Statistical Learning of Visual Feature Hierarchies

Fabien Scalzo and Justus H. Piater∗

Montefiore InstituteUniversity of Liege

B-4000 Liege, Belgium

Abstract

We propose an unsupervised, probabilistic method forlearning visual feature hierarchies. Starting from local,low-level features computed at interest point locations, themethod combines these primitives into high-level abstrac-tions. Our appearance-based learning method uses lo-cal statistical analysis between features and Expectation-Maximization (EM) to identify and code spatial correla-tions. Spatial correlation is asserted when two features tendto occur at the same relative position of each other. Thislearning scheme results in a graphical model that allows aprobabilistic representation of a flexible visual feature hier-archy. For feature detection, evidence is propagated usingNonparametric Belief Propagation (NBP), a recent gener-alization of particle filtering. In experiments, the proposedapproach demonstrates efficient learning and robust detec-tion of object models in the presence of clutter and occlu-sion and under view point changes.

1. IntroductionFeature representation is one of the most important issuesfor learning and recognition applications in computer vi-sion. In the present work, we propose a new approach torepresenting and learning visual feature hierarchies in anunsupervised manner. Our hierarchical representation is in-spired by the compositional nature of objects. Most objectsencountered in the world, which can be either man-made ornatural objects, are composed of a number of distinct con-stituent parts (e.g., a face contains a nose and two eyes, aphone possesses a keypad). If we examine these parts, itbecomes obvious that they are themselves recursively com-posed of other subparts (e.g., an eye contains an iris and eye-lashes, a keypad is composed of buttons). This ubiquitousobservation constitutes our main motivation for arguing thata hierarchical representation must be taken into account tomodel objects in more flexible and realistic way.

Our long-term goal is thus to learn visual feature hierar-chies that correspond to object/part hierarchies. The devel-

∗This work was supported by the Wallonian Ministry of Research(D.G.T.R.E.) under Contract No. 03/1/5438.

opment of a hierarchical and probabilistic framework thatis tractable in terms of complexity is a central problem formany computer vision applications such as visual track-ing, object recognition and categorization, face recognition,stereo matching and image retrieval. In this paper, the prin-cipal objective is to obtain a probabilistic framework thatallows the organization of complex feature models. Themain idea is to use a graphical model to represent the hi-erarchical feature structure. In this representation, which isdetailed in Section 2, the nodes correspond to the visual fea-tures. The edges model both the spatial arrangement and thestatistical dependence between nodes. The formulation interms of graphical models is attractive because it providesa statistical model of the variability of shape and appear-ance. These are specified separately by the edges and thelow-level nodes of the graph, respectively.

An unsupervised feature learning method that allows theconstruction of a hierarchy of visual features is introducedin Section 4. The proposed framework accumulates statisti-cal evidence from feature observations in order to find “con-spicuous coincidences” of visual feature co-occurrences.The structure of the graph is iteratively built by combin-ing correlated features into higher-level abstractions. Ourlearning method is best explained by first presenting the de-tection process, which is described in Section 3.

Level 0

1

2

Figure 1: Object part decomposition (left) and correspond-ing graphical model (right). In our representation, eachedge defines the relative location (defined by a distance andan orientation) from the subfeature to the center of the com-pound feature.

0-7695-2372-2/05/$20.00 (c) 2005 IEEE

For detection, we use Nonparametric Belief Propagation[7, 19], a message-passing algorithm, to propagate the ob-servations in the graph, thus inferring the belief associatedwith higher-level features that are not directly observable.The functioning and the efficacy of our method are illus-trated in Section 5. Finally, Section 6 provides a discussionof related work and biological analogies.

2. RepresentationIn this section, we introduce a new part-based and proba-bilistic representation of visual features. In the proposedgraphical model, nodes represent visual features and are an-notated with the detection information for a given scene.The edges represent two types of information: the relativespatial arrangement between features, and their hierarchicalcomposition. We employ the term “visual feature” in twodistinct contexts:

Primitive visual features are low-level features. They arerepresented by a local descriptor. For this work, weused simple descriptors constructed from normalizedpixel values and located at Harris interest points [5],but our system does not rely on any particular featuredetector. Any other feature detector can be used to de-tect and extract more robust information (such as SIFTkeys [10]).

Compound visual features consist of flexible geometricalcombinations of other subfeatures (primitive or com-pound features).

Formally, our graph G is a mathematical object made up oftwo sets: a vertex set V , and a directed edge set

−→E . Forany node s ∈ V , the set of parents and the set of childrenare respectively defined as U(s) = {ui ∈ V|(ui, s) ∈ −→E }and C(s) = {ci ∈ V|(s, ci) ∈ −→E }. Information aboutfeature types and their specific occurrences in an image willbe represented in the graph by annotations of vertices andedges, as described next.

2.1. The Vertex SetThe vertices s ∈ V of the graph represent features. Theycontain the feature activations for a given image. Our graph-ical model associates each node with a hidden random vari-able x ∈ R2 representing the spatial probability distributionof the feature in the image. This random variable is continu-ous and is defined in a two-dimensional space where the di-mensions X ,Y are the location in the image. For simplicity,we assume that the distribution of x can be approximated bya Gaussian mixture:

p(x; Θ) =N∑

i=1

wiG(x;µi,Σi) (1)

In order to retain information about feature orientation, weassociate a mean orientation θi ∈ [0, 2π[ to each com-ponent of the Gaussian mixture. All mixture componentsare collectively represented by the parameter vector Θ =(µ1, . . . , µN ; Σ1, . . . ,ΣN ;w1, . . . , wN ; θ1, . . . , θN ).

Primitive feature classes lie at the first level of the graph.They are represented by a local descriptor and are associ-ated with an observable random variable y, defined in thesame two-dimensional space (X ,Y) as the hidden variablesx, and are likewise approximated by a Gaussian mixture.

2.2. The Edge Set

An edge e ∈ −→E of the graph models two aspects of the fea-ture structure. First, whenever an edge links two features itsignifies that the destination feature is a part of the sourcefeature. Second, an edge describes the spatial relation be-tween the compound feature and its subfeature in terms ofrelative position. The annotation does not contain any spe-cific information about a given scene but represents a staticview of the feature hierarchy and shape.

Each edge e = xt, xs is associated with a parame-ter µst = [µx

st;µyst] that defines the relative, orientation-

normalized, position of the compound feature xt versus theother xs.

We also define a potential function ψs,t(xs, xt) that en-codes the relative distribution of the compound feature withrespect to the subfeature. The conditional density is approx-imated by a Gaussian mixture:

ψs,t(xs, xt) =Ns∑i=1

wis G(xt;ϑs,t(xi

s),Σis) (2)

where xis is the i-th component of the mixture xs, wi

s is thecomponent weight, ϑ is a function that computes the meanof the compound feature component xi

t given the subfeaturelocation µi

s and orientation θis:

ϑs,t(x(i)s ) = µi

s +[

cos(θis) − sin(θi

s)sin(θi

s) cos(θis)

] [µx

st

µyst

](3)

The use of Gaussian mixture models to represent spatial re-lations (Eq. 2) has been successfully applied in many works[17, 20]. It is especially useful for modeling relative posi-tion variability between features.

3. Nonparametric Belief PropagationIn the preceding section, we defined the structure of ourgraphical model used to represent visual features. We nowdescribe the detection process of a visual feature, given itsgraphical-model representation. In our framework, infer-ring the probability density function p(x|y) of features con-ditioned on the primitives detected amounts to estimatingthe belief of the nodes.

Nonparametric Belief Propagation (NBP) [7, 19] is per-fectly suitable for our continuous parameter space of eachfeature part and the non-Gaussian conditionals betweennodes. NBP is an inference algorithm for graphical mod-els that generalizes particle filtering. It has the advantageof allowing inference over general graphs and is not lim-ited to Markov chains. The main idea of this inferencealgorithm is to propagate information via a series of localmessage-passing operations. Each message is representedusing a sample-based density estimate which is defined non-parametrically as a Gaussian mixture. The conditional dis-tribution (or belief) of each node is defined by the prod-uct of the incoming messages of the node. Due to the in-tractable complexity of an exact computation, this productmust be approximated. Several techniques are possible. Inthis work, we use an efficient Gibbs sampling procedurewith 300 samples for each message.

Following the conventional NBP notation, a messagemij from node xi to xj is written

mi,j(xj) =∫ψi,j(xi, xj) φi(xi, yi)

∏k∈Ni\j

mk,i(xi) dxi

where Ni is the set of neighbors of node i, ψi,j is the pair-wise potential between nodes i, j, and φi(xi, yi) is the locallikelihood associated with the node i and obtained via theobservation function yi of a low-level feature.

The message update algorithm is summarized in Algo-rithm 1 and illustrated in Figure 2. After a few iterations,depending on the size of the graph, each node i can producean approximation of the conditional distribution p(xi|y) us-ing the product of incoming messages. For more detailsabout Nonparametric Belief Propagation, consult the origi-nal research papers [7, 19].

Algorithm 1 Message update using NBP

Draw samples {xit, w

it}M

i=1 from the product of inputmessages mkt(xt) using the Gibbs sampler,Compute values for each outgoing message mtu(xu)Means: xi

tu = ψt,u(xit, x

iu)

Orientations: θitu using mkt(xt) (Eq. 4)

As we mentioned in Section 2.1, we associate an orienta-tion to each sample. It is computed using the mean orienta-tion of the incoming messages and local evidence at the lo-cation of the sample. More precisely, we compute the meanorientation θxz (l), z = 1, . . . ,M of the available mixturecomponents weighted by their corresponding weights:

tan θxz (l) = Sxz (l)Cxz (l)

Cx(l) =∑Nx

i=1 vi × wi cos(θix) (4)

Sx(l) =∑Nx

i=1 vi × wi sin(θix)

Y1

X1

Y2

X2

X4

X3

Μx1,x3

Μx4,x3

Μx2,x3

← x1

← x2

60 70 80 90 100 110

10

20

30

40

50

60

M2,3

M1,3

x4 →M

4,3

75 76 77 78 79 80 81 82 83 84 85

35

36

37

38

39

40

41

42

43

44

4575 76 77 78 79 80 81 82 83 84 85

35

36

37

38

39

40

41

42

43

44

45

Figure 2: During an iteration of NBP, feature x3 receivedmessages from subfeatures x1, x2 and parent x4. Even if in-dividual messages may contain uncertain information aboutthe location of feature (bottom left), the product of the in-coming messages constrains the feature location (bottomright).

where l is a location of the sample, θix is the orientation of

the i-th component of message x,Nx is the number of avail-able components, vi = R(l, xi) is the response of compo-nent xi at point l andwi is the associated weight. This couldhave been integrated to NBP inference. To reduce the com-plexity and avoid the use of a linear-circular distribution, wecompute it separately.

In our representation, higher-level features are not di-rectly observable. Evidence is incorporated into the graphexclusively via the variables yi representing primitives atthe leaves of the graph as described next.

To initialize the observable variables yi of the graph andto compute the local evidence φi(xi, yi), we match the de-tected local descriptors in an image with the most similarexisting feature descriptor at the leaf nodes of the availablegraphs. We use the observed descriptor parameters (loca-tion, orientation, similarity) to annotate the variable yi ofthe corresponding node i ∈ V .

We illustrate the message-passing algorithm by present-ing in Figure 3 the NBP detection process on an object.The object model was learned during our experiments. Theproduct of incoming messages clearly allows a more preciselocalization of the features.

Figure 3: Illustration of an upward message-passing iter-ation during the NBP detection process. Starting from thefirst level (top left), the detection process uses the presenceof primitives to predict the location of the second level fea-tures (top center). The product of these messages (on theright) refines the belief to a more precise localization. Thisproduct is then used for the next level. The geometricalmodel for each level is shown on the left. At the end ofthis simple example, we obtain the final set of samples thatcorresponds to the localization of the object model.

4. Visual Feature LearningIn this section, we introduce our unsupervised feature learn-ing method that allows the construction of a hierarchy ofvisual features. The general idea of our algorithm is toaccumulate statistical evidence from the relative positionsof observed features in order to find “conspicuous coinci-dences” of visual feature co-occurrences. The structure ofour graphical model is iteratively built by combining corre-lated features into new visual feature abstractions. First, thelearning process votes to accumulate information on the rel-ative position of features and extracts the feature pairs thattend to be located in the same neighborhood. Second, itestimates the parameters of the geometrical relations usingeither Expectation-Maximization (EM) or a voting scheme.It finally creates new feature nodes in the graph by combin-ing spatially correlated features. In the following sections,we describe the three main steps of this unsupervised learn-ing procedure.

4.1. Spatial CorrelationThe objective of this first step of our learning process is tofind spatially correlated features. A spatial correlation ex-ists between two features if they are often detected in thesame neighborhood. The size of the neighborhood to con-sider is set proportionally to the space covered by the featureparts. Co-occurrence statistics are collected from multiplefeature occurrences within one or across many different im-ages. The procedure to find correlated features is summa-rized in Algorithm 2. After its completion, we obtain a votearray S concerning the relative locations of correlated fea-tures. It is important to notice that before the first iterationwe apply K-means clustering to the set of feature descrip-tors. This identifies primitive classes from the training setand is used to create the first level of the graphical model.

Algorithm 2 Spatial Correlation ExtractionSuccessively extract each image I from the training setApply NBP (Section 3) to detect all features fI ={fi0 . . . fin} ∈ G in image Ifor each pair [fi, fj ] where fj is observed in the neigh-borhood of fi do

Compute the relative position pr ∈ R2 of fj given fi

Vote for the corresponding observation [fi, fj , pr]end forKeep all pairs [fi, fj ] where

∑pr

S[fi, fj , pr] > tc

4.2. Spatial RelationsIn our framework, spatial relations are defined in terms ofrelative position between features. We implemented two so-lutions to estimate this parameter. The first method uses the

Expectation-Maximization (EM) algorithm, and the secondimplements a fast discrete voting scheme to find location ev-idence. The estimated geometrical relations are used duringfeature generation (Section 4.3) in order to create new fea-tures. First, however, we give some details on both methodsfor the estimation of spatial relations.

4.2.1 Expectation-Maximization

In principle, a sample of observed spatial relations xr be-tween two given features can be approximated by a Gaus-sian mixture, where each component k represents a clusterof relative positions µk of one of the two features fj withrespect to the other, the reference feature fi: f(xr; Θ) =∑K

k=1 wkGk(xr; (µk,Σk)).EM can be used to estimate the parameters of the spatial

relation between each correlated feature pair [fi, fj ] ∈ S.It maximizes the likelihood of the observed spatial relationsover the model parameters Θ = (w1...K ;µ1...K ; Σ1...K).The Expectation (E) and Maximization (M) steps of eachiteration of the algorithm are defined as follows:

Step E Compute the current expected values of the compo-nent indicators tk(xi), 1 ≤ i ≤ n, 1 ≤ k ≤ K, wheren is the number of observations, K is the number ofcomponents and q is the current iteration:

t(q)k (xi) =

w(q)k G

(xi; µ

(q)k , Σ(q)

k

)∑K

l=1 w(q)l G

(xi; µ

(q)l , Σ(q)

l

)

Step M Determine the value of parameters Θq+1 contain-ing the estimates wk, µk, Σk that maximize the likeli-hood of the data {x} given the tk(xi):

w(q+1)k =

∑ni=1 t

(q)k

nµ

(q+1)k =

∑ni=1 t

(q)k (xi)∑n

i=1 t(q)k

Σ(q+1)k =

∑ni=1 t

(q)k

(xi − µ

(q+1)k

) ((xi − µ

(q+1)k

)T

∑ni=1 t

(q)k

In our implementation, a mixture of only two Gaussiancomponents (K = 2) is used to model spatial relations. Thefirst component represents the most probable relative posi-tion, and the second is used to model the noise. When thelocation µ1 and variance Σ1 of the first component is esti-mated between features i and j, we store the correspondinginformation [fi, fj , µi,j ,Σi,j ] in a table T .

4.2.2 Voting

A faster method to estimate spatial-relation parameters isto discretize distance and direction between features. The

idea is to create a bi-dimensional histogram for every corre-lated feature pair [fi, fj ] ∈ S . The dimensions of these his-tograms are the distance d and the relative direction θ fromfeatures fi to fj . Each observation [fi, fj , pr] stored in ta-ble S is projected into a cylindrical space [d, θ] and votes forthe corresponding entry [d, θ] of histogram H[fi, fj ]. Afterthe completion of this voting procedure, we look for signif-icant local maxima in the 2D histograms and store them inthe table T . In our implementation, the distances are ex-pressed relative to the part size and are discretized into 36bins, while the directions are discretized into 72 bins (5-degree precision).

Algorithm 3 Discrete Voting

for each vote vi1...nof entry [fi, fj , pr] in table S do

Project pr(vi) ∈ R2 into a cylindrical space of therelative orientation θ(vi) and the distance d(vi)Vote for the Gaussian kernel at d(vi), θ(vi) in his-togram H[fi(vi), fj(vi)]

end forFind maxima [d, θ] in all histograms and store corre-sponding relative positions µi,j in T

4.3. Feature generation

When a reliable reciprocal spatial correlation is detected be-tween two features [fi, fj ], the generation of a new featurein our model is straightforward. We combine these featuresto create a new higher-level feature by adding a new nodefn in the graph. We connect it to its subfeatures [fi, fj ]by two edges ei, ej that are added to

−→E . Their parametersare computed using the spatial relation {µi,j , µj,i} obtainedfrom the preceding step, and are stored in table T .

The generated feature is located at the midpoint betweenthe subfeatures. Thus the distance from subfeatures tothe new feature is set to half distance between subfeatures[fi, fj ]; µ1 = µi,j/2, µ2 = µj,i/2 and is copied to the newedges; ei(fi, fn) = {µ1,Σ1}, ej(fj , fn) = {µ2,Σ2}.

5. Experiments

In this section, we illustrate our feature learning schemeon an object recognition task using several objects of theColumbia University Object Image Library (COIL-100)[12]. This library is very commonly used in object recog-nition research and contains color images of 100 differentobjects. Each image was captured by a fixed camera at poseintervals of 5 degrees. For our experiments, we used 5 viewsaround both sides of the frontal pose of an object to buildthe object model. When the learning process is completed,the model is thus tuned for a given view of the object.

As we mentioned before, our system does not dependon any particular feature detector. We used Harris inter-est points and rotation invariant descriptors comprising 256pixel values. Any other feature detector can be used to de-tect and extract more robust information (such as SIFT keys[10]). We deliberately used simple features to demonstratethe functioning of our method. To estimate the primitives ofeach object model, we used K-Means to cluster the featurespace. The number of classes (between 16 and 60 in our ex-periments) was selected according to the BIC criterion [16].

During learning, in order to avoid excessive growth ofthe graph due to the feature combinatorics, we only keptthe most salient spatial relations between features. In Fig-ure 5, for each object, the first level of the learned graphicalmodel is illustrated on the top center image. Each circlecorresponds to a low-level feature and is associated to an-other one to create a new compound feature. On the bot-tom of each image, we illustrate the detection results of ourmodels with NBP after only six iterations. During our ex-periments, each message is sampled with 300 values. Twographical models are illustrated in Figure 8. They corre-spond to the first two objects appearing in Figure 5. Weobserve that some low-level features are connected to twocompound features because they are involved in two differ-ent spatial relations.

A quantitative detection evaluation is presented in Fig-ure 4. The first graph (on the left) illustrates the viewpointinvariance of the five object models presented in Figure 5.To generate the graph, we ran the detection process on aseries of images differing in viewing angle by incrementsof 5 degrees. The models responded maximally around thetraining views. We observe that the response quickly fallsat ± 40 degrees. This is caused by the loss of pertinentfeatures in the view. We obtained an average viewpoint in-variance over 80 degrees. These results are remarkable con-sidering the fact that we did not use affine-invariant featuresat the leaf level. The second graph (right) demonstrates theconvergence of the detection process using NBP. This graphwas generated by measuring the uncertainty in the graphacross 23 message-passing iterations. In our experiments,we observe that NBP converges to the optimal solution inless than 7 iterations. In general, the number of iterationsrequired for convergence depends on the number of levelsin the graph.

We also demonstrate the robustness of our model in acluttered scene (Figure 6) and in the presence of occlusion(Figure 7). The first example contains two different learnedobject models. As we explained in Section 3, the detec-tion process starts with low-level feature detection and thenpropagate evidence in graph. Many primitives are detectedin the image. In this experiment, we add extra difficultyby assigning every interest point to the most similar fea-ture class, without requiring a minimum degree of similar-

−60 −50 −40 −30 −20 −10 −0 10 20 30 40 50 60

1

Viewing Angle

Res

pons

e

0 5 10 15 20 25

NBP Iterations

Unc

erta

inty

Figure 4: Maximum response of the five object models pre-sented in Figure 5 on a series of images differing in viewingangle (left). Convergence of NBP to the optimal solutionduring the detection on different views of the objects (right).

ity (left of Figure 6). This results in noisy detection data.However, the use of geometric relations to infer the pres-ence of higher-level features allows an unambiguous detec-tion. As we can see for the objects of the Figure 6, only afew features are needed to detect the objects. In the secondexample (Figure 7), NBP correctly infers the localization ofthe occluded features.

6. DiscussionDuring the past years, great attention has been paid to un-supervised model learning methods applied to object mod-els [13]. In theses techniques, objects are represented byparts, each modeled in terms of both shape and appear-ance by Gaussian probability density functions. This con-cept, which originally operated in batch mode, has beenimproved by introducing incremental learning [6] and byintroducing variational Bayesian methods incorporating in-formation obtained from previously learned models [3]. Inparallel, Agarwal et al. [1] presented a method for learningto detect objects that is based on sparse, part-based repre-sentations. The main limitation of these schemes lies in therepresentation because it only contains two levels, the fea-tures and the models.

A hierarchical model of recognition in the visual cor-tex was proposed by Riesenhuber and Poggio [15] wherespatial combinations of view-tuned units are constructed byalternately employing a maximum and sum pooling oper-ation. Wersing and Korner [21] used a similar hierarchyfor object recognition where invariance is achieved throughpooling over increasing receptive fields. Unfortunately, insuch models there exists no explicit representation of objectstructure.

A scheme for constructing visual features by hierarchicalcomposition of primitive features [14] has been introduced.It uses an incremental procedure for learning discriminativecomposite features using a Bayesian network classifier. Inthis framework, the spatial arrangement of primitives wasrigid and limits the robustness of the system.

Figure 5: Detection results on a series of COIL-100 ob-jects. The geometrical models between low-level featuresare shown in the top center images. The detection resultswith NBP are presented in the bottom images.

Figure 6: Cluttered scene containing two previously learnedobjects. Low-level features detected in the scene (left) anddetection responses for both object models (right).

Figure 7: Spatial relations between the features of the firstlevel are presented in the top left image. The results of thedetection process after 6 iterations can be observed (for thefirst level) in the bottom images. For the occluded object,NBP correctly inferred the presence of missing first levelfeatures.

Figure 8: Two graphical models learned on five differentviews of the first two objects presented in Figure 5.

Recently, Felzenszwalb et al. [4] and Kumar et al. [8]presented two different frameworks for modeling objectsboth inspired by pictorial structure models. These frame-works have some similarities to our work in the sense thatobjects are modeled by different parts arranged in a graph-ical model. The appearance and shape of each part arealso modeled separately. These techniques provide a goodtool to represent articulated structures. However, Felzen-szwalb’s framework requires labeled example images tolearn the models and both schemes are also limited to a sin-gle layer of features.

In neuroscience, recent evidence [2] reinforces the ideathat the coding of geometrical relations in high-level visualfeatures is essential. Moreover, recent work suggests thatthe visual cortex represents objects hierarchically by partsor subfeatures [9].

Even in the context of image structure analysis, hierar-chical and probabilistic models such as HIP [18] demon-strate encouraging results. However, in the case of objectrecognition, it is not clear how to integrate rotation or trans-lation invariance in such models.

The framework presented in this paper offers several sig-nificant improvements over current methods proposed inthe literature. Taking advantage of graphical models, werepresent shape and appearance separately. This allows usto deal with shape deformation and appearance variabil-ity. Moreover, our topology can deal with minor viewpointchanges without explicitly using affine-invariant features.Our scheme is also invariant to rotation and translation ofthe object. Scale invariance can easily be obtained by usingscale invariant features [11] and by normalizing the distancebetween features with the intrinsic feature scale.

Finally, the use of Nonparametric Belief Propagation al-lows a robust and flexible detection of higher-level visualfeatures. The proposed system spans the entire visual hi-erarchy from low-level features to object-level abstractionswithin a single, coherent framework. Learning occurs at alllevels. We argue that a hierarchical model is essential forrepresenting visual features. This appears to be a necessarystep towards more flexible learning methods applied to ob-ject categories.

Future research will focus on two main directions: theunsupervised discovery of visual categories and the integra-tion of the hierarchical model in a task-oriented learningenvironment. Many other applications are possible such asface recognition, object tracking, stereo matching and im-age retrieval.

References

[1] S. Agarwal and D. Roth. Learning a sparse representation forobject detection. In ECCV, volume 4, pages 113–130, 2002.

[2] E.E. Cooper and T.J. Wojan. Differences in the coding ofspatial relations in face identification and basic-level objectrecognition. In J. of Exp. Psychology, volume 26, pages 470–488, 2000.

[3] L. Fei-Fei, R. Fergus, and P. Perona. A Bayesian approachto unsupervised one-shot learning of object categories. InICCV, pages 1134–1141, 2003.

[4] P. Felzenszwalb and D. Huttenlocher. Pictorial structuresfor object recognition. Intl. Journal of Computer Vision,61(1):55–79, January 2005.

[5] C. Harris and M. Stephens. A combined corner and edgedetector. In ALVEY vision conference, pages 147–151, 1988.

[6] S. Helmer and D. G. Lowe. Object recognition with many lo-cal features. In Workshop on Generative Model Based Vision(GMBV), Washington, D.C., July 2004.

[7] M. Isard. Pampas: Real-valued graphical models for com-puter vision. In CVPR, pages 613–620, 2003.

[8] M. P. Kumar, P. H. S. Torr, and A. Zisserman. Extendingpictorial structures for object recognition. In Proceedings ofthe British Machine Vision Conference, 2004.

[9] Y. Lerner, T. Hendler, D. Ben-Bashat, M. Harel, andR. Malach. A hierarchical axis of object processing stagesin the human visual cortex. In Cerebral Cortex, volume 4,pages 287–97, 2001.

[10] D.G. Lowe. Object recognition from local scale-invariantfeatures. In ICCV, pages 1150–1157, 1999.

[11] K. Mikolajczyk and C. Schmid. Scale and affine invariantinterest point detectors. International Journal of ComputerVision, 60(1):63–86, 2004.

[12] S. A. Nene, S. K. Nayar, and H. Murase. Columbia objectimage library (coil-100), 1996.

[13] P. Perona, R. Fergus, and A. Zisserman. Object class recog-nition by unsupervised scale-invariant learning. In CVPR,volume 2, page 264, Madison, Wisconsin, June 2003.

[14] J. H. Piater and R. A. Grupen. Distinctive features should belearned. In BMCV, pages 52–61, 2000.

[15] M. Riesenhuber and T. Poggio. Hierarchical models of objectrecognition in cortex. In Nature Neuroscience, volume 2,pages 1019–1025, 1999.

[16] G. Schwartz. Estimating the dimension of a model. Ann.Stat., 6(2):461–464, 1978.

[17] L. Sigal, S. Bhatia, S. Roth, M. J. Black, and M. Isard. Track-ing loose-limbed people. In CVPR, pages 421–428, 2004.

[18] C. Spence, L. C. Parra, and P. Sajda. Hierarchical imageprobability (hip) models. In ICIP, 2000.

[19] E. B. Sudderth, A. T. Ihler, W. T. Freeman, and A. S. Willsky.Nonparametric belief propagation. In CVPR, pages 605–612,2003.

[20] V. Vogelhuber and C. Schmid. Face detection based ongeneric local descriptors and spatial constraints. In ICPR,volume 1, pages 1084–1087, 2000.

[21] H. Wersing and E. Koerner. Unsupervised learning of com-bination features for hierarchical recognition models. InICANN, pages 1225–1230, 2002.