zhu Wu and Mum-ford (1997) proposed the minimax entropy principle for mod-eling textures This principle seeks to find statistical summariesof the observed image so that the maximum entropy model con-strained by these statistical summaries has the minimum en-tropy The particular statistical summaries adopted by Zhu et alare marginal histograms of Gabor wavelet coefficients The re-sulting models are in the form of Markov random fields (Besag1974 Geman and Graffigne 1987) Realistic texture patternscan be generated by sampling from such random fields

153 Meaningful Alignment Principle Moisan Desol-neux and Morel (2000) proposed the meaningful alignmentprinciple for perceptual grouping This principle seeks to iden-tify coincidence patterns in the observed image data that other-wise would be extremely rare in a completely random imageSuch coincidences are said to be meaningful One particularexample of this is the detection of line segments in image databased on the alignment of local orientations along the potentialline segments The local orientations can be computed as theorientations of the best-tuned Gabor wavelets

These three principles correspond to the three regimes of im-age patterns discussed in Section 11 The sparse coding prin-ciple corresponds to the object regime where the image can bemodeled based on the composition of a small number of waveletelements at different positions scales and orientations Theminimax entropy principle corresponds to the texture regimewhere the image is modeled by pooling the wavelet coeffi-cients into marginal histograms while discarding the positioninformation The meaningful alignment principle correspondsto the geometry regime where the best-tuned Gabor waveletsare highly aligned in both the spatial and frequency domains

Although these three principles correspond to three differ-ent regimes of patterns with different complexities and scalesthey all seek to identify a small number of constraints on theGabor wavelet coefficients to restrict the observed image to animage ensemble of small volume As a result these constraintsconstitute a simple and informative description of the observedimage The uniform distribution on the constrained image en-semble can be linked to an unconstrained statistical model Thispoint of view suggests that it is possible to develop a unifiedstatistical model for all these three regimes of image patterns asthe composition or pooling of Gabor wavelet coefficients

The rest of the article is organized as follows Section 2 re-views Gabor wavelets Section 3 reviews the three statisticalprinciples and Section 4 investigates the relationships betweenthese principles Section 5 concludes with a discussion

2 GABOR WAVELETS EDGE AND SPECTRUM

Gabor wavelets are localized in both spatial and frequencydomains and can detect edges and bars in the spatial domainand extract spectra in the frequency domain In this section wereview Gabor wavelets first in the spatial domain and then inthe frequency domain For simplicity we assume that both theimage and the Gabor wavelets are functions defined on the two-dimensional real domain R

2

21 Edge-Bar Detector

A Gabor wavelet (Daugman 1985) is a sinusoidal wave mul-tiplied by a Gaussian density function The following functionis an example

G(x) = 1

2πσ1σ2exp

minus1

2

(x2

1

σ 21

+ x22

σ 22

)eix1 (1)

where x = (x1 x2) isin R2 and i = radicminus1 This complex-valued

function G(x) consists of two parts The first part is an elongateGaussian function with σ1 lt σ2 so the shorter axis is alongthe x1 direction and the Gaussian function is oriented along thex2 direction The second part is a pair of sine and cosine planewaves propagating along the x1 axismdashthat is the shorter axis ofthe Gaussian function Because of the Gaussian function G(x)is localized in the spatial domain The Gabor cosine componentor the real part of G(x) is an even-symmetric function and theGabor sine component or the imaginary part of G(x) is an odd-symmetric function (see Figs 3(b) and 3(c) for illustrations)

We can translate rotate and dilate the function G(x) of (1)to obtain a general form of Gabor wavelets such as those plot-ted in Figure 3(a) Gysθ (x) = G(xs)s2 where x = (x1 x2)x1 = (x1 minus y1) cos θ minus (x2 minus y2) sin θ and x2 = (x1 minus y1) sin θ +(x2 minus y2) cos θ The function Gysθ is centered at y with ori-entation θ The standard deviations of the two-dimensionalGaussian function in Gysθ (x) along its shorter and longer axesare sσ1 and sσ2 The sine and cosine waves propagate at thefrequency 1s along the shorter axis of the Gaussian function

For an image I(x) the projection coefficient of I onto Gysθ

is

rysθ = 〈IGysθ 〉 =int

I(x)Gysθ (x)dx (2)

where the integral is over R2 In engineering literature the Ga-

bor wavelets are also called Gabor filters rysθ obtained by (2)is called the filter response and the image J(y) = rysθ is calledthe filtered image The Gabor sine component or the imagi-nary part of Gysθ is odd-symmetric and sensitive to step-edgestructures If the image I has a step-edge structure at locationy with orientation θ then the imaginary part of rysθ will beof large magnitude The Gabor cosine component of Gysθ iseven-symmetric and sensitive to bar structures If the image Ihas a bar structure at location y with orientation θ then the realpart of rysθ will be of large magnitude

The Gabor wavelets can be used as edge-bar detectors (Wangand Jenkin 1992 Perona and Malik 1990 Mallat and Zhong1992 Canny 1986) Let rxsθ = axsθ + ibxsθ Here |rxsθ |2 =a2

xsθ + b2xsθ is the energy extracted by the Gabor wavelet

Gxsθ (Adelson and Bergen 1985) For fixed x and s let θlowast =TECHNOMETRICS AUGUST 2007 VOL 49 NO 3


(a) (b) (c)

Figure 4 The observed image (a) and edge maps at two different scales [(b) and (c)] The scale of (b) is finer than that of (c)

arg maxθ |rxsθ |2 We call Gxsθlowast the best-tuned Gabor waveletat (x s) The local orientation is defined as θ(x s) = θlowast Thelocal phase is defined as φ(x s) = arctan(bxsθlowastaxsθlowast) Foredge structure the local phase is π2 or minusπ2 for bar struc-ture the local phase is 0 or π The local energy is defined asA(x s) = |rxsθlowast |2 Here x is an edge-bar point at scale s ifA(x s) ge A(y s) for all y such that the direction of y minus x is per-pendicular to θ(x s) and |y minus x| lt d for predefined range dthat is A(x s) is a local maximum along the normal directionof x and we call Gxsθ(xs) a locally best-tuned Gabor wavelet atscale s Figures 4(b) and 4(c) display the edge-bar points of theobserved image in Figure 4(a) at two different scales where theintensity of an edge-bar point (x s) is proportional to A(x s)12The scale s for computing the edge map is smaller in Figure 4(b)than in Figure 4(c)

22 Spectral Analyzer

The Gabor wavelets also extract power spectra For an imageI(x) which is a function in L2 this can be represented as thesuperposition of sinusoidal waves

I(x) = 1

4π2

intI(ω)eiωx dω (3)

where x = (x1 x2) isin R2 ω = (ω1ω2) isin R

2 ωx = x1ω1 +x2ω2 and the integral is over the two-dimensional frequencydomain R

2 For fixed ω the plane wave eiωx in (3) propa-gates in the spatial domain R

2 along the orientation of the vec-tor ω = (ω1ω2) at frequency |ω| = (ω2

1 + ω22)

12 The coef-

ficient I(ω) in (3) can be obtained by the Fourier transformI(ω) = int

I(x)eminusiωx dxIdeally we can extract certain frequency content of I by cal-

culating 4π2J = intF I(ω)eiωx dω for a certain set F sub R

2 of fre-quencies If F contains only those frequencies ω such that |ω|are below a threshold then J is called a low-pass image If Fcontains only those frequencies ω such that |ω| are within a cer-tain finite interval then J is called a bandpass image In prac-tice such images can be approximately extracted by convolv-ing I with a kernel function g The convoluted image is J(x) =int

I(y)g(y minus x)dy In the frequency domain J(ω) = I(ω)g(ω)thus 4π2J(x) = int

I(ω)g(ω)eiωx dω One may consider g(ω) ageneralized version of the indicator function for F and designg(ω) to be a function localized in the frequency domain to ex-tract the corresponding frequency content of I

The Gabor wavelets can serve this purpose Let Gsθ (x) =G0sθ (x) [See the previous section for the definition of Gysθ here y = (00)] The Fourier transform of Gsθ is

Gsθ (ω) = exp

minus1

2

[(sσ1

(ω1 minus 1

s

))2

+ (sσ2ω2)2]

(4)

where ω = (ω1 ω2) with ω1 = ω1 cos θ + ω2 sin θ and ω2 =minusω1 sin θ + ω2 cos θ Gs0 is a Gaussian function centered at(1s0) with standard deviations 1(sσ1) and 1(sσ2) along ω1and ω2 and Gsθ is a Gaussian function obtained by rotatingGs0 by an angle minusθ and Gsθ is localized in the frequency do-main Figure 5 shows an example of Gsθ in the frequency do-main for a sample of (s θ) corresponding to the sample of Ga-bor wavelets displayed in Figure 3(a) Each Gsθ is illustratedby an ellipsoid whose two axes are proportional to 1(sσ1) and1(sσ2) One may intuitively imagine each ellipsoid as the fre-quency band covered by Gsθ and together these Gsθ can paveor tile the entire frequency domain

Let J = I lowast Gsθ J(x) = 〈I(y)Gxsθ (minusy)〉 can be ex-pressed as the projection of I onto the Gabor wavelet cen-tered at x We have that 4π2J(x) = int

J(ω)eiωx dω whereJ(ω) = I(ω)Gsθ (ω) Thus J extracts the frequency contentof I within the frequency band covered by Gsθ Accordingto the Parseval identity 4π2J2 = 4π2

int |J(x)|2 dx = J2 =int |I(ω)|2|Gsθ (ω)|2 dω Thus J2 extracts the local average ofthe power spectrum of I within the frequency band covered byGsθ Section 32 provides more discussion on this issue

Figure 5 A sample of Gsθ in the frequency domain Each Gsθis illustrated by an ellipsoid whose two axes are proportional to thecorresponding standard deviations of Gsθ



In this section we explain that Gabor wavelets can be usedto extract edge-bar points as well as spectra However how touse Gabor wavelets to represent and recognize image patternsremains unclear In the next section we review three statisticalprinciples that shed light on this issue

3 THREE STATISTICAL PRINCIPLES

31 Sparse Coding Principle Beyond Edge Detection

The sparse coding principle can be used to justify the Ga-bor wavelets as linear basis elements for coding natural imagesSpecifically let Bn(x)n = 1 N be a set of linear basisfunctions Then we can code I by

I(x) =Nsum

n=1

cnBn(x) + ε(x) (5)

where cn are the coefficients and ε is the residual image Thedictionary of the basis functions Bn in the representation (5)should be designed or learned so that for any typical natural im-age I only a small number of coefficients cn need to be nonzeroto code I with small residual error The dictionary of the basiselements Bn can be overcomplete that is the number of basiselements N can be greater than the number of pixels in I

Using this principle Olshausen and Field (1996) learned adictionary of basis elements from a random sample of naturalimage patches Specifically they collected a large number of12 times 12 image patches Imm = 1 M from pictures ofnatural scenes and learned Bnn = 1 N (with N gt 144)by minimizing the objective function

Msumm=1

[∥∥∥∥∥Im minusNsum

n=1

cmnBn

∥∥∥∥∥2

+ λ

Nsumn=1

S(cmn)

](6)

over both Bn and cmn Here λ is a tuning parameter andsumn S(cmn) measures the sparsity of the coefficients cmnn =

1 N The most natural sparsity measure is the numberof nonzero elements in cmn that is S(c) = 0 if c = 0 andS(c) = 1 if c = 0 This is the l0-norm of the sequence cmnn =1 N For computational convenience one may choose l1-norm (Chen Donoho and Saunders 1999) or other functionsto approximate the l0-norm so that gradient-based algorithmscan be used for minimization The basis elements learned byOlshausen and Field (1996) resemble the Gabor wavelets thatis for each learned Bn we can approximate it by the sine orcosine component of a Gabor wavelet Gxsθ for some (x s θ)

with small approximation errorSuppose that the dictionary of basis functions is already

given [eg a dictionary of Gabor wavelets Gxsθ ] then com-puting the sparse coding of an image I amounts to selecting asmall number of basis functions from the dictionary so that theleast squares regression of I on these selected basis functionshas a very small residual error This is the variable selectionproblem in linear regression

A statistical formulation is the generative model (Lewickiand Olshausen 1999)

I =Nsum

n=1

cnBn + ε C = (cnn = 1 N) sim p(C) (7)

where p(C) is often assumed to have independent componentsand each cn is assumed to follow a mixture of a normal distrib-ution and a point mass at 0 (Pece 2002 Olshausen and Millman2000) This is the Bayesian variable selection model in linearregression (George and McCulloch 1997)

An efficient algorithm for selecting the basis elements tocode a given image is the matching-pursuit algorithm (Mal-lat and Zhang 1993) which is a stepwise forward-regressionalgorithm Assume that all of the basis elements are normal-ized to have unit L2 norm The algorithm starts from the emptyset of basis elements At the kth step let B1 Bk bethe set of elements selected with coefficients c1 ck Letε = I minus (c1B1 + middot middot middot + ckBk) Then we choose an element Bk+1

from the dictionary so that the inner product 〈εBk+1〉 is maxi-mum among all of the elements in the dictionary Then we addBk+1 to the current set of elements and record its coefficientck+1 = 〈εBk+1〉 We repeat this process until ε2 is less thana prefixed threshold Chen et al (1999) have provided a moresophisticated computational method

Figure 6 illustrates the sparse coding representation usingthe matching-pursuit algorithm Figure 6(a) is an observed185 times 250 image Figure 6(b) is the image reconstructed byc1B1 + middot middot middot + cKBK using the matching-pursuit algorithm HereK = 343 and the 343 elements are selected from a dictio-nary similar to those Gabor wavelets illustrated in Figure 3(Young 1987 Wu Zhu and Guo 2002) Thus there are 185 times250343 asymp 135 folds of dimension reduction in this sparse cod-ing Part (c) displays a symbolic representation of the selected343 basis elements with each element represented by a bar ofthe same location orientation and length The intensity of thebar is proportional to the magnitude of the corresponding coef-ficient

From Figure 6(c) it is evident that the selected basis ele-ments target the edge-bar structures in the image Recall that in

(a) (b)

(c) (d)

Figure 6 The sparse coding representation using the match-ing-pursuit algorithm (a) An observed 185 times 250 image (b) The im-age reconstructed by 343 linear basis elements selected by the match-ing-pursuit algorithm (c) A symbolic sketch in which each selectedbasis element is represented by a bar of the same location orienta-tion and length (d) An edge-bar map created by the edge-bar detectordescribed in Section 21 There are 4099 edge-bar points



Section 21 the edge-bar points are detected by the locally best-tuned Gabor wavelets One may ask whether the sparse codingprinciple offers anything more than edge detection The answerlies in sparsity Figure 6(d) displays the map of edge-bar pointsdetected at a small scale This edge map may be more visuallypleasing than Figure 6(c) however there are 4099 edge pointsfar more than the number of elements in sparse coding (343)To represent a local shape in Figure 6(a) a few basis elementsare sufficient but a large number of edge points is requiredfor this task From a modeling perspective it is much easierto model a low-dimensional structure than a high-dimensionalstructure Such models have been studied by Wu et al (2002)Guo Zhu and Wu (2003ab) Wang and Zhu (2004) and ZhuGuo Wang and Xu (2005) We discuss object modeling in Sec-tion 41 (See also Candes and Donoho 1999 and Huo and Chen2005 for curvelet and beamlet systems for sparse coding)

32 Minimax Entropy Principle BeyondSpectral Analysis

The minimax entropy principle was adopted by Zhu et al(1997) for modeling textures For convenience we assume inthis section that I is a stationary texture image defined on a fi-nite lattice D with |D| pixels for instance D = 1 M times1 N so |D| = M times N A popular choice of texture statis-tics is the marginal histograms of wavelet coefficients (Heegerand Bergen 1995) Let Gk k = 1 K be a set of kernelfunctionsmdashfor example the Gabor sine and cosine functionsGsθ defined in Section 22 Let Hk be the marginal histogramof the convoluted image IlowastGk Specifically we divide the rangeof [I lowast Gk](x) into T bins 1 T so that

Hkt(I) =sum

x

δ([I lowast Gk](x) isin t

) t = 1 T (8)

where δ(middot) is the indicator function Let hkt(I) = Hkt(I)|D| bethe normalized marginal histogram of I lowastGk For simplicity wewrite Hk = (Hkt t = 1 T) and hk = (hkt t = 1 T)

Wu Zhu and Liu (2000) defined the following image ensem-ble = I hk(I) = hk(Iobs) k = 1 K If the set of mar-ginal histograms hk captures the texture information in Iobsthen all the images in the ensemble should share the sametexture pattern Thus we can model the image Iobs as a randomsample from the uniform distribution over Figure 7 shows anexample of this Figure 7(a) is the observed image Iobs whereasFigure 7(b) is an image randomly sampled from where Gk

(a) (b)

Figure 7 Observed image Iobs (a) and a random image sampledfrom the image ensemble = I hk(I) = hk(Iobs) k = 1 K (b)

is a set of Gabor kernel functions Gsθ Although the sampledimage in Figure 7(b) is a different image than the observed onein Figure 7(a) the two images share identical texture patternsjudged by human visual perception (see Zhu Liu and Wu 2000for details)

In terms of the image ensemble the meaning of the mini-max entropy principle can be stated as follows

1 For a fixed set of kernel functions Gk the maximumentropy model is the uniform distribution over

2 To select the set of kernel functions Gk k = 1 K(with prespecified K) from a dictionary (eg a dictionaryof Gabor functions) we want to select the set Gk k =1 K to minimize the volume of (ie ||)

For the uniform distribution over its entropy is log ||which is also the negative log-likelihood By minimizing thevolume of we are minimizing the entropy of the correspond-ing uniform distribution or maximizing its log-likelihood

Although intuitively simple the constrained image ensemble or its entropy log || is computationally difficult to handleAs pointed out by Wu et al (2000) if the image size |D| is largethen the uniform distribution over the constrained ensemble

can be approximated by an unconstrained statistical model

p(I) = 1

Z()exp

Ksum

k=1

〈λkHk(I)〉

(9)

where = (λk k = 1 K) λk = (λkt t = 1 T) andZ() is the normalizing constant to make p(I) integrate to1 Here can be estimated by maximizing the log-likelihoodlog p(Iobs) which amounts to solving the estimating equa-tion E[Hk(I)] = Hk(Iobs) for k = 1 K The entropylog || can be approximated by the negative log-likelihood ofthe fitted model log p(Iobs) Thus the minimax entropy prob-lem is equivalent to selecting K kernel functions Gk and com-puting the corresponding λk to maximize the log-likelihoodlog p(Iobs)

An intuitive explanation for the approximation to the uni-form distribution over by a statistical model (9) is that un-der p(I) hkt(I) = Hkt(I)|D| = sum

x δ([IlowastGk](x) isin t)|D| rarrPr([I lowast Gk](x) isin t) as |D| rarr infin if ergodicity holds Here er-godicity means that the spatial averages go to the correspondingfixed expectations that is the random fluctuations in the spa-tial averages diminish in the limit Because p(I) assigns equalprobabilities to images with fixed values of hk(I) p(I) goes toa uniform distribution over an ensemble of images I with fixedhk(I)

The model (9) generalizes early versions of Markov random-field models (Besag 1974 Geman and Graffigne 1987) An-other way to express (9) is p(I) prop prod

kx fk([I lowast Gk](x)) wherefk is a step function over the bins of the histogram Hk such thatfk(r) prop expλkt if r isin t This is similar to the form of projec-tion pursuit density estimation (Friedman 1987) except that foreach fk there is the product over all pixels x

Zhu et al (1997) proposed a filter pursuit procedure to addone filter or kernel function Gk at a time to approximately max-imize the log-likelihood log p(Iobs) or minimize the entropylog || Figure 8 displays an example of filter pursuit proce-dure (a) is the observed image and (b)ndash(e) are images sampled



(a) observed (b) k = 0 (c) k = 2 (d) k = 3 (e) k = 7

Figure 8 Filter pursuit Adding one filter at a time to reduce the entropy (a) The observed image (b)ndash(e) Images sampled from the fittedmodels

from the fitted models p(I) corresponding to an increasingsequence of the filter set G1 Gk for k = 0 7 Specifi-cally for each k after fitting the model p(I) to the observedimage for filters G1 Gk we sample an image from the fit-ted model p(I) With k = 0 filter the sampled image is whitenoise With k = 7 filters the sampled image in (e) is perceptu-ally equivalent to the input image Della Pietra Della Pietraand Lafferty (1997) described earlier work on introducing fea-tures into the exponential models

One may ask what is in the marginal histograms defined in(8) that is beyond the spectrum information extracted by theGabor kernel functions Without loss of generality let J = IlowastGwhere G is the Gabor function defined in Section 21 Supposethat we normalize the sine and cosine parts of G so that bothhave mean 0 and unit L2 norm Let us consider the marginaldistribution of |J(x)|2 under the following three hypotheses

Hypothesis 0 I(x) is a white noise process that is I(x) simN(0 σ 2) independently

Hypothesis 1 I(x) is a plane wave I(x) = cos(x1 + φ) prop-agating along the x1 direction at the unit frequency whichis the frequency of the wave component of G

Hypothesis 2 I(x) is a step-edge function I(x) = δ(x1 gt 0)where δ is the indicator function

Under hypothesis 0 the real and imaginary parts of J(x) fol-low independent normal distribution N(0 σ 2) so |J(x)|2 fol-lows σ 2χ2

2 or the exponential distribution with E[|J(x)|2] = σ 2

and var[|J(x)|2] = σ 4 This conclusion still holds as long as I(x)is a ldquolocally whiterdquo stationary Gaussian process in the sensethat the spectrum of the process is a constant within the fre-quency band covered by G This is because J does not con-tain frequency content of I outside this frequency band There-fore for the marginal distribution of |J(x)|2 E[|J(x)|2] cap-tures the average spectrum over the frequency band of G andvar[|J(x)|2] = E2[|J(x)|2] if the process is locally white

Under hypothesis 1 I(x) = [eiφeix1 + eminusiφeminusix1]2 that is ithas two frequency components at (10) and (minus10) G(ω) iscentered at (10) with standard deviations 1σ1 and 1σ2 As-sume that 1σ1 is sufficiently small so that G does not cover(minus10) Then J(x) = G(10)eiφeix1 Thus |J(x)|2 = G(10)2which is a constant and var[|J(x)|2] = 0 lt E2[|J(x)|2] There-fore a small marginal variance of J(x) indicates the possiblepresence of a periodic component within the frequency bandcovered by G

Under hypothesis 2 it is clear that |J(x)|2 gt 0 only if |x1|is small compared with σ1 and |J(x)|2 achieves its maximum

at the edge point x1 = 0 If |x1σ1| is large so that the corre-sponding Gabor wavelet is away from the edge then J(x) = 0The marginal histogram of |J(x)|2 has a high probability to be 0and a low probability to be significantly greater than 0 For sucha distribution var[|J(x)|2] gt E2[|J(x)|2] for a random point x

We may consider hypothesis 0 a null hypothesis of locallywhite Gaussian process Hypothesis 1 is an alternative periodichypothesis and hypothesis 2 is an alternative edge hypothe-sis The two alternative hypotheses are extremes of two direc-tions of departure from the null hypothesis Dunn and Higgins(1995) derived a class of marginal distributions for image pat-terns along the direction toward hypothesis 1 and SrivastavaGrenander and Liu (2002) derived a class of marginal distrib-utions for image patterns along the direction toward hypothesis2 Biological vision recognizes patterns along both directionsThere are V1 cells that are sensitive to edge and bar patterns(Hubel and Wiesel 1962) There is also a small portion of V1cells that respond to periodic grating patterns but not to edgepatterns (Petkov and Kruizinga 1997) Both types of cells canbe modeled by combining Gabor wavelets

For a set of Gsθ that paves the frequency domain (as illus-trated in Fig 5) let Jsθ = I lowast Gsθ Then the set of marginalaverages E[|Jsθ (x)|2] captures the spectrum of I over the fre-quency bands covered by these Gsθ If we constrain the setof E[|Jsθ (x)|2] then the corresponding image ensemble is ap-proximately a Gaussian process with a smooth spectrum If wefurther constrain var[|Jsθ (x)|2] then we add new informa-tion about the two directions of departures from the smooth-spectrum Gaussian process Spectrum and non-Gaussian statis-tics of natural images also have been studied by Ruderman andBialek (1994) Simoncelli and Olshausen (2001) and Srivas-tava Lee Simoncelli and Zhu (2003) Portilla and Simoncelli(2002) described the use of joint statistics for texture modeling

For an image patch the issue of whether to summarize it bymarginal histograms or to represent it by sparse coding arisesFigure 9 displays an example of this situation Here Figure 9(a)is the observed 256 times 256 image Figure 9(b) is a random im-age from the ensemble constrained by the marginal histogramsof Gabor wavelets and Figure 9(c) is the image reconstructedby the sparse coding model using 1265 wavelet elements Al-though Figure 9(b) captures some edge information the mar-ginal histograms are implicit in the sense that by pooling themarginal histograms we discard all of the position informationThe sparse coding model captures the local edges at differentpositions explicitly and preserves the local shapes It would beinteresting to study the transition between the implicit marginal



(a) (b) (c)

Figure 9 (a) The observed 256 times 256 image (b) A random sample from the image ensemble constrained by the marginal histograms ofGabor wavelets (c) Reconstructed image by the sparse coding model using 1265 wavelet elements

histograms and the explicit sparse coding Guo et al (2003ab2007) have attempted to explain this issue

33 Meaningful Alignment Principle BeyondMarginal Properties

The sparsity and marginal histograms studied in the previoustwo sections are marginal properties of the wavelet coefficientsThey do not capture the joint properties of wavelet coefficientsThe meaningful alignment principle studied by Moisan et al(2000) is intended to capture these joint properties

One particular type of joint property studied by Moisan etal (2000) is the alignment of local orientations on line seg-ments in natural images For a pixel x its orientation θ(x s)at scale s is defined in Section 21 The alignment event is de-fined in terms of the coincidence of the orientations of a setof pixels Specifically consider any two pixels a and b andlet θ(ab) be the orientation of the line segment that connectsa and b We can sample a number of equally spaced pixelsa = x1 x2 xl = b on this line segment (ab) and computetheir orientations θ1 θ2 θl Here θi is said to be alignedwith θ(ab) if |θi minus θ(ab)| le α where α is a prespecifiedthreshold We then compute S = suml

i=1 δ(|θi minus θ | le α) that isthe number of sampled pixels whose orientations are alignedwith θ(ab)

Then we want to determine whether S is sufficiently largefor (ab) to be a meaningful line segment For this purposewe may choose a threshold k(l) = mink Pr(S ge k) le ε|D|2where Pr(A) is the probability of an event A under the hypoth-esis that the image is white noise and |D| is the total num-ber of pixels so that |D|2 bounds the total number of poten-tial line segments (ab) If S ge k(l) then (ab) is said to bean ε-meaningful line segment In a completely random imagethe expected number of falsely declared ε-meaningful line seg-ments is less that ε For computational convenience we cansample the equally spaced pixels a = x1 x2 xl = b so thatin the random image θ1 θ2 θl are independent Then S fol-lows a binomial distribution and relevant probabilities can becalculated or approximated in closed form

Generally speaking an ε-meaningful event A is such that ina white noise image the expected number of such events isless than ε This means that Pr(A) is small under the complete-randomness hypothesis The meaningful alignment principle isclosely related to the hypothesis testing (or more preciselymultiple hypothesis testing) problem in statistics

4 CONNECTIONS AMONG THREESTATISTICAL PRINCIPLES

The three statistical principles reviewed in the previous sec-tion can be connected by a common perspective where the goalis to identify a small number of most powerful constraints to re-strict the observed image into an image ensemble of a minimumvolume Also see the recent article by Wu et al (2007) whichunifies the three statistical principles in a theoretical frameworkcombining hypothesis testing and statistical modeling

41 Pooling Projections

First we consider the sparse coding principle Given the dic-tionary of basis elements this principle seeks to identify a smallnumber of basis elements B1 BK from the dictionary tocode the observed image I with small error Geometricallythis means that I projects most of its squared norm onto thesubspace spanned by B1 BK Thus we can view sparsecoding as constraining the projection of I onto the subspacespanned by B1 BK

Specifically for any image I defined on a finite grid D of|D| pixels we may vectorize I to make it a |D|-dimensionalvector Similarly we also can vectorize B1 BK into |D|-dimensional vectors Let rk = 〈IBk〉 and let R = (r1 rK)primebe the K-dimensional vector of projection coefficients LetB = (B1 BK) be the (|D| times K)-dimensional matrix whosekth column is Bk Then R = BprimeI Let H = B(BprimeB)minus1Bprime be theprojection matrix Then HI = B(BprimeB)minus1R is the projection ofI onto the subspace spanned by B1 BK and HI2 =Rprime(BprimeB)minus1R

Let B be an (|D| times (|D| minus K))-dimensional matrix whosecolumns are orthonormal vectors that are also orthogonal toall of the Bkrsquos Let R = BprimeI Then R2 = I2 minus HI2 =I2 minus Rprime(BprimeB)minus1R

Consider the image ensemble

= I I2 = |D|σ 2 = Iobs2BprimeI = R = BprimeIobs

(10)

This ensemble is a sphere viewed in the (|D| minus K)-dimensionalspace spanned by B

= R R2 = |D|σ 2 minus Rprime(BprimeB)minus1R (11)

Let

γ 2 = |D|σ 2 minus Rprime(BprimeB)minus1R

|D| minus K (12)



Then using the formula for computing the volume of high-dimensional sphere and Stirlingrsquos formula as |D| rarr infin

1

|D| minus Klog || rarr log(

radic2πeγ ) (13)

where || is the volume of Thus to find the K basis functions B to constrain Iobs within

a minimum-volume we should minimize the radius of thesphere or equivalently the least squares error Iobs2 minusHIobs2 This is clearly the goal of the sparse coding modelpresented in Section 31

Now we can connect the sparse coding model and the texturemodel In the sparse coding model we restrict a small numberof projection coefficients as well as the marginal variance Inthe texture model we restrict the marginal histograms of theprojection coefficients pooled over positions Both models seekto restrict the observed image into an image ensemble of smallvolume The two models differ in their respective constraints Inthe sparse coding model the constraints are on individual pro-jection coefficients of Gabor wavelets of particular positionsscales and orientations In the texture models the constraintsare on the marginal histograms of projection coefficients of par-ticular scales and orientations but pooled over positions that isthe position information is discarded

Just as the constrained texture ensemble can be approxi-mated by an unconstrained Markov random field the con-strained sphere also can be approximated by an unconstrainedstatistical model which is nothing but a white noise modelR sim N(0 γ 2I|D|minusK) where I|D|minusK is the (|D|minus K)-dimensionalidentity matrix This is because under the white noise modelR2(|D| minus K) rarr γ 2 as |D| rarr infin according to the law oflarge numbers and the white noise model assigns equal proba-bilities to all of the Rrsquos with fixed R2 The log-volume of thesphere in (13) is the same as the negative log-likelihood ofthe white noise model

This leads to a statistical model for image patterns inthe object regime where we can select the basis elementsB1 BK and estimate the joint distribution f (r1 rK) ofthe projection coefficients by learning from a sample of train-ing image patches of the same class of objects for example asample of 40 times 40 image patches of cars where the scales andlocations of the objects in these image patches are roughly fixedand the objects are roughly aligned

Specifically consider the change of variable(RR

)= (B B)primeI (14)

We can model (R R) by f (R R) = f (R)f (R|R) where f (R|R) simN(0 γ 2I|D|minusK) with γ 2 computed according to (12) Thenthe probability distribution of I is p(I) = f (R R)det(BprimeB)12where det(BprimeB)12 is the Jocobian of the change of variable(14) The log-likelihood for an image patch I is

log p(I) = log[f (R R)det(BprimeB)12]

= log f (R) minus 1

2(|D| minus K) log(2πeγ 2)

+ 1

2log det(BprimeB) (15)

Figure 10 Difference between the object model and the texturemodel In the object model projection coefficients of certain basis el-ements (subject to local shifting) are pooled over multiple training im-ages In the texture model projection coefficients of basis elements arepooled over positions in a single image

The goal is to select B = (B1 BK) from the dictionaryand estimate f (R) from the training images by maximizing thelog-likelihood (15) over independent training images In otherwords for each class of objects we want to model it as a com-position of a small number of basis elements We may allowthese basis elements to change their positions scales and ori-entations slightly to account for local deformations Toleranceof such deformations has been observed in the complex cells inV1 (Lampl et al 2004 Riesenhuber and Poggio 1999)

The foregoing model is different from the more commonlyassumed form in sparse coding I = BC + ε C sim p(C) ε simN(0 I|D|) (Lewicki and Olshausen 1999 Pece 2002 Olshausenand Millman 2000 Zhu et al 2005) which models the recon-struction coefficients C instead of the projection coefficientsR Modeling C is technically difficult because C is a latentvariable that needs to be inferred and p(I) must be obtainedby integrating out C In contrast R can be obtained deter-ministically through R = BprimeI and the distribution of p(I) canbe obtained explicitly by the change of variable The modelbears some similarity to independent component analysis (Belland Sejnowski 1997) and projection pursuit density estimation(Friedman 1987) but has a different mathematical form and hasthe capability of modeling the composition and deformation ofthe basis elements

We conclude this section by comparing the foregoing objectmodel with the texture model In the object model we pool thedistribution of projections coefficients f (R) for a number of ba-sis elements at certain positions (subject to local shifting) overmultiple training image patches In the texture model we poolthe marginal distributions of projection coefficients over the po-sitions of a single texture image The difference is illustrated inFigure 10

42 Identifying Alignments

In this section we consider the meaningful alignment princi-ple which also can be understood as searching for constraintsthat restrict the observed image into an image ensemble of smallvolume

Let 0 be the set of images defined on a grid D with fixedmarginal variance 0 = I I2 = |D|σ 2 Let A1 AK be



Figure 11 In the object regime the basis elements (illustrated assmall bars) around the vertical arrow are aligned (subject to local shift-ing) across multiple images In the geometry regime those basis ele-ments that are away from the vertical arrow are aligned within a singleimage

K statements about the alignment events in the observed imageIobs isin 0 Mathematically each Ak is a subset of 0 that con-tains all of the images satisfying the statement Ak To find infor-mative A1 AK we want the volume of = A1 cap middot middot middot cap AK

to be as small as possible Under the uniform distribution over0 the probability is p() = |||0| For a large image lat-tice the uniform distribution over is equivalent to a Gaussianwhite noise model thus p() is also the probability of underthe white noise hypothesis Because || = p()|0| searchingfor alignment events that are very rare under the white noisehypothesis is equivalent to searching for alignment events toconstrain the observed image into an image ensemble of smallvolume

The alignment patterns can be found in the geometry regimewhere the orientations of locally best-tuned Gabor wavelets arehighly aligned The probability for such alignment is small un-der the white noise hypothesis

The alignments also can be found in the object regime exceptthat alignments of the locally best-tuned Gabor wavelets are ob-served across multiple training image patches of the same classof objects instead of in alignment within a single image Fig-ure 11 illustrates the difference The basis elements (shown assmall bars) around the vertical arrow can be aligned across mul-tiple images but within a single image such as the one at thebottom those basis elements near the vertical arrow do not havesignificant alignment and thus the pattern cannot be detectedby meaningful alignment without the knowledge learned frommultiple training images But the alignment pattern of the basiselements that are away from the vertical arrow can be detectedfrom this image without any specifically learned knowledge

Both the geometry and object regimes can be modeled ascompositions of Gabor wavelets Patterns in the geometryregime are compositions of highly aligned Gabor waveletsand patterns in the object regime are compositions of Gaborwavelets that are not highly aligned within a single image Inthis sense edge and region patterns are just simple generic ob-ject patterns In comparison images in the texture regime ex-hibit no meaningful alignments or a priori known compositionsand are summarized by pooling the marginal statistics that dis-card the position information

(a)

(b)

(c)

(d)

Figure 12 Alignment over scale (a) A horizontal slice of an imageof a vertical bar I(x) versus x1 (b) Energy A(x s) versus x1 (c) Phaseφ(x s) versus x1 Each curve corresponds to a scale s (d) Positions ofedge-bar points over the scale The two edge points merge into a barpoint at a certain scale

Another type of alignment in the geometry regime also canbe very important the alignment over scales (Witkin 1983Lindberg 1993) or frequencies (Morrone Ross Burr andOwens 1986) Such an alignment was used by Kovesi (1999)to detect edges and corners Figure 12 illustrates the basic ideaFigure 12(a) shows a horizontal slice of an image I(x) of a ver-tical bar The horizontal axis is x1 and the vertical axis is I(x)Figures 12(b) and 12(c) display the energy A(x s) and phaseφ(x s) on this slice The definitions of A(x s) and φ(x s) wereprovided in Section 21 In Figures 12(b) and 12(c) each curvecorresponds to A(x s) or φ(x s) for a fixed s It is evident thatan edge-bar point x [ie a local maximum in energy A(x s)]can exist over a range of scales Within this range the phaseφ(x s) and orientation θ(x s) remain constant over differentscales s For an edge point the energy A(x s) also remainsconstant (subject to discretization error) if the Gabor sine andcosine wavelets are normalized to have unit L1 norm If the Ga-bor wavelets are normalized to have unit L2 norm then A(x s)increases over s within the range of alignment Figure 12(d)traces the positions of the two edge points over scale s In thisplot the horizontal axis is x1 and the vertical axis is s We cansee that at a certain scale the two edge points merge into a barpoint Beyond this scale the orientation and the phase continueto be aligned at this bar point

The alignment over scale can be important for modeling thecoarse-to-fine generative model shown in Figure 2 For exam-ple an edge pattern at a certain scale can be expanded intolarger edge patterns at a finer scale A bar pattern can be ex-panded into two edge patterns at a finer scale More generallyan object pattern may be composed of several Gabor waveletseach of which may be expanded into an edge or bar patternThis point of view can be useful for modeling the relationshipbetween schemes 1 and 2 discussed in Section 13



Multiscale hierarchical models of wavelet coefficients havebeen developed by De Bonet and Viola (1997) for texture syn-thesis and recognition and by Buccigrossi and Simocelli (1999)for image compression and denoising What we are propos-ing here is to add the labels of the recognized patterns in thecoarse-to-fine generative model Needless to say we remain faraway from such a model and much theoretical and experimen-tal work needs to be done A preliminary attempt toward such amodel has been made by Wang et al (2005)

5 DISCUSSION

51 Classification Rule or Generative Model

There are two major schools of thoughts on statistical learn-ing in computer vision One school emphasizes learning gen-erative models (Cootes Edwards and Taylor 2001 DorettoChiuso Wu and Soatto 2003 Isard and Blake 1998 Wang andZhu 2004) and deriving the likelihood or the posterior distri-bution Pr(labels|image patches) from the learned models Theother school emphasizes learning Pr(labels|image patches) ormore simply the classification rules Label = f (image patch)directly (Amit and Geman 1997 Viola and Jones 2004) with-out learning the generative models Attempts have been madeto combine these two schools of methods (Tu and Zhu 2002Tu Chen Yuille and Zhu 2005)

Comparing these two schools shows that the second school ismore direct than the first and is very powerful for classificationtasks in which the variable ldquolabelrdquo is a category that takes valuesin a finite or binary set But for general vision tasks the variableldquolabelsrdquo has a much more complex structure than a category forexample it can have a hierarchical structure with compositionaland spatial organizations as illustrated in Figure 2(b) For sucha complex structure learning Pr(labels|image patches) may notbe practical learning the coarse-to-fine generative model maybe simpler and more natural

52 Low-Level and Midlevel Vision

Vision tasks are roughly classified into low-level mid-leveland high-level tasks Low-level tasks involve detecting localimage structures (eg edges corners junctions) or detectingwhat Julesz called ldquotextonsrdquo (Julesz 1981) or what Marr calledldquosymbolsrdquo or ldquotokensrdquo (Marr 1982) Midlevel tasks involve ex-tracting curves and contours from the image or segmenting theimage into regions of coherent intensity patterns High-leveltasks involve recognizing objects and their shapes and poses

Detecting local image structures in low-level tasks can be dif-ficult because natural images are full of ambiguities if we lookonly at a local image patch at a fixed resolution Extracting con-tours or regions from images also can be difficult even withthe help of regularity terms or generic prior distributions (KassWitkin and Terzopoulos 1987 Mumford and Shah 1989) to en-force smoothness and continuity The low-level and midleveltasks are often considered the foundation for the high-leveltasks of object recognition but at the same time the ambigu-ities in low-level and midlevel tasks can be resolved only afterthe specific objects are recognized

The Bayesian framework discussed in Section 13 can be anelegant framework for resolving the mutual dependencies be-tween the tasks at different levels In this framework object pat-terns and edge patterns are labeled simultaneously at differentlayers The edge patterns are not treated as being more funda-mental than the object patterns Although an object pattern canbe composed of edge patterns at a fine resolution at a coarseresolution an object pattern occupies an image patch that is nolarger than the image patch of an edge pattern at a fine resolu-tion Prior knowledge of how the object patterns are composedof the edge patterns aids the labeling of both types of patternsIntuitively this prior knowledge serves as a constraint to helpeliminate the ambiguities in the local image patches at differentlayers

ACKNOWLEDGEMENT

The work presented in this article was supported by NSFDMS 0707055 NSF IIS 0413214 ONR C2Fusse

[Received January 2006 Revised September 2006]

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Adelson E H and Bergen J R (1985) ldquoSpatiotemporal Energy Models forthe Perception of Motionrdquo Journal Optical Society of America Ser A 2284ndash299

Amit Y and Geman D (1997) ldquoShape Quantization and Recognition WithRandomized Treesrdquo Neural Computation 9 1545ndash1588

Bell A and Sejnowski T J (1997) ldquoThe lsquoIndependent Componentsrsquo of Nat-ural Scenes Are Edge Filtersrdquo Vision Research 37 3327ndash3338

Besag J (1974) ldquoSpatial Interaction and the Statistical Analysis of LatticeSystemsrdquo (with discussion) Journal of the Royal Statistical Society Ser B36 192ndash236

Buccigrossi R W and Simoncelli E P (1999) ldquoImage Compression via JointStatistical Characterization in the Wavelet Domainrdquo IEEE Transactions onImage Processing 8 1688ndash1701

Burt P and Adelson E H (1983) ldquoThe Laplacian Pyramid as a CompactImage Coderdquo IEEE Transactions on Communication 31 532ndash540

Candes E J and Donoho D L (1999) ldquoCurvelets A Surprisingly EffectiveNonadaptive Representation for Objects With Edgesrdquo in Curves and Sur-faces ed L L Schumakeretal Nashville TN Vanderbilt University Press

Canny J (1986) ldquoA Computational Approach to Edge Detectionrdquo IEEE Trans-actions on Pattern Analysis and Machine Intelligence 8 679ndash698

Chen S Donoho D and Saunders M A (1999) ldquoAtomic Decomposition byBasis Pursuitrdquo SIAM Journal on Scientific Computing 20 33ndash61

Cootes T F Edwards G J and Taylor C J (2001) ldquoActive AppearanceModelsrdquo IEEE Transactions on Pattern Analysis and Machine Intelligence23 681ndash685

Daugman J (1985) ldquoUncertainty Relation for Resolution in Space SpatialFrequency and Orientation Optimized by Two-Dimensional Visual CorticalFiltersrdquo Journal of the Optical Society of America 2 1160ndash1169

De Bonet J S and Viola P A (1997) ldquoA Non-Parametric Multi-Scale Statis-tical Model for Natural Imagesrdquo Advances in Neural Information ProcessingSystems

Della Pietra S Della Pietra V and Lafferty J (1997) ldquoInducing Featuresof Random Fieldsrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence 19 380ndash393

Doretto G Chiuso A Wu Y N and Soatto S (2003) ldquoDynamic TexturesrdquoInternational Journal of Computer Vision 51 91ndash109

Dunn D and Higgins W E (1995) ldquoOptimal Gabor Filters for Texture Seg-mentationrdquo IEEE Transactions on Image Processing 4 947ndash964

Friedman J H (1987) ldquoExploratory Projection Pursuitrdquo Journal of the Amer-ican Statistical Association 82 249

Geman S and Geman D (1984) ldquoStochastic Relaxation Gibbs Distribu-tions and the Bayesian Restoration of Imagesrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence 6 721ndash741

Geman S and Graffigne C (1987) ldquoMarkov Random Field Image Modelsand Their Applications to Computer Visionrdquo Proceedings of the Interna-tional Congress of Mathematicians 1 1496ndash1517



Geman S Potter D F and Chi Z (2002) ldquoComposition Systemrdquo Quarterlyof Applied Mathematics 60 707ndash736

George E I and McCulloch R E (1997) ldquoApproaches to Bayesian VariableSelectionrdquo Statistica Sinica 7 339ndash373

Grenander U (1993) General Pattern Theory Oxford UK Oxford Univer-sity Press

Grenander U and Miller M I (1994) ldquoRepresentation of Knowledge in Com-plex Systemsrdquo Journal of the Royal Statistics Society Ser B 56 549ndash603

Guo C Zhu S C and Wu Y N (2003a) ldquoVisual Learning by Integrating De-scriptive and Generative Modelsrdquo International Journal of Computer Vision53 5ndash29

(2003b) ldquoA Mathematical Theory of Primal Sketch and Sketchabil-ityrdquo in Proceedings of the International Conference of Computer Visionpp 1228ndash1235

(2007) ldquoPrimal Sketch Integrating Structure and Texturerdquo ComputerVision and Image Understanding 106 5ndash19

Heeger D J and Bergen J R (1995) ldquoPyramid-Based Texture Analy-sisSynthesisrdquo Computer Graphics Proceedings 229ndash238

Hubel D and Wiesel T (1962) ldquoReceptive Fields Binocular Interaction andFunctional Architecture in the Catrsquos Visual Cortexrdquo Journal of Physiology160 106ndash154

Huo X and Chen J (2005) ldquoJBEAM Multiscale Curve Coding via Beam-letsrdquo IEEE Transactions on Image Processing 14 1665ndash1677

Isard M and Blake A (1998) ldquoCondensation-Conditional Density Propa-gation for Visual Trackingrdquo International Journal of Computer Vision 295ndash28

Julesz B (1981) ldquoTextons the Elements of Texture Perception and Their In-teractionsrdquo Nature 290 91ndash97

Kass M Witkin A and Terzopoulos D (1987) ldquoSnakes Active ContourModelsrdquo International Journal of Computer Vision 1 321ndash331

Kovesi P (1999) ldquoImage Features From Phase Congruencyrdquo Videre Journalof Computer Vision Research 1

Lampl I Ferster D Poggio T and Riesenhuber M (2004) ldquoIntracellularMeasurements of Spatial Integration and the MAX Operation in ComplexCells of the Cat Primary Visual Cortexrdquo Journal of Neurophysiology 922704ndash2713

Lee T S (1996) ldquoImage Representation Using 2D Gabor Waveletsrdquo IEEETransactions on Pattern Analysis and Machine Intelligence 10 959ndash971

Lewicki M S and Olshausen B A (1999) ldquoProbabilistic Framework for theAdaptation and Comparison of Image Codesrdquo Journal of the Optical Societyof America 16 1587ndash1601

Lindberg T (1993) ldquoEffective Scale A Natural Unit for Measuring Scale-Space Lifetimerdquo IEEE Transactions on Pattern Analysis and Machine Intel-ligence 15 1068ndash1074

(1994) Scale-Space Theory in Computer Vision Dordrecht KluwerAcademic Publishers

Mallat S (1989) ldquoA Theory of Multiresolution Signal Decomposition TheWavelet Representationrdquo IEEE Transactions on Pattern Analysis and Ma-chine Intelligence 11 674ndash693

Mallat S and Zhang Z (1993) ldquoMatching Pursuit in a Time-Frequency Dic-tionaryrdquo IEEE Transactions on Signal Processing 41 3397ndash3415

Mallat S and Zhong Z (1992) ldquoCharacterization of Signals From MultiscaleEdgesrdquo IEEE Transactions on Pattern Analysis and Machine Intelligence14 710ndash732

Marr D (1982) Vision Gordonsville VA W H FreemanMoisan L Desolneux A and Morel J-M (2000) ldquoMeaningful Alignmentsrdquo

International Journal of Computer Vision 40 7ndash23Morrone M C Ross J Burr D C and Owens R A (1986) ldquoMach Bands

Are Phase Dependentrdquo Nature 324 250ndash253Mumford D B (1994) ldquoPattern Theory A Unifying Perspectiverdquo in Proceed-

ings of the First European Congress of Mathematics Boston BirkhauserMumford D and Shah J (1989) ldquoOptimal Approximations by Piecewise

Smooth Functions and Associated Variational Problemsrdquo Communicationson Pure and Applied Mathematics 42 577ndash685

Olshausen B A and Field D J (1996) ldquoEmergence of Simple-Cell ReceptiveField Properties by Learning a Sparse Code for Natural Imagesrdquo Nature 381607ndash609

Olshausen B A and Millman K J (2000) ldquoLearning Sparse Codes Witha Mixture-of-Gaussians Priorrdquo Advances in Neural Information ProcessingSystems 12 841ndash847

Pece A (2002) ldquoThe Problem of Sparse Image Codingrdquo Journal of Mathe-matical Imaging and Vision 17 89ndash108

Perona P and Malik J (1990) ldquoDetecting and Localizing Composite Edges inImagesrdquo in Proceedings of the International Conference of Computer Visionpp 52ndash57

Petkov N and Kruizinga P (1997) ldquoComputational Models of Visual Neu-rons Specialised in the Detection of Periodic and Aperiodic Oriented VisualStimuli Bar and Grating Cellsrdquo Biological Cybernetics 76 83ndash96

Portilla J and Simoncelli E P (2000) ldquoA Parametric Texture Model Basedon Joint Statistics of Complex Wavelet Coefficientsrdquo International Journalof Computer Vision 40 49ndash71

Riesenhuber M and Poggio T (1999) ldquoHierarchical Models of ObjectRecognition in Cortexrdquo Nature Neuroscience 2 1019ndash1025

Ruderman D L and Bialek W (1994) ldquoStatistics of Natural Images Scalingin the Woodsrdquo Physics Review Letters 73 814ndash817

Simoncelli E P Freeman W T Adelson E H and Heeger D J (1992)ldquoShiftable Multiscale Transformsrdquo IEEE Transactions on Information The-ory 38 587ndash607

Simoncelli E P and Olshausen B A (2001) ldquoNatural Image Statistics andNeural Representationrdquo Annual Review of Neuroscience 24 1193ndash1216

Srivastava A Grenander U and Liu X (2002) ldquoUniversal Analytical Formsfor Modeling Image Probabilitiesrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence 24 1200ndash1214

Srivastava A Lee A Simoncelli E and Zhu S (2003) ldquoOn Advances inStatistical Modeling of Natural Imagesrdquo Journal of Mathematical Imagingand Vision 18 17ndash33

Tu Z W and Zhu S C (2002) ldquoImage Segmentation by Data Driven Markovchain Monte Carlordquo IEEE Transactions on Pattern Analysis and MachineIntelligence 24 657ndash673

Tu Z W Chen X R Yuille A L and Zhu S C (2005) ldquoImage ParsingUnifying Segmentation Detection and Recognitionrdquo International Journalof Computer Vision 63 113ndash140

Viola P A and Jones M J (2004) ldquoRobust Real-Time Face Detectionrdquo In-ternational Journal of Computer Vision 57 137ndash154

Wang Y Z and Zhu S C (2004) ldquoAnalysis and Synthesis of Textured Mo-tion Particles and Wavesrdquo IEEE Transactions on Pattern Analysis and Ma-chine Intelligence 26 1348ndash1363

Wang Y Z Bahrami S and Zhu S C (2005) ldquoPerceptual Scale Space andIts Applicationsrdquo in Proceedings of the International Conference of Com-puter Vision pp 58ndash65

Wang Z and Jenkin M (1992) ldquoUsing Complex Gabor Filters to Detect andLocalize Edges and Barsrdquo Advances in Machine Vision 32 151ndash170

Witkin A (1983) ldquoScale-Space Filteringrdquo in Proceedings of the InternationalJoint Conference on Artificial Intelligence pp 1019ndash1021

Wu Y N Zhu S C and Guo C (2002) ldquoStatistical Modeling of TextureSketchrdquo in Proceedings of the European Conference of Computer Visionpp 240ndash254

(2007) ldquoFrom Information Scaling to Regimes of Statistical ModelsrdquoQuarterly of Applied Mathematics to appear

Wu Y N Zhu S C and Liu X W (2000) ldquoEquivalence of Julesz Ensem-ble and FRAME Modelsrdquo International Journal of Computer Vision 38245ndash261

Young R A (1987) ldquoThe Gaussian Derivative Model for Spatial Vision IRetinal Mechanismrdquo Spatial Vision 2 273ndash293

Zhu S C and Mumford D (2007) ldquoQuest for a Stochastic Grammar of Im-agesrdquo Foundations and Trends in Computer Graphics and Vision to appear

Zhu S C Guo C E Wang Y Z and Xu Z J (2005) ldquoWhat Are TextonsrdquoInternational Journal of Computer Vision 62 121ndash143

Zhu S C Liu X and Wu Y N (2000) ldquoExploring Texture Ensembles byEfficient Markov Chain Monte Carlo Towards a lsquoTrichromacyrsquo Theory ofTexturerdquo IEEE Transactions on Pattern Analysis and Machine Intelligence22 554ndash569

Zhu S C Wu Y N and Mumford D (1997) ldquoMinimax Entropy Prin-ciple and Its Applications in Texture Modelingrdquo Neural Computation 91627ndash1660



(a) (b) (c)

Figure 1 Examples from three regimes of image patterns (a) texture regime (b) object regime and (c) geometry regime The three regimesare connected by the change in viewing distance or equivalently the change in camera resolution

then we can recognize different image patterns within the win-dow and these patterns belong to different regimes describedin the previous section We recognize lines and regions at highresolution or small scale shapes and objects at medium reso-lution or scale and textures and clutters at low resolution orlarge scale In addition these patterns are highly organizedThe large-scale texture patterns are composed of medium-scaleshape patterns which in turn are composed of small-scale edgeand corner patterns Figure 2(b) provides an illustration wherethe image patches at different resolutions are labeled as variouspatterns such as foliage leaves and edges or corners The ver-tical and diagonal arrows indicate the compositional relation-ships and the horizontal arrows indicate the spatial arrange-ments Thus a complete interpretation of an image must con-sist of the labels of image patches at all of these layers It alsoshould include the compositional and spatial organizations ofthe labels

Figure 2 illustrates the multiscale structures of both the im-age data and the recognized patterns The multiscale structureof the image data has been extensively studied in wavelet the-ory (Mallat 1989 Simoncelli Freeman Adelson and Heeger1992) and scale-space theory (Witkin 1983 Lindberg 1994)

however the multiscale structure of the recognized patterns hasnot received as much treatment Recent attempts to understandthis issue have been made by Wu Zhu and Guo (2007) andWang Bahrami and Zhu (2005) The compositional relation-ships have been studied by Geman Potter and Chi (2002) TheANDndashOR grammars for the multilayer organizations of the im-age patterns have been studied by Zhu and Mumford (2007)

13 Bayesian Generative Modeling andPosterior Inference

An elegant approach to recognizing the multiscale patternsfrom the multiscale image data is to construct a Bayesian gen-erative model and use the posterior distribution to guide the in-ferential process (Grenander 1993 Grenander and Miller 1994Geman and Geman 1984 Mumford 1994) Given the multilayerstructures of both the image data and the labels of the recog-nized patterns as shown in Figure 2(b) a natural form of thismodel is a recursive coarse-to-fine generative process consist-ing of the following two schemes

(a) (b)

Figure 2 Multiscale representations of data and patterns (a) A Gaussian pyramid The image at each layer is a zoomed-out version of theimage at the layer beneath The Laplacian pyramid can be obtained by computing the differences between consecutive layers of the Gaussianpyramid (b) For the same image different patterns are recognized at different scales These patterns are organized in terms of compositionaland spatial relationships The vertical and diagonal arrows represent compositional relationships the horizontal arrows spatial relationships



(a) (b) (c)




























2



G(x) = 1

2πσ1σ2exp

minus1

2

(x2

1

σ 21

+ x22

σ 22

)eix1 (1)





is


I(x)Gysθ (x)dx (2)







(a) (b) (c)





I(x) = 1

4π2






1 + ω22)

12 The coef-








Gsθ (ω) = exp

minus1

2

[(sσ1

(ω1 minus 1

s

))2

+ (sσ2ω2)2]

(4)












I(x) =Nsum

n=1

cnBn(x) + ε(x) (5)



Msumm=1


n=1

cmnBn

∥∥∥∥∥2

+ λ

Nsumn=1

S(cmn)

](6)






I =Nsum

n=1







(a) (b)

(c) (d)







Hkt(I) =sum

x


) t = 1 T (8)



(a) (b)









p(I) = 1

Z()exp

Ksum

k=1

〈λkHk(I)〉

(9)



























(a) (b) (c)

















(10)



Let


|D| minus K (12)




1


radic2πeγ ) (13)







)= (B B)primeI (14)



= log f (R) minus 1


+ 1

















(a)

(b)

(c)

(d)







5 DISCUSSION








ACKNOWLEDGEMENT



REFERENCES













































































(a) (b) (c)




























2



G(x) = 1

2πσ1σ2exp

minus1

2

(x2

1

σ 21

+ x22

σ 22

)eix1 (1)





is


I(x)Gysθ (x)dx (2)







(a) (b) (c)





I(x) = 1

4π2






1 + ω22)

12 The coef-








Gsθ (ω) = exp

minus1

2

[(sσ1

(ω1 minus 1

s

))2

+ (sσ2ω2)2]

(4)












I(x) =Nsum

n=1

cnBn(x) + ε(x) (5)



Msumm=1


n=1

cmnBn

∥∥∥∥∥2

+ λ

Nsumn=1

S(cmn)

](6)






I =Nsum

n=1







(a) (b)

(c) (d)







Hkt(I) =sum

x


) t = 1 T (8)



(a) (b)









p(I) = 1

Z()exp

Ksum

k=1

〈λkHk(I)〉

(9)



























(a) (b) (c)

















(10)



Let


|D| minus K (12)




1


radic2πeγ ) (13)







)= (B B)primeI (14)



= log f (R) minus 1


+ 1

















(a)

(b)

(c)

(d)







5 DISCUSSION








ACKNOWLEDGEMENT



REFERENCES























































































2



G(x) = 1

2πσ1σ2exp

minus1

2

(x2

1

σ 21

+ x22

σ 22

)eix1 (1)





is


I(x)Gysθ (x)dx (2)







(a) (b) (c)





I(x) = 1

4π2






1 + ω22)

12 The coef-








Gsθ (ω) = exp

minus1

2

[(sσ1

(ω1 minus 1

s

))2

+ (sσ2ω2)2]

(4)












I(x) =Nsum

n=1

cnBn(x) + ε(x) (5)



Msumm=1


n=1

cmnBn

∥∥∥∥∥2

+ λ

Nsumn=1

S(cmn)

](6)






I =Nsum

n=1







(a) (b)

(c) (d)







Hkt(I) =sum

x


) t = 1 T (8)



(a) (b)









p(I) = 1

Z()exp

Ksum

k=1

〈λkHk(I)〉

(9)



























(a) (b) (c)

















(10)



Let


|D| minus K (12)




1


radic2πeγ ) (13)







)= (B B)primeI (14)



= log f (R) minus 1


+ 1

















(a)

(b)

(c)

(d)







5 DISCUSSION








ACKNOWLEDGEMENT



REFERENCES













































































(a) (b) (c)





I(x) = 1

4π2






1 + ω22)

12 The coef-








Gsθ (ω) = exp

minus1

2

[(sσ1

(ω1 minus 1

s

))2

+ (sσ2ω2)2]

(4)












I(x) =Nsum

n=1

cnBn(x) + ε(x) (5)



Msumm=1


n=1

cmnBn

∥∥∥∥∥2

+ λ

Nsumn=1

S(cmn)

](6)






I =Nsum

n=1







(a) (b)

(c) (d)







Hkt(I) =sum

x


) t = 1 T (8)



(a) (b)









p(I) = 1

Z()exp

Ksum

k=1

〈λkHk(I)〉

(9)



























(a) (b) (c)

















(10)



Let


|D| minus K (12)




1


radic2πeγ ) (13)







)= (B B)primeI (14)



= log f (R) minus 1


+ 1

















(a)

(b)

(c)

(d)







5 DISCUSSION








ACKNOWLEDGEMENT



REFERENCES

















































































I(x) =Nsum

n=1

cnBn(x) + ε(x) (5)



Msumm=1


n=1

cmnBn

∥∥∥∥∥2

+ λ

Nsumn=1

S(cmn)

](6)






I =Nsum

n=1







(a) (b)

(c) (d)







Hkt(I) =sum

x


) t = 1 T (8)



(a) (b)









p(I) = 1

Z()exp

Ksum

k=1

〈λkHk(I)〉

(9)



























(a) (b) (c)

















(10)



Let


|D| minus K (12)




1


radic2πeγ ) (13)







)= (B B)primeI (14)



= log f (R) minus 1


+ 1

















(a)

(b)

(c)

(d)







5 DISCUSSION








ACKNOWLEDGEMENT



REFERENCES
















































































Hkt(I) =sum

x


) t = 1 T (8)



(a) (b)









p(I) = 1

Z()exp

Ksum

k=1

〈λkHk(I)〉

(9)



























(a) (b) (c)

















(10)



Let


|D| minus K (12)




1


radic2πeγ ) (13)







)= (B B)primeI (14)



= log f (R) minus 1


+ 1

















(a)

(b)

(c)

(d)







5 DISCUSSION








ACKNOWLEDGEMENT



REFERENCES































































































(a) (b) (c)

















(10)



Let


|D| minus K (12)




1


radic2πeγ ) (13)







)= (B B)primeI (14)



= log f (R) minus 1


+ 1

















(a)

(b)

(c)

(d)







5 DISCUSSION








ACKNOWLEDGEMENT



REFERENCES













































































(a) (b) (c)

















(10)



Let


|D| minus K (12)




1


radic2πeγ ) (13)







)= (B B)primeI (14)



= log f (R) minus 1


+ 1

















(a)

(b)

(c)

(d)







5 DISCUSSION








ACKNOWLEDGEMENT



REFERENCES














































































1


radic2πeγ ) (13)







)= (B B)primeI (14)



= log f (R) minus 1


+ 1

















(a)

(b)

(c)

(d)







5 DISCUSSION








ACKNOWLEDGEMENT



REFERENCES



















































































(a)

(b)

(c)

(d)







5 DISCUSSION








ACKNOWLEDGEMENT



REFERENCES














































































5 DISCUSSION








ACKNOWLEDGEMENT



REFERENCES




































































































































Documents

Statistical Principles in Image Modeling - UCLA Statistics