PAMPAS: Real-Valued Graphical Models for Computer Vision. Abstract. 1. Introduction. Anonymous

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1 PAMPAS: Real-Value Graphical Moels for Computer Vision Anonymous Abstract Probabilistic moels have been aopte for many computer vision applications, however inference in high-imensional spaces remains problematic. As the state-space of a moel grows, the epenencies between the imensions lea to an exponential growth in computation when performing inference. Many common computer vision problems naturally map onto the graphical moel framework; the representation is a graph where each noe contains a portion of the state-space an there is an ege between two noes only if they are not conitionally inepenent. When this graph is sparsely connecte, belief propagation algorithms can turn an exponential inference computation into one which is linear in the size of the graph. However belief propagation is only applicable when the variables in the noes are iscrete-value or jointly represente by a single multivariate Gaussian istribution, an this rules out many computer vision applications. This paper combines belief propagation with ieas from particle filtering; the resulting algorithm performs inference on graphs containing both cycles an continuousvalue latent variables with general conitional probability istributions. Such graphical moels have wie applicability in the computer vision omain an we test the algorithm on example problems of low-level ege linking an locating jointe structures in clutter. 1. Introuction Probabilistic techniques for moelling uncertainty have foun wiesprea success in computer vision. Their application ranges from pixel-level image generation moels [9, 5] to tracking in high-level parametric spaces [7, 18] however as the size of the state-space grows the curse of imensionality can cause an exponential growth in the computations necessary to perform inference. Many probabilistic vision moels naturally ecompose into a graphical moel structure [10]; basic components become noes in a graph where each noe is conitionally inepenent of all but its near neighbours. When the noes are image patches the neighbours may be spatially close in an image, ajacent levels in a multi-scale representation or nearby time instants in an image sequence. Alternatively many complex objects can be ecompose into graphs where now the noes are subparts of the object an an ege is forme in the graph when two subparts are physically connecte. The avantage of this graphical moel representation is that some formerly exponential inference computations become linear in the size of the graph. Exact inference on graphical moels is possible only in restricte circumstances [10]; the (irecte) graph must be acyclic an the joint istribution over those latent variables which are not iscrete must be a single multivariate Gaussian. When a moel violates these restrictions, Gibbs sampling can be use to raw approximate samples from the joint istribution [6] but for most computer vision applications this technique remains intractable. Approximate inference methos can be use for Conitional Linear Gaussian moels [1] incluing time series []. Recently two methos have been wiely stuie in the computer vision literature to perform approximate inference on more general classes of graph; loopy belief propagation [1] (LBP) can be applie to graphs with cycles, though it still only applies to iscrete or jointly Gaussian ranom variables; an particle filtering [4] allows the use of very general istributions over continuous-value ranom variables but applies only to graphs with a simple linear chain structure. The restriction to Gaussian latent variables is particularly onerous in computer vision applications because clutter in image generation moels invariably leas to highly non-gaussian, multi-moel posterior istributions. This effect largely accounts for the popularity of particle filtering in the computer vision tracking community. In this paper we escribe the PAMPAS algorithm (PArticle Message PASsing) which combines LBP with ieas from particle filtering. It has wie applicability in the computer vision omain, for the first time allowing tractable probabilistic inference over complex moels incluing high-imensional jointe structures an low-level ege chains. We emonstrate the behaviour of the algorithm on test problems incluing illusory-contour fining an locating a jointe structure. Aitionally we argue that an algorithm which performs tractable inference over continuous-value graphical moels is a promising tool for research into unifie computer vision moels implementing a singe probabilistic framework all the way from the pixel level to a semantic escription. We note recent work in the biological vision community which favours a similar architecture to moel human vision. 1

2 . Belief propagation Belief propagation (BP) has been use for many years to perform inference in Bayesian networks [10] an has recently been applie to graphs with cycles uner the name of loopy belief propagation [1]. The metho consists in passing messages between the noes of the graph where each message can be interprete as a conitional probability istribution (cp). When all the latent variables are iscretevalue the cps can be encoe as matrices, an when the continuous-value variables are jointly Gaussian the istributions can be summarise using a mean an covariance matrix. For moels with more complex continuous-value cps the ata representation becomes problematic since as messages are propagate the complexity of the cps grows exponentially [14]. Techniques exist for pruning the representations when the cps an likelihoos are exponentional istributions [1, 15] (e.g. Gaussians), an when the graph is a chain, particle filters aress the issue by aopting a non-parametric representation, the particle set [4]. Particle filtering has been wiely use in computer vision since it is well suite to the complex likelihoo functions which result from natural images. This section begins with a brief escription of stanar BP an goes on to escribe the PAMPAS algorithm which moifies BP to use particle sets as messages an thus permits approximate inference over general continuous-value graphical moels. Suerth et al [19] have inepenently evelope an almost ientical algorithm an a comparison is presente in section Discrete Belief Propagation Belief propagation can in general be state using a pairwise MRF formulation [1]. Consier a set of latent-variable noes X = {X 1,..., X M } paire with a set of observe noes Y = {Y 1,..., Y M }. The conitional inepenence structure of the problem is expresse in the neighbourhoo set P. A pair of inices (i, j) P if the hien noe X j is not conitionally inepenent of X i, an in this case we say X i is a parent of X j an by slight abuse of notation i P(j). For tutorial simplicity, when the X i are iscrete-value ranom variables we assume each takes on some value x i {1,..., L} L. Given a fixe assignment to Y the observe an hien noes are relate by observation functions φ i (x i, y i ) φ i (x i ) : L R. Ajacent hien noes are relate by the correlation functions ψ ij (x j, x i ) : L L R. The joint probability over X is given by P (X ) = 1 Z (i,j) P ( ) ψ ij (x j, x i ) φ i (x i ). (1) i The belief propagation algorithm introuces variables such as m ij (x j ) which can intuitively be unerstoo as a message from hien noe i to hien noe j about what state noe j shoul be in. Messages are compute iteratively using the upate algorithm m ij (x j ) φ i (x i )ψ ij (x i, x j ) m ki (x i ) x i L k P(i)\j () an the marginal istribution of X i (the noe s belief ) is given by b i (x i ) = kφ i (x i ) j P(i) m ji (x i ). (3).. Belief propagation in particle networks When a problem calls for continuous-value ranom variables X i R D then φ i (x i ) : R D R an ψ ij (x i, x j ) : R D R D R an (1) now efines a probability ensity. In the continuous analogues of () an (3) now m ij ( ) an b i ( ) are functions rather than L-element arrays; in fact we will assume that they are normalisable probability ensity functions. When aitionally every ψ ij (x j, x i ) has the form of a cp ψ ij (x j, x i ) p ij (x j x i ) then () becomes m ij (x j ) p ij (x j x i )φ i (x i ) m ki (x i )x i. R D k P(i)\j (4) If the structure of the network is such that the marginal istribution at each noe is Gaussian, this integral can be performe exactly [15]. As note above, if any of the m ij or φ i violate this conitional linear Gaussian structure exact inference is intractable an the m ij must be replace by an approximation; the PAMPAS algorithm uses particle sets. At the heart of any algorithm for belief propagation with particle sets is a Monte-Carlo approximation to the integral in (4). We set out here a general algorithmic framework for performing this approximation an in section.4 escribe the PAMPAS algorithm which is a specific implementation suitable for common computer vision moels. The function p M ij (x i ) 1 Z ij φ i (x i ) k P(i)\j m ki (x i ) (5) is enote the founation of message m ij ( ) where Z ij is a constant of proportionality which turns p M ij ( ) into a probability ensity. A Monte-Carlo integration of (4) yiels m ij, an approximation of m ij, by rawing N samples from the founation s n ij p M ij ( ) (6)

3 an setting m ij (x j ) = 1 N N p ij (x j x i = s n ij). (7) n=1 In general it is possible to pick an importance function g ij (x i ) from which to generate samples with (unnormalise) importance weights s n ij g ij ( ) (8) π n ij p M ij (s n ij)/g(s n ij) (9) in which case the m ij is a weighte mixture m ij (x j ) = 1 N k=1 πk ij N πijp n ij (x j x i = s n ij). (10) n=1 When the marginal istribution over x i is require, samples s n i can be rawn from the estimate belief istribution bi (x i ) = 1 Z i φ i (x i ) j P(i) m ji (x i ) (11) either irectly or by importance sampling as above an these samples can be use to compute expectations over the belief, for example its mean an higher moments. By fixing a set of samples s n i at each noe a Gibbs sampler can be use to generate samples from the joint istribution over the graph an this is escribe in section Choice of importance function Some limitations of the approach in section. can be seen by consiering a simple example. Suppose that we wish to perform inference on a three-noe network (X 1, X, X 3 ) which has the chain structure: P(1) = (), P() = (1, 3), P(3) = (). Two istinct inference chains are being sample: X 1 X X 3 an X 3 X X 1 but the information is not combine in any of the messages; there is no mixing between the forwar an backwar irections. This situation also arises whenever there is a layere structure to the graph with a set of low-level noes passing information up to higher levels an high-level information propagating back own. Of course the belief oes correctly incorporate information from both irections, but the efficiency of a Monte Carlo algorithm is strongly epenent on the sample positions so if possible we woul like to use all the available information when choosing these positions. One solution is to use importance sampling for some of the particles an a natural choice of function is the current belief estimate g(x i ) = b i (x i ) in which case the importance weights are π n ij = 1/ m ji (s n ij). (1) In the context of the simple chain given above this algorithm amounts to smoothing the istribution. Smoothing has been propose for particle filters [8] but previous algorithms merely upate the particle weights in a backwar pass rather than generating new particle positions localise in regions of high belief..4. The PAMPAS algorithm A property of many computer vision moels is that the cps p ij (x j x i ) can be written as a mixture of Gaussians an that the likelihoos φ i (x i ) are complex an ifficult to sample from but can be evaluate up to a multiplicative constant. We now escribe the PAMPAS algorithm which is specialise to perform belief propagation for this type of moel; section.5 inclues a brief sketch of a general algorithm for which these Gaussian restrictions o not apply. For notational simplicity we escribe the case that each cp is a single Gaussian plus a Gaussian outlier process though the extension to mixtures of Gaussians is straightforwar. Setting λ o to be the fixe outlier probability an µ ij an Λ ij the parameters of the outlier process, p ij (x j x i ) =(1 λ o )N (x j ; f ij (x i ), G ij (x i ))+ λ o N (x j ; µ ij, Λ ij ) (13) where f ij ( ) an G ij ( ) are eterministic functions, potentially non-linear, respectively computing the mean an covariance matrix of the conitional Gaussian istribution. Each message approximation m ij is now a mixture of N +1 Gaussians: m ij (x j ) = (1 λo ) N K πijn n (x j ; f ij (s n ij), G ij (s n ij)) k=1 πk ij n=1 + λ o N (x j ; µ ij, Λ ij ) (14) an is summarise by N + 1 triplets: m ij = {(π n ij, s n ij, Λ n ij) : 1 n N + 1}. (15) Note that π N+1 ij, s N+1 ij an Λ N+1 ij correspon to the outlier process an are fixe throughout the algorithm. As inicate in section.3 it makes sense to importancesample some fraction of the message partices from the approximate belief istribution. In fact, because φ i is assume to be har to sample from but easy to evaluate, we will use importance sampling for all the particles but with two istinct importance istributions an corresponing weight 3

4 functions. A fraction (1 ν)n of the particles, by slight abuse of terminology, are enote irect samples: s n ij 1 Zij π n ij = φ i (s n ij) k P(i)\j m ki (x i ) (16) an the remaining νn particles we will refer to as importance samples: s n ij 1 Zij k P(i) m ki (x i ) π n ij = φ i (s n ij)/ m ji (s n ij). (17) Note that if the network is a forwar chain where P(i) i 1 an ν = 0 the algorithm reuces exactly to a stanar form of particle filtering [7]. Both (16) an (17) require sampling from a founation F which is the prouct of D mixtures m = N+1 n=1 π n N ( ; µ n, Λ n ). (18) This prouct F is a Gaussian mixture with O(N D ) componenents so irect sampling rapily becomes infeasible for large N or D. Suerth et al [19] propose a Gibbs sampler which performs approximate sampling from F in O(DN) operations per sample which is escribe next. Each component F L of F is a prouct of D Gaussians, one rawn from each of the m an is inexe by a label vector L = (l 1,..., l D ), so F ( ) = 1 Z η L N ( ; µ L, Λ L ) (19) L I where from the stanar Gaussian prouct formula Λ 1 L D = (Λ l =1 η L = ) 1 Λ 1 L µ L = D (Λ l ) 1 µ l (0) =1 D =1 ηl N ( ; µ l, Λ l N ( ; µ L, Λ L ) ). (1) A sample is rawn from F by first choosing a label vector L using a Gibbs sampler an then generating a ranom sample from the Gaussian F L. The Gibbs sampler works by sampling l with all the other labels in L hel fixe for each in turn. With all of L but l fixe the marginal prouctcomponent weights are given by η n πn N ( ; µn ; Λn ) N ( ; µ L n, Λ L n) () where L n = (l 1,..., l 1, n, l +1,..., l D ); the full Gibbs sampler algorithm is given in figure 1. Note that the normal To generate a sample from the prouct of D mixtures m : 1. Initialise L. For 1 N + 1: (a) Choose l uniformly from {1,..., N + 1}.. Iterate: (a) Choose uniformly from {1,..., D}. (b) For 1 n N + 1: i. Set L n = (l 1,..., l 1, n, l +1,..., l D ). ii. Compute Λ L n an µ L n from (0). iii. Compute η n from (). 3. Sample a value n where P (l = n) η n. 4. Upate L = (l 1,..., l 1, n, l +1,..., l D ). Figure 1: Gibbs sampling a prouct of mixtures. istributions in (1) an () can be evaluate at any convenient point, e.g. the origin. Generating N samples using the Gibbs sampler takes O(DN ) operations so for D < 3 it is preferable to compute the exact prouct istribution since a stratifie sampler can then be use [3]. As an implementation etail, we fin that the Gibbs sampler for s n ij can be noisy unless run for many iterations, in the sense that it regularly generates particles which have very low probability. This can lea to problems when importance sampling since m ji (s n ij ) in (17) may be very small leaing to a high weight πij n which suppresses the other samples in the set. This may suggest an aaptive sampling algorithm which continues to iterate the Gibbs sampler until the variance of the πij n falls below some threshol value. In practice, in experiments below we simply smooth the πij n by replacing any πn ij > 0.5ν by πij n (ν πij)/6 n (3) an renormalising. The full PAMPAS message upate algorithm is given in figure. When N belief samples s n i have been rawn for each noe in X they can be use to sample from the joint istribution over the graph. This also relies on a Gibbs sampler but now the label vector L = (l 1,..., l M ) inexes a sample s lm m at each of the M noes an the marginal probability of choosing label l when the other labels are fixe is P (l = n) = φ (s n i ) p ji (s n i x j = s l j j ). (4) j P(i).5. Comparison with the NBP algorithm Suerth et al [19] have inepenently evelope an almost ientical algorithm for performing belief propagation with 4

5 Algorithm to upate message m ij. The algorithm for sampling from an incoming message prouct is given in figure Draw samples from the incoming message prouct. (a) For 1 n (1 ν)n: i. Draw s n ij k P(i)\j m ki( ). ii. Set π n ij = 1/N. (b) For (1 ν)n < n N: i. Draw s n ij k P(i) m ki( ). ii. Set γ n ij = 1/ m ji( s n ij ). (c) For (1 ν)n < n N: i. Set π n ij = νγn ij / N k=1+(1 ν)n γk ij. () Smooth importance weights using (3) if necessary.. Apply importance correction from likelihoo. For 1 n N: (a) Set π n ij = πn ij φ i( s n ij ) 3. Store normalise weights an mixture components. For 1 n N: (a) Set πij n = (1 πn+1 ij ) π ij n / N k=1 πk ij (b) Set s n ij = f ij( s n ij ). (c) Set Λ n ij = G ij( s n ij ). Figure : The PAMPAS message upate algorithm. the ai of particle sets, which they call Non-Parametric Belief Propagation, or NBP. An innovation of theirs which we have aopte is the use of a Gibbs sampler to generate approximate samples from a prouct of mixture istributions. The alternative woul be to approximate the prouct with an importance istribution g( ) perhaps chosen using a variational metho. This woul also be possible if a cp were not well represente by mixtures of Gaussians. The main remaining ifference between the algorithms is the way in which the message founations are interprete. In [19] N samples r n ij are generate from pm ij (x i) an then for each r n ij a single sample sn ij is generate using the conitional istribution s n ij p ij( x i = r n ij ). The message mixture m ij (x j ) is then forme by placing ientical iagonalcovariance Gaussian kernels about the s n ij. This aitional smoothing kernel leas to variance estimates which are biase upwars of their true values. This step is unnecessary when the cps are mixtures of Gaussians so we omit it; thus we achieve unbiase kernel estimates as well as generating less noisy samples from the message istribution, an in aition we o not have the problem of selecting a kernel with. The focus of [19] is also quite ifferent so they o not concentrate on importance sampling or inclue the Gibbs sampler for sampling the joint belief over the graph. 3. Results 3.1. Ege linking moels We exercise the algorithm using a simple ege moel. In these experiments each noe in the graph correspons to an egel x i = (x i, y i, α i, λ i ) R 4 where (x i, y i ) is an image coorinate, α i is an orientation angle an λ i is the istance between ajacent egels. We use a simple iagonal Gaussian conitional istribution moel for j = i+1 where p ij (x j x i ) = N (x j ; x i + λ i cos α i, σ p) N (y j ; y i + λ i sin α i, σ p) N (α j ; α i, σ α) N (λ j ; λ i, σ λ) (5) an the moel for j = i 1 is the same but in reverse. A similar moel has been use for pixel-base illusory contour completion [1, 0] as well as fining segmentation eges using particle filtering [17], an if the φ i are Gaussian it is very similar to the stanar snake moel [11]. Figure 3 emonstrates the effectiveness of importance sampling using the belief istribution. The top images shows messages an the bottom beliefs, an the φ i are uniform to emphasise the sampling behaviour in the absence of image ata. The left column shows a run of the PAMPAS algorithm with no importance sampling (ν = 0) on a 16-link chain where each en of the chain is initialise from a ifferent prior istribution. Since there is no interaction between the forwar an backwar sample positions the particles sprea away from each en in a ranom walk, proucing the noisy belief istribution shown below. The remaining images show 5 consecutive iterations of the algorithm with the same parameters but now using importance sampling with ν = 0.5. The message particles become concentrate in areas of high belief making the belief estimates below successively less noisy. In the absence of complex image measurements the above moel is Gaussian an so the belief coul be estimate by a variety of methos. The cluttere image ata use in figure 4 causes the belief istribution to become highly non-gaussian; the algorithm converges on a globally plausible solution espite strong local istractors. Here the φ i use are simple robust graient-affinity operators φ i (x i ) = η + (1 η) x (x i, y i )sin(α i ) y (x i, y i )cos(α i ) (6) 5

6 Figure 3: Importance sampling mixes information from forwar an backwar chains. The top images show messages passe forwars an backwars along a simple chain, in re an green respectively. The bottom images show the resulting beliefs. The left han column shows the output when no importance sampling is use an the forwar an backwar messages o not interact. When importance sampling is use with ν = 0.5 the message particles are guie to areas of high belief; the right han columns show 5 successive iterations of the algorithm. where x ( ) an y ( ) are respectively horizontal an vertical image graient functions (Sobel operators) an η = 0.1 is the probability that no ege is visible. The incorporation of information along the chain in both irections can be expecte to improve the performance of e.g. the Jetstream algorithm [17] which is a particle-base ege finer. 3.. Illusory contour matching We teste the algorithm on a illusory-contour fining application shown in figure 5. The input image has C pacman like corner points where the open mouth efines two ege irections leaing away from the corner separate by an angle Cα. The task is to match up the eges with smooth contours. We construct C chains each of four links, an each is anchore at one en with a prior starting at its esignate corner pointing in one of the two ege irections. At the untethere en of a chain from corner C is a connector egel which is attache to (conitionally epenent on) the connectors at the ens of the C other chains not starting from corner C. We use the same egel moel as in section 3.1 but now each noe state-vector xi is augmente with a iscrete label υi enoting which of the C partner chains it matches, an the cps at the are moifie so that pij (xj xi ) = 0 when υi an υj o not match. Figure 5 shows the moel being applie to two scenarios with C = 6 in both cases. First, Cα = 11π/36 which leas to two overlapping triangles with slightly concave eges when the chains are linke by the algorithm. Seconly, Cα = 7π/9 which leas to a slightly convex regular hexagon. The same parameters are use for the moels in both cases, so the algorithm is force to aapt the egel lengths to cope with the iffering istances between target Figure 4: An ege linking moel fins global solutions. The images show (in raster orer) the belief estimates after 1, 5, an 10 iterations, an 7 samples from the joint istribution after 10 iterations. The ens of the chain are sample from prior istributions centre on the mile-left an mileright of the image respectively. The istractors in the top half of the image have stronger eges an are closer to the fixe prior than the continuous ege below. After a few iterations the belief has stabilise on a bimoal istribution. As is clear from the samples rawn from the joint graph, most of the weight of the istribution is on the continuous ege. 6

7 corners. In the other irection the arm process is not Gaussian but we approximate it with the following: x i = x j h j cos(α i ) + N ( ; 0, h j σ s sin((i )π/) σp) y i = y j h j sin(α i ) + N ( ; 0, h j σ s cos((i )π/) σp) α i = α j + N ( ; 0, σα) w i = w j + N ( ; 0, σs) h i = h j + N ( ; 0, σs). The p i1 (x 1 x i ) are straightforwar: Figure 5: The PAMPAS algorithm performs illusory contour completion. A moel consisting of twelve 4-link chains tries to link pairs of chains to form smooth contours. Depening on the angle at which the chains leave the corner positions, the belief converges on either two overlapping triangles (left) or a single hexagon (right). Links inclue a iscrete label etermining which partner chain they are attache to; the isplaye link colour is the colour of the partner chain s corner. See text for more etails Fining a jointe object in clutter Finally we constructe a graph representing a jointe object an applie the PAMPAS algorithm to locating the object in a cluttere scene. The moel consists of 9 noes; a central circle with four jointe arms each mae up of two rectangular links. The circle noe x 1 = (x 0, y 0, r 0 ) encoes a position an raius. Each arm noe x i = (x i, y i, α i, w i, h i ), i 9 encoes a position, angle, with an height an prefers one of the four compass irections (to break symmetry). The arms pivot aroun their inner joints, so for the inner arms i 5, p 1i (x i x 1 ) encoes the process x i = x 1 + N ( ; 0, σ p) y i = y 1 + N ( ; 0, σ p) α i = (i )π/ + N ( ; 0, σ β) w i = r 1 δ w + N ( ; 0, σ s) h i = r 1 δ h + N ( ; 0, σ s). The cp to go from an inner arm i 5 to the outer arms j = i + 4 is given by x j = x i + h i cos(α i ) + N ( ; 0, σ p) y j = y i + h i sin(α i ) + N ( ; 0, σ p) α j = α i + N ( ; 0, σ α) w j = w i + N ( ; 0, σ s) h j = h i + N ( ; 0, σ s). x 1 = x i + N ( ; 0, σ p) y 1 = y i + N ( ; 0, σ p) r 1 = 0.5(w i /δ w + h i /δ h ) + N ( ; 0, σ s). The object has been place in image clutter in figure 6. The clutter is mae up of 1 circles an 100 rectangles place ranomly in the image. The messages are initialise with a simulate specialise feature etector: x 1 is sample from positions near the circles in the image an the arms are sample to be near the rectangles, with rectangles closer to the preferre irection of each arm sample more frequently. Each noe contains N = 75 particles so the space is unersample. After initialisation the PAMPAS algorithm is iterate using a single Gaussian outlier covering the whole image (i.e. the feature etector information is not use after initialisation). After two or three iterations the belief istribution has converge on the object. In this case it might be argue that it woul be more straightforwar to simply etect the circles an perform an exhaustive search near each. Figure 7 emonstrates the power of the proba- Figure 6: A jointe object is locate in clutter. The left han image shows a sample from the belief istribution overlai on a cluttere scene. The right han image shows the full belief istribution after 10 iterations. See text for more etails. bilistic moelling approach, however, since now the circle at the centre of the object has not been isplaye, simulating occlusion. Of course no x 1 samples are generate near the occlue circle, but even after one iteration the belief at that noe has high weight at the correct location ue to the agreement of the nearby arm noes. After a few iterations the belief has converge on the correct object position espite the lack of the central circle. 7

8 Figure 7: An occlue object is locate in clutter. The left han image shows a sample from the belief istribution overlai on a cluttere scene. Note that there is no circle present in the image where the belief has place it. This moels an occlue object which has still been correctly locate. The right han image shows the full belief istribution after 10 iterations. See text for more etails. 4. Conclusion We have concentrate in this paper on examples with synthetic ata in orer to highlight the working of the PAMPAS algorithm. The applications we have chosen are highly suggestive of real-worl problems, however, an we hope the promising results will stimulate further research into applying continuous-value graphical moels to a variety of computer vision problems. Sections 3.1 an 3. suggest a variety of ege completion applications. Section 3.3 inicates that the PAMPAS algorithm may be effective in locating articulate structures in images, in particular people. Person-fining algorithms exist alreay for locating an grouping subparts [16] but they o not naturally fit into a probalistic framework. One obvious avantage of using a graphical moel representation is that it extens straightforwarly to tracking, simply by increasing the size of the graph an linking noes in ajacent timesteps. Another area of great interest is linking the continuous-value graph noes to lower-level noes which may be iscrete-value as in [9]. In fact, researchers in the biological vision omain have recently propose a hierarchical probabilistic structure to moel human vision which maps very closely onto our algorithmic framework [13]. References of Statistics, University of Oxfor, Available from cliffor/inex.html. [4] A. Doucet, N. e Freitas, an N. Goron, eitors. Sequential Monte Carlo Methos in Practice. Springer-Verlag, 001. [5] W.T. Freeman, T.R. Jones, an E.C. Pasztor. Example-base superresolution. IEEE Computer Graphics an Applications, March/April 00. [6] S. Geman an D. Geman. Stochastic relaxation, Gibbs istributions, an the Bayesian restoration of images. IEEE Trans. on Pattern Analysis an Machine Intelligence, 6(6):71 741, [7] M. Isar an A. Blake. Conensation conitional ensity propagation for visual tracking. Int. J. Computer Vision, 8(1):5 8, [8] M.A. Isar an A. Blake. A smoothing filter for Conensation. In Proc. 5th European Conf. Computer Vision, volume 1, pages , [9] N. Jojic, N. Petrovic, B. J. Frey, an T. S. Huang. Transforme hien markov moels: Estimating mixture moels of images an inferring spatial transformations in vieo sequences. In cvpr, 000. [10] M.I. Joran, T.J. Sejnowski, an T. Poggio, eitors. Graphical Moels : Founations of Neural Computation. MIT Press, 001. [11] M. Kass, A. Witkin, an D. Terzopoulos. Snakes: Active contour moels. In Proc. 1st Int. Conf. on Computer Vision, pages 59 68, [1] D. Koller, U. Lerner, an D. Angelov. A general algorithm for approximate inference an its application to hybri bayes nets. In Proceeings of the 15th Annual Conference on Uncertainty in Artificial Intelligence, pages , [13] T.S. Lee an D. Mumfor. Hierarchical bayesian inference in the visual cortex. Journal of the Optical Society of America A, 00 (submitte). [14] U. Lerner. Hybri Bayesian Networks for Reasoning about Complex Systems. PhD thesis, Stanfor University, October 00. [15] T.P. Minka. A family of algorithms for approximate Bayesian inference. PhD thesis, MIT, January 001. [16] G. Mori an J. Malik. Estimating human boy configurations using shape context matching. In eccv7, pages , 00. [17] P. Perez, A. Blake, an M. Gangnet. Jetstream: Probabilistic contour extraction with particles. In iccv8, volume II, pages , 001. [18] H. Sienblah, M.J. Black, an L. Sigal. Implicit probabilistic moels of human motion for synthesis an tracking. In eccv7, volume 1, pages , 00. [19] E.B. Suerth, A.T. Ihler, W.T. Freeman, an A.S. Willsky. Nonparametric belief propagation. Technical Report P-551, MIT Laboratory for Information an Decision Systems, 00. [0] L.R. Williams an D.W. Jacobs. Stochastic completion fiels: A neural moel of illusory contour shape an salience. Neural Computation, 9(4): , [1] J.S. Yeiia, W.T. Freeman, an Y. Weiss. Unerstaning belief propagation an its generalizations. Technical Report TR001-, MERL, 001. [1] J. August an S.W. Zucker. The curve inicator ranom fiel: curve organization via ege correlation. In Perceptual Organization for Artificial Vision Systems. Kluwer Acaemic, 000. [] X. Boyen an D. Koller. Tractable inference for complex stochastic processes. In Proceeings of the 14th Annual Conference on Uncertainty in Artificial Intelligence, [3] J. Carpenter, P. Cliffor, an P. Fearnhea. An improve particle filter for non-linear problems. Technical report, Dept. 8