{"title": "Distributed Occlusion Reasoning for Tracking with Nonparametric Belief Propagation", "book": "Advances in Neural Information Processing Systems", "page_first": 1369, "page_last": 1376, "abstract": null, "full_text": " Distributed Occlusion Reasoning for Tracking\n with Nonparametric Belief Propagation\n\n\n\n Erik B. Sudderth, Michael I. Mandel, William T. Freeman, and Alan S. Willsky\n Department of Electrical Engineering and Computer Science\n Massachusetts Institute of Technology\n esuddert@mit.edu, mim@alum.mit.edu, billf@mit.edu, willsky@mit.edu\n\n\n\n\n Abstract\n\n We describe a threedimensional geometric hand model suitable for vi-\n sual tracking applications. The kinematic constraints implied by the\n model's joints have a probabilistic structure which is well described by\n a graphical model. Inference in this model is complicated by the hand's\n many degrees of freedom, as well as multimodal likelihoods caused by\n ambiguous image measurements. We use nonparametric belief propaga-\n tion (NBP) to develop a tracking algorithm which exploits the graph's\n structure to control complexity, while avoiding costly discretization.\n While kinematic constraints naturally have a local structure, self\n occlusions created by the imaging process lead to complex interpenden-\n cies in color and edgebased likelihood functions. However, we show\n that local structure may be recovered by introducing binary hidden vari-\n ables describing the occlusion state of each pixel. We augment the NBP\n algorithm to infer these occlusion variables in a distributed fashion, and\n then analytically marginalize over them to produce hand position esti-\n mates which properly account for occlusion events. We provide simula-\n tions showing that NBP may be used to refine inaccurate model initializa-\n tions, as well as track hand motion through extended image sequences.\n\n\n1 Introduction\n\nAccurate visual detection and tracking of threedimensional articulated objects is a chal-\nlenging problem with applications in humancomputer interfaces, motion capture, and\nscene understanding [1]. In this paper, we develop a probabilistic method for tracking a\ngeometric hand model from monocular image sequences. Because articulated hand mod-\nels have many (roughly 26) degrees of freedom, exact representation of the posterior dis-\ntribution over model configurations is intractable. Trackers based on extended and un-\nscented Kalman filters [2, 3] have difficulties with the multimodal uncertainties produced\nby ambiguous image evidence. This has motived many researchers to consider nonparamet-\nric representations, including particle filters [4, 5] and deterministic multiscale discretiza-\ntions [6]. However, the hand's high dimensionality can cause these trackers to suffer catas-\ntrophic failures, requiring the use of models which limit the hand's motion [4] or sophisti-\ncated prior models of hand configurations and dynamics [5, 6].\n\nAn alternative way to address the high dimensionality of articulated tracking problems is\nto describe the posterior distribution's statistical structure using a graphical model. Graph-\n\n\f\nFigure 1: Projected edges (left block) and silhouettes (right block) for a configuration of the 3D\nstructural hand model matching the given image. To aid visualization, the model is also projected\nfollowing rotations by 35 (center) and 70 (right) about the vertical axis.\n\n\nical models have been used to track viewbased human body representations [7], con-\ntour models of restricted hand configurations [8], viewbased 2.5D \"cardboard\" models\nof hands and people [9], and a full 3D kinematic human body model [10]. Because the\nvariables in these graphical models are continuous, and discretization is intractable for\nthreedimensional models, most traditional graphical inference algorithms are inapplica-\nble. Instead, these trackers are based on recently proposed extensions of particle filters\nto general graphs: mean field Monte Carlo in [9], and nonparametric belief propagation\n(NBP) [11, 12] in [10].\n\nIn this paper, we show that NBP may be used to track a threedimensional geometric model\nof the hand. To derive a graphical model for the tracking problem, we consider a redun-\ndant local representation in which each hand component is described by its own three\ndimensional position and orientation. We show that the model's kinematic constraints,\nincluding selfintersection constraints not captured by joint angle representations, take a\nsimple form in this local representation. We also provide a local decomposition of the\nlikelihood function which properly handles occlusion in a distributed fashion, a significant\nimprovement over our earlier tracking results [13]. We conclude with simulations demon-\nstrating our algorithm's robustness to occlusions.\n\n\n2 Geometric Hand Modeling\n\nStructurally, the hand is composed of sixteen approximately rigid components: three pha-\nlanges or links for each finger and thumb, as well as the palm [1]. As proposed by [2, 3],\nwe model each rigid body by one or more truncated quadrics (ellipsoids, cones, and cylin-\nders) of fixed size. These geometric primitives are well matched to the true geometry of\nthe hand, allow tracking from arbitrary orientations (in contrast to 2.5D \"cardboard\" mod-\nels [5, 9]), and permit efficient computation of projected boundaries and silhouettes [3].\nFigure 1 shows the edges and silhouettes corresponding to a sample hand model configu-\nration. Note that only a coarse model of the hand's geometry is necessary for tracking.\n\n\n2.1 Kinematic Representation and Constraints\n\nThe kinematic constraints between different hand model components are well described\nby revolute joints [1]. Figure 2(a) shows a graph describing this kinematic structure, in\nwhich nodes correspond to rigid bodies and edges to joints. The two joints connecting the\nphalanges of each finger and thumb have a single rotational degree of freedom, while the\njoints connecting the base of each finger to the palm have two degrees of freedom (cor-\nresponding to grasping and spreading motions). These twenty angles, combined with the\npalm's global position and orientation, provide 26 degrees of freedom. Forward kinematic\ntransformations may be used to determine the finger positions corresponding to a given set\nof joint angles. While most modelbased hand trackers use this joint angle parameteriza-\ntion, we instead explore a redundant representation in which the ith rigid body is described\nby its position qi and orientation ri (a unit quaternion). Let xi = (qi, ri) denote this local\ndescription of each component, and x = {x1, . . . , x16} the overall hand configuration.\n\nClearly, there are dependencies among the elements of x implied by the kinematic con-\n\n\f\n (a) (b) (c) (d)\n\nFigure 2: Graphs describing the hand model's constraints. (a) Kinematic constraints (EK ) de-\nrived from revolute joints. (b) Structural constraints (ES) preventing 3D component intersections.\n(c) Dynamics relating two consecutive time steps. (d) Occlusion consistency constraints (EO).\n\n\nstraints. Let EK be the set of all pairs of rigid bodies which are connected by joints, or\nequivalently the edges in the kinematic graph of Fig. 2(a). For each joint (i, j) EK ,\ndefine an indicator function K (x\n i,j i, xj ) which is equal to one if the pair (xi, xj ) are valid\nrigid body configurations associated with some setting of the angles of joint (i, j), and zero\notherwise. Viewing the component configurations xi as random variables, the following\nprior explicitly enforces all constraints implied by the original joint angle representation:\n\n pK(x) K (x\n i,j i, xj ) (1)\n (i,j)EK\n\nEquation (1) shows that pK (x) is an undirected graphical model, whose Markov structure\nis described by the graph representing the hand's kinematic structure (Fig. 2(a)).\n\n\n2.2 Structural and Temporal Constraints\n\nIn reality, the hand's joint angles are coupled because different fingers can never occupy\nthe same physical volume. This constraint is complex in a joint angle parameterization, but\nsimple in our local representation: the position and orientation of every pair of rigid bodies\nmust be such that their component quadric surfaces do not intersect.\n\nWe approximate this ideal constraint in two ways. First, we only explicitly constrain those\npairs of rigid bodies which are most likely to intersect, corresponding to the edges ES of the\ngraph in Fig. 2(b). Furthermore, because the relative orientations of each finger's quadrics\nare implicitly constrained by the kinematic prior pK (x), we may detect most intersections\nbased on the distance between object centroids. The structural prior is then given by\n\n 1 ||q\n p i - qj || > i,j\n S (x) S (x (x\n i,j i, xj ) S\n i,j i, xj ) = (2)\n 0 otherwise\n (i,j)ES\n\nwhere i,j is determined from the quadrics composing rigid bodies i and j. Empirically, we\nfind that this constraint helps prevent different fingers from tracking the same image data.\n\nIn order to track hand motion, we must model the hand's dynamics. Let xt denote the\n i\nposition and orientation of the ith hand component at time t, and xt = {xt1, . . . , xt16}. For\neach component at time t, our dynamical model adds a Gaussian potential connecting it to\nthe corresponding component at the previous time step (see Fig. 2(c)):\n 16\n pT xt | xt-1 = N xt - xt-1; 0, \n i i i (3)\n i=1\nAlthough this temporal model is factorized, the kinematic constraints at the following time\nstep implicitly couple the corresponding random walks. These dynamics can be justified as\nthe maximum entropy model given observations of the nodes' marginal variances i.\n\n\f\n3 Observation Model\n\nSkin colored pixels have predictable statistics, which we model using a histogram distribu-\ntion pskin estimated from training patches [14]. Images without people were used to create\na histogram model pbkgd of nonskin pixels. Let (x) denote the silhouette of projected\nhand configuration x. Then, assuming pixels are independent, an image y has likelihood\n p\n p skin(u)\n C (y | x) = pskin(u) pbkgd(v) (4)\n pbkgd(u)\n u(x) v\\(x) u(x)\n\nThe final expression neglects the proportionality constant p\n v bkgd(v), which is inde-\npendent of x, and thereby limits computation to the silhouette region [8].\n\n\n3.1 Distributed Occlusion Reasoning\n\nIn configurations where there is no selfocclusion, pC (y | x) decomposes as a product of\nlocal likelihood terms involving the projections (xi) of individual hand components [13].\nTo allow a similar decomposition (and hence distributed inference) when there is occlu-\nsion, we augment the configuration xi of each node with a set of binary hidden variables\nzi = {zi } = 0 if pixel u in the projection of rigid body i is occluded\n (u) u. Letting zi(u)\nby any other body, and 1 otherwise, the color likelihood (eq. (4)) may be rewritten as\n 16 p z 16\n i(u)\n p skin(u)\n C (y | x, z) = = p\n p C (y | xi, zi) (5)\n bkgd(u)\n i=1 u(xi) i=1\n\nAssuming they are set consistently with the hand configuration x, the hidden occlusion\nvariables z ensure that the likelihood of each pixel in (x) is counted exactly once.\n\nWe may enforce consistency of the occlusion variables using the following function:\n 0 if x = 1\n (x j occludes xi, u (xj ), and zi(u)\n j , zi ; x (6)\n (u) i) = 1 otherwise\n\nNote that because our rigid bodies are convex and nonintersecting, they can never take\nmutually occluding configurations. The constraint (xj, zi ; x\n (u) i) is zero precisely when\npixel u in the projection of xi should be occluded by xj, but zi is in the unoccluded state.\n (u)\nThe following potential encodes all of the occlusion relationships between nodes i and j:\n\n O (x (x ; x ; x\n i,j i, zi, xj , zj ) = j , zi(u) i) (xi, zj(u) j ) (7)\n u\nThese occlusion constraints exist between all pairs\nof nodes. As with the structural prior, we enforce\nonly those pairs EO (see Fig. 2(d)) most prone to xj\nocclusion:\n pO(x, z) O (x\n i,j i, zi, xj , zj ) (8)\n (i,j)EO z y\n i(u) xi\nFigure 3 shows a factor graph for the occlusion\nrelationships between xi and its neighbors, as\nwell as the observation potential pC (y | xi, zi). x u \n k \nThe occlusion potential (xj, zi ; x\n (u) i) has a very Figure 3: Factor graph showing\nweak dependence on xi, depending only on\n p(y | xi, zi), and the occlusion con-\nwhether xi is behind xj relative to the camera. straints placed on xi by xj , xk. Dashed\n lines denote weak dependencies. The\n3.2 Modeling Edge Filter Responses plate is replicated once per pixel.\n\nEdges provide another important hand tracking\ncue. Using boundaries labeled in training images,\nwe estimated a histogram pon of the response of a derivative of Gaussian filter steered to\nthe edge's orientation [8, 10]. A similar histogram poff was estimated for filter outputs at\n\n\f\nrandomly chosen locations. Let (x) denote the oriented edges in the projection of model\nconfiguration x. Then, again assuming pixel independence, image y has edge likelihood\n p 16 p z 16\n i(u)\n p on(u) on(u)\n E (y | x, z) = = p\n p E (y | xi, zi) (9)\n off (u) poff(u)\n u(x) i=1 u(xi) i=1\n\nwhere we have used the same occlusion variables z to allow a local decomposition.\n\n\n4 Nonparametric Belief Propagation\n\nOver the previous sections, we have shown that a redundant, local representation of the\ngeometric hand model's configuration xt allows p (xt | yt), the posterior distribution of the\nhand model at time t given image observations yt, to be written as\n 16\n p xt | yt pK(xt)pS(xt)pO(xt, zt) pC(yt | xt, zt)p , zt)\n i i E (yt | xti i (10)\n zt i=1\n\nThe summation marginalizes over the hidden occlusion variables zt, which were needed to\nlocally decompose the edge and color likelihoods. When video frames are observed, the\noverall posterior distribution is given by\n \n\n p (x | y) p xt | yt pT (xt | xt-1) (11)\n t=1\nExcluding the potentials involving occlusion variables, which we discuss in detail in\nSec. 4.2, eq. (11) is an example of a pairwise Markov random field:\n\n p (x | y) i,j (xi, xj) i (xi, y) (12)\n (i,j)E iV\n\nHand tracking can thus be posed as inference in a graphical model, a problem we propose to\nsolve using belief propagation (BP) [15]. At each BP iteration, some node i V calculates\na message m (x\n ij j ) to be sent to a neighbor j (i) {j | (i, j) E}:\n\n mn (x mn-1 (x\n ij j ) j,i (xj , xi) i (xi, y) ki i) dxi (13)\n xi k(i)\\j\n\nAt any iteration, each node can produce an approximation ^\n p(xi | y) to the marginal distri-\nbution p (xi | y) by combining the incoming messages with the local observation:\n\n ^\n pn(xi | y) i (xi, yi) mn (x\n ji i) (14)\n j(i)\n\nFor treestructured graphs, the beliefs ^\n pn(xi | y) will converge to the true marginals\np (xi | y). On graphs with cycles, BP is approximate but often highly accurate [15].\n\n4.1 Nonparametric Representations\n\nFor the hand tracking problem, the rigid body configurations xi are sixdimensional con-\ntinuous variables, making accurate discretization intractable. Instead, we employ nonpara-\nmetric, particlebased approximations to these messages using the nonparametric belief\npropagation (NBP) algorithm [11, 12]. In NBP, each message is represented using either a\nsamplebased density estimate (a mixture of Gaussians) or an analytic function. Both types\nof messages are needed for hand tracking, as we discuss below. Each NBP message update\ninvolves two stages: sampling from the estimated marginal, followed by Monte Carlo ap-\nproximation of the outgoing message. For the general form of these updates, see [11]; the\nfollowing sections focus on the details of the hand tracking implementation.\n\nThe hand tracking application is complicated by the fact that the orientation component ri\nof xi = (qi, ri) is an element of the rotation group SO(3). Following [10], we represent\n\n\f\norientations as unit quaternions, and use a linearized approximation when constructing den-\nsity estimates, projecting samples back to the unit sphere as necessary. This approximation\nis most appropriate for densities with tightly concentrated rotational components.\n\n\n4.2 Marginal Computation\n\nBP's estimate of the belief ^\n p(xi | y) is equal to the product of the incoming messages from\nneighboring nodes with the local observation potential (see eq. (14)). NBP approximates\nthis product using importance sampling, as detailed in [13] for cases where there is no\nselfocclusion. First, M samples are drawn from the product of the incoming kinematic\nand temporal messages, which are Gaussian mixtures. We use a recently proposed multi-\nscale Gibbs sampler [16] to efficiently draw accurate (albeit approximate) samples, while\navoiding the exponential cost associated with direct sampling (a product of d M Gaussian\nmixtures contains M d Gaussians). Following normalization of the rotational component,\neach sample is assigned a weight equal to the product of the color and edge likelihoods\nwith any structural messages. Finally, the computationally efficient \"rule of thumb\" heuris-\ntic [17] is used to set the bandwidth of Gaussian kernels placed around each sample.\n\nTo derive BP updates for the occlusion masks zi, we first cluster (xi, zi) for each hand\ncomponent so that p (xt, zt | yt) has a pairwise form (as in eq. (12)). In principle, NBP\ncould manage occlusion constraints by sampling candidate occlusion masks zi along with\nrigid body configurations xi. However, due to the exponentially large number of possible\nocclusion masks, we employ a more efficient analytic approximation.\n\nConsider the BP message sent from xj to (zi, xi), calculated by applying eq. (13) to the\nocclusion potential (x ; x\n u j , zi(u) i). We assume that ^\n p(xj | y) is well separated from\nany candidate xi, a situation typically ensured by the kinematic and structural constraints.\nThe occlusion constraint's weak dependence on xi (see Fig. 3) then separates the message\ncomputation into two cases. If xi lies in front of typical xj configurations, the BP message\nj,i(u)(zi ) is uninformative. If x\n (u) i is occluded, the message approximately equals\n\n j,i(u)(zi = 0) = 1 = 1) = 1 - Pr [u (x\n (u) j,i(u)(zi(u) j )] (15)\nwhere we have neglected correlations among pixel occlusion states, and where the prob-\nability is computed with respect to ^\n p(xj | y). By taking the product of these messages\nk,i(u)(zi ) from all potential occluders x\n (u) k and normalizing, we may determine an ap-\n\nproximation to the marginal occlusion probability i Pr[z = 0].\n (u) i(u)\n\nBecause the color likelihood pC (y | xi, zi) factorizes across pixels u, the BP approximation\nto pC (y | xi) may be written in terms of these marginal occlusion probabilites:\n p\n p skin(u)\n C (y | xi) i + (1 - ) (16)\n (u) i(u) pbkgd(u)\n u(xi)\n\nIntuitively, this equation downweights the color evidence at pixel u as the probability of\nthat pixel's occlusion increases. The edge likelihood pE(y | xi) averages over zi similarly.\nThe NBP estimate of ^\n p(xi | y) is determined by sampling configurations of xi as before,\nand reweighting them using these occlusionsensitive likelihood functions.\n\n\n4.3 Message Propagation\n\nTo derive the propagation rule for nonocclusion edges, as suggested by [18] we rewrite\nthe message update equation (13) in terms of the marginal distribution ^\n p(xi | y):\n ^\n pn-1(x\n mn (x i | y) dx\n ij j ) = j,i (xj , xi) i (17)\n x mn-1 (x\n i ji i)\n\nOur explicit use of the current marginal estimate ^\n pn-1(xi | y) helps focus the Monte Carlo\napproximation on the most important regions of the state space. Note that messages sent\n\n\f\n 1 2 1 2\nFigure 4: Refinement of a coarse initialization following one and two NBP iterations, both without\n(left) and with (right) occlusion reasoning. Each plot shows the projection of the five most significant\nmodes of the estimated marginal distributions. Note the difference in middle finger estimates.\n\n\nalong kinematic, structural, and temporal edges depend only on the belief ^\n p(xi | y) follow-\ning marginalization over occlusion variables zi.\n\nDetails and pseudocode for the message propagation step are provided in [13]. For kine-\nmatic constraints, we sample uniformly among permissable joint angles, and then use\nforward kinematics to propagate samples from ^\n pn-1(xi | y) /mn-1 (x\n ji i) to hypothesized\nconfigurations of xj. Following [12], temporal messages are determined by adjusting the\nbandwidths of the current marginal estimate ^\n p(xi | y) to match the temporal covariance i.\nBecause structural potentials (eq. (2)) equal one for all state configurations outside some\nball, the ideal structural messages are not finitely integrable. We therefore approximate the\nstructural message m (x\n ij j ) as an analytic function equal to the weights of all kernels in\n^\np(xi | y) outside a ball centered at qj, the position of xj.\n\n5 Simulations\n\nWe now present a set of computational examples which investigate the performance of\nour distributed occlusion reasoning; see [13] for additional simulations. In Fig. 4, we\nuse NBP to refine a coarse, usersupplied initialization into an accurate estimate of the\nhand's configuration in a single image. When occlusion constraints are neglected, the NBP\nestimates associate the ring and middle fingers with the same image pixels, and miss the\ntrue middle finger location. With proper occlusion reasoning, however, the correct hand\nconfiguration is identified. Using M = 200 particles, our Matlab implementation requires\nabout one minute for each NBP iteration (an update of all messages in the graph).\n\nVideo sequences demonstrating the NBP hand tracker are available at\nhttp://ssg.mit.edu/nbp/. Selected frames from two of these sequences are\nshown in Fig. 5, in which filtered estimates are computed by a single \"forward\" sequence\nof temporal message updates. The initial frame was approximately initialized manually.\nThe first sequence shows successful tracking through a partial occlusion of the ring finger\nby the middle finger, while the second shows a grasping motion in which the fingers\nocclude each other. For both of these sequences, rough tracking (not shown) is possible\nwithout occlusion reasoning, since all fingers are the same color and the background is\nunambiguous. However, we find that stability improves when occlusion reasoning is used\nto properly discount obscured edges and silhouettes.\n\n6 Discussion\n\nSigal et. al. [10] developed a threedimensional NBP person tracker which models the\nconditional distribution of each linkage's location, given its neighbor, via a Gaussian mix-\nture estimated from training data. In contrast, we have shown that an NBP tracker may\nbe built around the local structure of the true kinematic constraints. Conceptually, this has\nthe advantage of providing a clearly specified, globally consistent generative model whose\nproperties can be analyzed. Practically, our formulation avoids the need to explicitly ap-\nproximate kinematic constraints, and allows us to build a functional tracker without the\nneed for precise, labelled training data.\n\n\f\nFigure 5: Four frames from two different video sequences: a hand rotation containing finger occlu-\nsion (top), and a grasping motion (bottom). We show the projections of NBP's marginal estimates.\n\n\nWe have described the graphical structure underlying a kinematic model of the hand, and\nused this model to build a tracking algorithm using nonparametric BP. By appropriately\naugmenting the model's state, we are able to perform occlusion reasoning in a distributed\nfashion. The modular state representation and robust, local computations of NBP offer a\nsolution particularly well suited to visual tracking of articulated objects.\n\nAcknowledgments\n\nThe authors thank C. Mario Christoudias and Michael Siracusa for their help with video data collec-\ntion, and Michael Black, Alexander Ihler, Michael Isard, and Leonid Sigal for helpful conversations.\nThis research was supported in part by DARPA Contract No. NBCHD030010.\nReferences\n\n [1] Y. Wu and T. S. Huang. Hand modeling, analysis, and recognition. IEEE Signal Proc. Mag.,\n pages 5160, May 2001.\n [2] J. M. Rehg and T. Kanade. 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Constructing free energy approximations and\n generalized belief propagation algorithms. Technical Report 2004-040, MERL, May 2004.\n[16] A. T. Ihler, E. B. Sudderth, W. T. Freeman, and A. S. Willsky. Efficient multiscale sampling\n from products of Gaussian mixtures. In NIPS, 2003.\n[17] B. W. Silverman. Density Estimation for Statistics and Data Analysis. Chapman & Hall, 1986.\n[18] D. Koller, U. Lerner, and D. Angelov. A general algorithm for approximate inference and its\n application to hybrid Bayes nets. In UAI 15, pages 324333, 1999.\n\n\f\n", "award": [], "sourceid": 2748, "authors": [{"given_name": "Erik", "family_name": "Sudderth", "institution": null}, {"given_name": "Michael", "family_name": "Mandel", "institution": null}, {"given_name": "William", "family_name": "Freeman", "institution": null}, {"given_name": "Alan", "family_name": "Willsky", "institution": null}]}