Hierarchical Kernel Stick-Breaking Process For Multi-Task .
Hierarchical Kernel Stick-Breaking Processfor Multi-Task Image AnalysisQi An, Chunping Wang, Ivo Shterev, Eric Wang, † David Dunson and Lawrence CarinDepartment of Electrical and Computer Engineering†Institute of Statistics and Decision SciencesDuke UniversityDurham, NC 27708-0291{qa,cw36,is33,ew28,lcarin}@ece.duke.edu, dunson@stat.duke.eduAbstractThe kernel stick-breaking process (KSBP) is employed to segment general imagery, imposing thecondition that patches (small blocks of pixels) that are spatially proximate are more likely to beassociated with the same cluster (segment). The number of clusters is not set a priori and is inferredfrom the hierarchical Bayesian model. Further, KSBP is integrated with a shared Dirichlet processprior to simultaneously model multiple images, inferring their inter-relationships. This latter applicationmay be useful for sorting and learning relationships between multiple images. The Bayesian inferencealgorithm is based on a hybrid of variational Bayesian analysis and local sampling. In addition toproviding details on the model and associated inference framework, example results are presented forseveral image-analysis problems.I. I NTRODUCTIONThe segmentation of general imagery is a problem of longstanding interest. There have beennumerous techniques developed for this purpose, including K-means and associated vector quan1
tization methods [7, 21], statistical mixture models [25], graph-diffusion techniques [26], aswell as spectral clustering [27]. This list of existing methods is not exhaustive, although thesemethods share attributes associated with most existing algorithms. First, the clustering is basedon the features of the image (at the pixel level or for small contiguous patches of pixels), andwhen clustering these features one typically does not account for their physical location withinthe image (although the location may be appended as a feature component). Secondly, thesegmentation or clustering of images is typically performed one image at a time, and thereforethere is no attempt to relate the segments of one image to segments in other images (i.e., tolearn inter-relationships between multiple images). The latter concept may be important whenconsidering a new unlabeled image with the goal of relating components of it to images labeledpreviously, and it may also be of interest when sorting or searching a database of images. Finally,in many of the techniques cited above one must a priori set the number of anticipated segmentsor clusters. The techniques developed in this paper seek to perform clustering or segmentation ina manner that explicitly accounts for the physical locations of the features within the image, andmultiple images are segmented simultaneously (termed “multi-task learning”) to infer their interrelationships. Moreover, the analysis is performed in a semi-parametric manner, in the sense thatthe number of segments or clusters is not set a priori, and is inferred from the data. There hasbeen recent research wherein spatial information has been exploited when clustering [14, 15, 28],but that segmentation has been performed one image at a time, and therefore not in a multi-tasksetting. Another recent paper of relevance to the method considered here is [32].To address these goals within a statistical setting, we employ a class of hierarchical models relatedto the Dirichlet process (DP) [2, 12, 13, 23]. The Dirichlet process is a statistical prior that maybe summarized succinctly as follows. Assume that the nth patch is represented by feature vector2
xn , and the total image is composed of N such feature vectors {xn }Nn 1 . The feature vectorassociated with each patch is assumed drawn from a parametric distribution f (φn ), where φnrepresents the parameters associated with the nth feature vector. In a DP-based hierarchicalmodel for generation of the feature vectors we havexn φnind f (φn )iidφn G GG α, G0(1) DP (αG0 )The DP is characterized by the non-negative parameter α and the “base” measure G0 . To makethe clustering properties of (1) more apparent, we recall the stick-breaking construction of DPdeveloped by Sethuraman [31]. Specifically, in [31] it was shown that a draw G from DP (αGo )may be expressed asG Xπh δθhh 1πh Vhh 1Y(1 Vl )(2)l 1iidVh Beta(1, α)iidθh G0This is termed a “stick-breaking” representation of DP because one sequentially breaks off“sticks” of length πh from an original stick of unit length (P h 1πh 1). As a consequenceof the properties of the distribution Beta(1, α), for relatively small α it is likely that only a3
relatively small set of sticks πh will have appreciable weight/size, and therefore when drawingparameters φn from the associated G it is probable multiple φn will share the same “atoms” θh(those associated with the large-amplitude sticks). The parameter α therefore plays an importantrole in defining the number of clusters that are constituted, and therefore in practice one typicallyplaces a non-informative Gamma prior on α [36].The form of the model in (1) and (2) imposes the prior belief that the feature vectors {xn }Nn 1associated with an image should cluster, and the data are used to infer the most probableclustering distribution, via the posterior distribution on the parameters {φn }n 1,N . Such semiparametric clustering has been studied successfully in many settings [11, 12, 30, 36]. However,there are two limitations of such a model, with these defining the focus of this paper. First,while the model in (1) and (2) captures our belief that the feature vectors should cluster,it does not impose our additional belief that the probability that two feature vectors are inthe same cluster should increase as their physical location within the image becomes moreproximate; this is an important factor when one is interested in segmenting an image intocontiguous regions. Secondly, typical semi-parametric clustering has been performed one imageor dataset at a time, and here we wish to cluster multiple images simultaneously, to infer theinter-relationships between clusters in different images, thereby inferring the inter-relationshipsbetween the associated multiple images themselves.As an extension of the DP-based mixture model, we here consider the recently developed kernelstick-breaking process (KSBP) [10], introduced by Dunson and Park. As detailed below, thismodel is similar to that in (2), but now the stick-breaking process is augmented to impose theprior belief associated with spatially proximate patches (that the associated feature vectors are4
more likely to be contained within the same cluster). As the name suggests, a kernel is employedto quantify the closeness of patches spatially, and it is also used to infer a representative setof atoms that capture the variation in the feature vectors {xn }n 1,N . While this is a relativelysophisticated model, its component parts are represented in terms of simple/standard distributions,and therefore inference is relatively efficient. In [10] a Markov chain Monte Carlo (MCMC)sampler was used to estimate the posterior on the model parameters. In the work consideredhere we are interested in relatively large data sets, and therefore we develop an inference enginethat exploits ideas from variational Bayesian analysis [3, 4, 16, 36].There are problems for which one may wish to perform segmentation on multiple imagessimultaneously, with the goal of inferring the inter-relationships between the different images.This is referred to as multi-task learning (MTL) [34, 36, 37], where here each “task” correspondsto clustering feature vectors from a particular image. There are at least three applications of MTLin the context of image analysis: (i) one may have a set of images, some of which are labeled,and others of which are unlabeled, and by performing an MTL analysis on all of the images onemay infer labels for the unlabeled image segmentation, by drawing upon the relationships to thelabeled imagery; (ii) by inferring the inter-relationships between the different images, one maysort the images as well as sort components within the images; (iii) one may identify abnormalimages and locations within an image in an unsupervised manner, by flagging those locationsthat are allocated to a texture that is locally rare, based on a collection of images.As indicated above the DP is a convenient framework for semi-parametric clustering of multiple“tasks”. Moreover, the KSBP algorithm is also in the same family of hierarchical statisticalmodels. Therefore, as presented below, it is convenient to simultaneously cluster/segment multiple5
images by linking the multiple associated KSBP models by an overarching DP. Note that weargued against using the DP for analysis of the individual images because the DP doesn’t imposeour additional belief about the relationship of feature vectors as a function of their location withinthe image. In the same manner, if one had additional information about the multiple images (e.g.,the times and/or locations at which they were measured), this may also be incorporated by using amore-sophisticated model than the DP, to link the task-dependent KSBPs. For example, methodsthat have taken into account the time-dependence of the observed data include [9, 22]. Methodsof this type could be used to replace the DP if spatial or temporal prior knowledge was availablewith regard to the multiple images, but here such is not assumed.Concerning the feature vectors xn , we adopt the independent feature subspace analysis (ISA)method proposed by Hyvarinen and Hoyer [19], which projects the pixel values to a subspaceexpanded by basis vectors, and uses the norm of the projection coefficients as features. The ISAfeatures have proven to be an appealing representation for image analysis, and encouraging resultsare also presented here. However, the basic statistical framework developed here is applicableto general features.The remainder of the paper is organized as follows. In Section II we review the basic kernelstick-breaking process (KSBP) model, and in Section III we extend this to multi-task KSBP(hierarchical KSBP, or H-KSBP) for analysis of multiple images. Example results are presentedin Section IV. Conclusions and future research directions are discussed in Section V.6
II. K ERNEL S TICK -B REAKING P ROCESSA. KSBP prior for image processingThe stick-breaking representation of the Dirichlet process (DP) was summarized in (2), and thishas served as the basis of a number of generalizations of the DP. The limitation of the DP for theimage-processing application of interest is that the clustering or segmentation assumes that thefeature vectors are exchangeable. This means that the location of the features within the imagemay be exchanged arbitrarily, and the DP clustering will not change. It is desirable to exploitthe known spatial position of the features within the image, and impose that it is more likelyfor feature vectors to be clustered together if they are physically proximate (this correspondsto removing the exchangeability assumption in DP). Goals of this type have motivated manymodifications to the DP. The dependent DP (DDP) proposed by MacEachern [22] assumes afixed set of weights, π, while allowing the atoms θ {θ1 , · · · , θN } to vary with the predictorx according to a stochastic process. Alternatively, rather than employing fixed weights, Griffinand Steel [17] and Duan et al. [8] developed models to allow the weights to change with thepredictor. Griffin and Steel’s approach incorporates the dependency by allowing a predictordependent ordering of the weights in the stick-breaking construction, while Duan et al. use amultivariate beta distribution.Dunson and Park [10] have proposed the kernel stick-breaking process (KSBP), which is particularly attractive for image-processing applications. In [10] the KSBP was presented in a moregeneral setting than considered here; the following discussion focuses specifically on the imageprocessing application. Rather than simply considering the feature vectors {xn }n 1,N , we nowconsider {xn , rn }n 1,N , where rn is tied to the location of the pixel or block of pixels used7
to constitute feature vector xn . We let K(r, r 0 , ψ) [0, 1] define a bounded kernel functionwith parameter ψ, where r and r 0 represent general locations in the image of interest. Onemay choose to place a prior on the kernel parameter ψ; this issue is revisited below. A drawGr KSBP (a, b, ψ, Go , H) from a KSBP prior is a function of position r (and valid for allr), and is represented asGr Xπh (r; Vh , Γh , ψ)δθhh 1πh (r; Vh , Γh , ψ) Vh K(r, Γh , ψ)h 1Y[1 Vl K(r, Γl , ψ)]l 1Vh Beta(a, b)Γh Hθh Go(3)Dunson and Park [10] prove the validity of Gr as a probability measure. Comparing (2) and(3), both priors take the general form of a stick-breaking representation, while the KSBP priorpossesses several interesting properties. For example, the stick weights πh (r; Vh , Γh , ψ) are afunction of r. Therefore, although the atoms {θh }h 1, are the same for all r, the weightsπh (r; Vh , Γh , ψ) effectively shift the probabilities of different θh based on r. The Γh serve tolocalize in r regions (clusters) in which the weights πh (r; Vh , Γh , ψ) are relatively constant, withthe size of these regions tied to the kernel parameter ψ.If f (φn ) is the parametric model (with parameter φn ) responsible for the generation of xn , wenow assume that the augmented data {xn , rn }n 1,N are generated as8
xnindφnindGr KSBP (a, b, ψ, Go , H) f (φn ) Grn(4)As (but one) example, f (φn ) may represent a Gaussian distribution, and φn may represent theassociated mean vector and covariance matrix. In this representation the vector Γh defines thelocalized region in the image at which particular atoms θh are likely to be drawn, with theprobability of a given atom tied to πh (r; Vh , Γh , ψ). The generative model in (4) states that twofeature vectors that come from the same region in the image (defined via r) will have similarπh (r; Vh , Γh , ψ), and therefore they are likely to share the same atoms θh . The settings of a andb control how much similarity there will be in drawn atoms for a given spatial cluster centeredabout a particular Γh . If we set a 1 and b α, analogous to the DP, small α will imposethat Vh is likely to be near one, and therefore only a relatively small number of atoms θh arelikely to be dominant for a given cluster spatial center Γh . On the other hand, if two features aregenerated from distant parts of a given image, the associated atoms θh that may be prominentfor each feature vector are likely to be different, and therefore it is of relatively low probabilitythat these feature vectors would have been generated via the same parameters φ. It is possiblethat the model may infer two distinct and widely separated clusters/segments with the similardominant parameters (atoms); if the KSBP base distribution Go is a draw from a DP (as it will bebelow when we consider processing of multiple images), then two distinct and widely separatedclusters/segments may have identical atoms.9
For the case a 1 and b α, which we consider below, we employ the notation Gr KSBP (α, ψ, Go , H); we emphasize that this draw is not meant to be valid for any one r, butfor emphall r. In practice we may also place a non-informative Gamma prior on α. Below wewill also assume that f (φ) corresponds to a multivariate Gaussian distribution, in the mannerindicated above.B. Spatial correlation propertiesAs indicated above, the functional form of the kernel function is important and needs to bechosen carefully. A commonly used kernel is given as K(r, Γ, ψ) exp ( ψkr Γk2 ) forψ 0, which allows the associated stick weight to change continuously from VhQh 1l 1(1 Vl )to 0 conditional on the distance between r and Γ. By choosing a kernel we are also implicitlyimposing the dependency between the priors of two samples, Gr and Gr0 . Specifically, both priorsare encouraged to share the same atoms θh if r and r 0 are close, with this discouraged otherwise.Dunson and Park [10] derive the correlation coefficient between two probability measures Grand Gr0 to beP πh (r; Vh , Γh , ψ)πh (r 0 ; Vh , Γh , ψ)o1/2 n Po1/2 , 20 ; V , Γ , ψ)2π(r;V,Γ,ψ)π(rhhhhh 1 hh 1 hcorr{Gr , Gr0 } nP h 1(5)The coefficient in (5) approaches unity in the limit as r r 0 . Since the correlation is a strongfunction of the kernel parameter ψ, below we will consider a distinct ψh for each Γh , with thesedrawn from a prior distribution. This implies that the spatial extent within the image over whicha given component is important will vary as a function of the location (to accommodate texturalregions of different sizes).10
III. M ULTI -TASK I MAGE S EGMENTATION WITH A H IERARCHICAL KSBPWe now consider the problem for which we wish to jointly segment M images, where eachimage has an associated set of feature vectors with location information, in the sense discussed above. Aggregating the data across the M images, we have the set of feature vectors{xnm , rnm }n 1,Nm ;m 1,M .The image sizes may be different, and therefore the number of featurevectors Nm may vary between images. The premise of the model discussed below is that thecluster or segment characteristics may be similar between multiple images, and the inference ofthese inter-relationships may be of value. Note that the assumption is that sharing of clustersmay be of relevance for the feature vectors xnm , but not for the associated locations rnm (thecharacteristics of feature-vector clusters may be similar between two images, but the associatedclusters may reside in different locations within the respective images).A. ModelA relatively simple means of sharing feature-vector clusters between the different images is tolet each image be processed with a separate KSBP (αm , ψm , Gm , Hm ). To achieve the desiredsharing of feature-vector clusters between the different images, we impose that Gm G and G isdrawn G DP (γ, Go ). Recalling the stick-breaking form of a draw from DP (γ, Go ), we haveG P h 1πh δθh , in the sense summarized in (2). The discrete form of G is very important, for itimplies that the different Gm will share the same set of discrete atoms {θh }h 1, . It is interestingto note that for the case in which the kernel parameter ψ is set such that K(r, Γh , ψ) 1,this model reduces to the hierarchical Dirichlet process (HDP) [33]. Therefore, the principaldifference between the HDP and the hierarchical KSBP (H-KSBP) is that for the latter weimpose that it is probable that feature vectors extracted from proximate regions in the image are11
likely to be clustered together.Assuming that f (φ) corresponds to a Gaussian distribution, the H-KSBP model is representedasxnmindφnmindGrnmiid N (φnm ) Grnm(6) KSBP (αm , ψm , G, Hm )G DP (γ, Go )Assume that G is composed of the atoms {θh }h 1, , from the perspective of the stick-breakingrepresentation in (2). These same atoms are shared across all {Grnm }m 1,M drawn from theassociated KSBPs, but with respective stick weights unique to the different images, and afunction of position within a given image; in (6) we again emphasize that the draw fromKSBP (αm , ψm , G, Hm ) is valid for all r, and we here evaluate it for all {rnm }m 1,M . Thesharing of atoms {θh }h 1, imposes the belief that feature vectors from clusters associated withmultiple images may have been drawn from the same Gaussian distribution N (φ), where φrepresents the mean vector and covariance of the Gaussian; in practice the base distributionGo in (6) corresponds to a normal-Wishart distribution. The posterior inference allows one toinfer which clusters of features are unique to a particular image, and which clusters are sharedbetween multiple images. The density functions Hm are tied to the support of the mth image,and in practice this is set as uniform across the image extent. The distinct αm , for each of whicha Gamma hyper-prior may be imposed, encourages that the number of clusters (segments) may12
vary between the different images, although one may simply wish to set αm α for all Mtasks.For notational convenience, in (6) it was assumed that the kernel parameter ψm varied betweentasks, but was fixed for all sticks within a given task; this is overly restrictive. In the implementation that follows the parameter ψ may vary across tasks and across the task-specific KSBPsticks.B. Posterior inferenceFor inference purposes, we truncate the number of sticks in the KSBP to T , and the number ofsticks in the truncated DP to K (the truncation properties of the stick-breaking representation arediscussed in [20], where here we must also take into account the properties of the kernel). Dueto the discreteness of G PKonly take atoms {φhm }h 1,T ;k 1βk δθk , each draw of the KSBP, Grnm m 1,MPTh 1πhm δφhm , canfrom K unique possible values {θk }k 1,K ; when drawingatoms φhm from G, the respective probabilities for {θk }k 1,K are given by {βk }k 1,K , andfor a given rnm the respective probabilities for different {φhm }h 1,T ;{πhm }h 1,T ;m 1,M .m 1,Mare defined byIn order to reflect the correspondences between the data and atoms explic-itly, we further introduce two auxiliary indicator variables. One is znm , this indicating whichcomponent of the KSBP the feature vector xnm is associated, and the other is thm , this indicatingwhich mixing component θk the atom φhm is associated.With this specification we can represent our H-KSBP mixture model via a stick-breaking characterization13
xnmindφnm φznm mznmind N (φnm ) TXπnm,h δhh 1πnm,h Vhm K(rnm , Γhm , ψhm )h 1Y[1 Vlm K(rnm , Γlm , ψhm )]l 1φhmthm θthmind KX(7)βk δkk 1βk βk0k 1Y(1 βl0 )l 1VhmiidΓhmiidβk0iidθkiid Beta(1, αm ) Hm Beta(1, γ) G0for i 1, · · · , M ; j 1, · · · , Nm ; h 1, · · · , T and k 1, · · · , K. Note that we employ distinctkernel widths ψhm , the details for which are addressed in the Appendix. In the expressions aboveand hereafter we use a bold letter to denote a set of variables with different indices, for exampleπnm {πnm,h }h 1,T and β {βk }k 1,K . With the conditional distributions above, we providea graphical representation of the proposed H-KSBP model in Figure 1.The Markov chain Monte Carlo (MCMC) method is a powerful and general simulation toolfor Bayesian inference, and one may readily implement the H-KSBP with Gibbs sampling,14
JGoEkTkDHVhm*hmrnmt hmzijKTx nm NmMFig. 1.A graphical representation of the H-KSBP mixture model.as an extension of the formulation in [10]. However, for the large-scale problems of interesthere we alternatively employ variational Bayesian (VB) inference. The VB method has provento be relatively fast (compared to MCMC) and accurate inference tool for many models andapplications [1, 4, 24]. To employ VB, a conjugate prior is required for all variables in the model.In the proposed model, we however cannot obtain a closed form for the variational posteriordistribution of the node Vhm , because of the the kernel function. Alternatively, motivated bythe Monte Carlo Expectation Maximization (MCEM) algorithm [35], where the intractable Estep in the Expectation-Maximization (EM) algorithm is approximated with sets of Monte Carlosamples, we develop a Monte Carlo Variational Bayesian (MCVB) inference algorithm. Wegenerally follow the variational Bayesian inference steps to iteratively update the variationaldistributions, and therefore push the lower bound closer to the model evidence. For thosenodes that are not available in closed form, we obtain Monte Carlo samples in each iterationthrough MCMC routines such as Gibbs and Metropolis-Hastings samplers. The resulting MCVBalgorithm combines the benefits of both MCMC and VB, and has proven to be effective for theexamples we have considered (some of which are presented here).15
Given the H-KSBP mixture model detailed in (III-A), we derive a Monte Carlo variationalBayesian approach to infer the variables of interests. Following standard variational Bayesianinference [16], we seek a lower bound for the log model evidence log p(Data) by integratingover all hidden variables and model parameters:Zlog p({xnm }, {rnm }) log p({xnm }, {rnm }, β, θ, V , Γ, t, z)dΘZp({xnm }, {rnm }, β, φ, V , Γ, t, z) q(β, φ, V , Γ, t, z) logdΘ (8)q(β, φ, V , Γ, t, z)Z q(β)q(φ)q(V )q(Γ)q(t)q(z)p({xnm }, {rnm }, β, φ, V , Γ, t, z)dΘq(β)q(φ)q(V )q(Γ)q(t)q(z)³ LB q(Θ)log(9)where Θ is a set including all variables of interest, q(·)’s are variational posterior distributions³ with specified functional form, LB q(Θ) is defined to be the lower bound of the log modelevidence (sometimes referred as negative variational free energy), (8) comes from Jensen’sinequality and (9) results from the independence assumption over all variational distributions.Since the log model evidence is independent of variational distributions, we can iteratively updatethe variational distributions to increase the lower bound, which is equivalent to minimizing theKullback-Leibler distance between q(β, φ, V , Γ, t, z) and p(β, φ, V , Γ, t, z {xij }, {lxij }). Theequality is achieved if and only if the variational distributions are equal to the true posteriordistributions.³ The lower bound, LB q(Θ) , is a functional of the variational distributions. To make thecomputation analytical, we assume the variational distributions take specified forms and iterate16
over every variable until convergence. All the updates are analytical except for Vhm , which isestimated with the samples from its conditional posterior distributions. The update equations forthe proposed model are given in the Appendix.C. Convergence³ To monitor the convergence of our MCVB algorithm, we computed the lower bound LB q(Θ)of log model evidence at each iteration. Because of the sampling of some variables (see Appendix), the lower bound does not in general increase monotonically, but we observed in allexperiments that the lower bound increases sequentially for the first several iterations, withgenerally small fluctuations after it has converged to the local optimal solution. An example ofthis phenomenon is presented below.IV. E XPERIMENTAL R ESULTSWe have applied the H-KSBP multi-task image-segmentation algorithm to both synthetic andreal images. We first present results on synthesized imagery, wherein we compare KSBP-basedclustering of a single image with associated DP-based clustering. This comparison shows ina simple manner the advantage of imposing spatial-proximity constraints when performingclustering and segmentation. We next consider H-KSBP as applied to actual imagery, taken froma widely utilized database. In that analysis we consider the class of textures/clusters inferred bythe algorithm, and how these are distributed across different classes of imagery. We also examinethe utility of H-KSBP for sorting this database of imagery. These H-KSBP results are comparedwith analogous results realized via HDP. The hyper-parameters in the model for the examplesthat follow are set as follows: τ10 1e 2 , τ20 1e 2 , τ30 3e 2 , τ40 3e 2 and µ 0,17
η0 1, w d 2, Σ 5 I. The discrete priors for Γ and ψ are set to be uniform over allcandidates.A. Single-image segmentation, comparing KSBP and DPIn this simple illustrative example, each feature vector is associated with a particular pixel, andthe feature is simply a real number, corresponding to its intensity; the pixel location is theauxiliary information within the KSBP, while this information is not employed by the DP-basedsegmentation algorithm. Since both DP and KSBP are semi-parametric, we do not a priori setthe number of clusters. Figure 2 shows the original image and the segmentation results of bothalgorithms. In Figure 2(a) we note that there are five contiguous regions for which the intensitiesare similar. There is a background region with a relatively fixed intensity, and within this arefour distinct contiguous sub-regions, and of these there are pairs for which the intensities arecomparable. The data in Figure 2(a) were generated as follows. Each pixel in each region isgenerated independently as a draw from a Gaussian distribution; the standard deviation of eachof the Gaussians is 10, and the background has mean intensity 5, and the two pairs are generatedwith mean intensities of 40 and 60. The color bar in Figure 2(a) denotes the pixel amplitudes.The DP and KSBP segmentation results are shown in Figures 2(b) and 2(c), respectively. In thelatter results a distinct color is associated with distinct cluster parameters (mean and precision ofa Gaussian). In the DP results we note that the four subregions are generally properly segmented,but there is significant speckle in the background region. The KSBP segmentation algorithm isbeset by far less speckle. Further, in the KSBP results there are five distinct clusters (dominantKSBP sticks), where in the DP results there are principally three distinct sticks (in the DP,the spatially separated segments with the same features are treated as one region, while in the18
30405060(a)60102030(b)405060102030405060(c)Fig. 2. A synthetic image example. (a) Original synthetic image, (b) image-segmentation results of DP-based model, and (c)image-segmentation results of KSBP-based model.KSBP each contiguous region is represented by its own stick). While the KSBP yields fivedistinct prominent sticks, there are only three distinct atoms, consistent with the representationin Figure 2(a).In the next set of results, on real imagery, we employ the H-KSBP algorithm, and therefore atthe task level segmentation is performed as in Figure 2(c). Alternatively, using HDP model [33],at the task level one employs clustering of the form in Figure 2(b). The relative performance ofH-KSBP and HDP is analyzed.B. Feature extraction for real imageryWithin the subsequent image a
Hierarchical Kernel Stick-Breaking Process for Multi-Task Image Analysis Qi An, Chunping Wang, Ivo Shterev, Eric Wang, yDavid Dunson and Lawrence Carin Department of Electrical and Computer Engineering . to lin
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nmap (Lyon, 2009) and nessus (Lampe, 2005)) enables the penetration tester to select particular exploits pertaining to the specific OS kernel version. One major problem during kernel version identification is the ability to find the location of the base address of the kernel binary in a memory dump. During the process of booting, kernel base
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