{"title": "The Correlated Correspondence Algorithm for Unsupervised Registration of Nonrigid Surfaces", "book": "Advances in Neural Information Processing Systems", "page_first": 33, "page_last": 40, "abstract": null, "full_text": " The Correlated Correspondence Algorithm for\n Unsupervised Registration of Nonrigid Surfaces\n\n\n\n Dragomir Anguelov1, Praveen Srinivasan1, Hoi-Cheung Pang1,\n Daphne Koller1, Sebastian Thrun1, James Davis2 \n 1 Stanford University, Stanford, CA 94305\n 2 University of California, Santa Cruz, CA 95064\n e-mail:{drago,praveens,hcpang,koller,thrun,jedavis}@cs.stanford.edu\n\n\n\n Abstract\n\n We present an unsupervised algorithm for registering 3D surface scans of\n an object undergoing significant deformations. Our algorithm does not\n need markers, nor does it assume prior knowledge about object shape, the\n dynamics of its deformation, or scan alignment. The algorithm registers\n two meshes by optimizing a joint probabilistic model over all point-to-\n point correspondences between them. This model enforces preservation\n of local mesh geometry, as well as more global constraints that capture\n the preservation of geodesic distance between corresponding point pairs.\n The algorithm applies even when one of the meshes is an incomplete\n range scan; thus, it can be used to automatically fill in the remaining sur-\n faces for this partial scan, even if those surfaces were previously only\n seen in a different configuration. We evaluate the algorithm on several\n real-world datasets, where we demonstrate good results in the presence\n of significant movement of articulated parts and non-rigid surface defor-\n mation. Finally, we show that the output of the algorithm can be used for\n compelling computer graphics tasks such as interpolation between two\n scans of a non-rigid object and automatic recovery of articulated object\n models.\n\n\n1 Introduction\n\nThe construction of 3D object models is a key task for many graphics applications. It is\nbecoming increasingly common to acquire these models from a range scan of a physical\nobject. This paper deals with an important subproblem of this acquisition task -- the\nproblem of registering two deforming surfaces corresponding to different configurations of\nthe same non-rigid object.\n The main difficulty in the 3D registration problem is determining the correspondences of\npoints on one surface to points on the other. Local regions on the surface are rarely distinc-\ntive enough to determine the correct correspondence, whether because of noise in the scans,\nor because of symmetries in the object shape. Thus, the set of candidate correspondences to\na given point is usually large. Determining the correspondence for all object points results\nin a combinatorially large search problem. The existing algorithms for deformable surface\n\n A results video is available at http://robotics.stanford.edu/drago/cc/video.mp4\n\n\f\nFigure 1: A) Registration results for two meshes. Nonrigid ICP and its variant augmented with spin\nimages get stuck in local maxima. Our CC algorithm produces a largely correct registration, although\nwith an artifact in the right shoulder (inset). B) Illustration of the link deformation process C) The\nCC algorithm which uses only deformation potentials can violate mesh geometry. Near regions can\nmap to far ones (segment AB) and far regions can map to near ones (points C,D).\n\n\nregistration make the problem tractable by assuming significant prior knowledge about the\nobjects being registered. Some rely on the presence of markers on the object [1, 20], while\nothers assume prior knowledge about the object dynamics [16], or about the space of non-\nrigid deformations [15, 5]. Algorithms that make neither restriction [18, 12] simplify the\nproblem by decorrelating the choice of correspondences for the different points in the scan.\nHowever, this approximation is only good in the case when the object deformation is small;\notherwise, it results in poor local maxima as nearby points in one scan are allowed to map\nto far-away points in the other.\n Our algorithm defines a joint probabilistic model over all correspondences, which ex-\nplicitly model the correlations between them -- specifically, that nearby points in one mesh\nshould map to nearby points in the other. Importantly, the notion of \"nearby\" used in our\nmodel is defined in terms of geodesic distance over the mesh. We define a probabilistic\nmodel over the set of correspondences, that encodes these geodesic distance constraints as\nwell as penalties for link twisting and stretching, and high-level local surface features [14].\nWe then apply loopy belief propagation [21] to this model, in order to solve for the entire\nset of correspondences simultaneously. The result is a registration that respects the surface\ngeometry. To the best of our knowledge, the algorithm we present in this paper is the first\nalgorithm which allows the registration of 3D surfaces of an object where the object config-\nurations can vary significantly, there is no prior knowledge about object shape or dynamics\nof deformation, and nothing whatsoever is known about the object alignment. Moreover,\nunlike many methods, our algorithm can be used to register a partial scan to a complete\nmodel, greatly increasing its applicability.\n We apply our approach to three datasets containing 3D scans of a wooden puppet, a\nhuman arm and entire human bodies in different configurations. We demonstrate good\nregistration results for scan pairs exhibiting articulated motion, non-rigid deformations, or\nboth. We also describe three applications of our method. In our first application, we show\nhow a partial scan of an object can be registered onto a fully specified model in a dif-\nferent configuration. The resulting registration allows us to use the model to \"complete\"\nthe partial scan in a way that preserves the local surface geometry. In the second, we use\nthe correspondences found by our algorithm to smoothly interpolate between two different\nposes of an object. In our final application, we use a set of registered scans of the same\nobject in different positions to recover a decomposition of the object into approximately\nrigid parts, and recover an articulated skeleton linking the parts. All of these applications\nare done in an unsupervised way, using only the output of our Correlated Correspondence\nalgorithm applied to pairs of poses with widely varying deformations, and unknown initial\nalignments. These results demonstrate the value of a high-quality solution to the registra-\ntion problem to a range of graphics tasks.\n\n\f\n2 Previous Work\n\nSurface registration is a fundamental building block in computer graphics. The classical so-\nlution for registering rigid surfaces is the Iterative Closest Point algorithm (ICP) [4, 6, 17].\nRecently, there has been work extending ICP to non-rigid surfaces [18, 8, 12, 1]. These\nalgorithms treat one of the scans (usually a complete model of the surface) as a deformable\ntemplate. The links between adjacent points on the surface can be thought of as springs,\nwhich are allowed to deform at a cost. Similarly to ICP, these algorithms iterate between\ntwo subproblems -- estimating the non-rigid transformation and estimating the set of\ncorrespondences C between the scans. The step estimating the correspondences assumes\nthat a good estimate of the nonrigid transformation is available. Under this assumption,\nthe assignments to the correspondence variables become decorrelated: each point in the\nsecond scan is associated with the nearest point (in the Euclidean distance sense) in the\ndeformed template scan. However, the decomposition also induces the algorithm's main\nlimitation. By assigning points in the second scan to points on the deformed model inde-\npendently, nearby points in the scan can get associated to remote points in the model if the\nestimate of is poor (Fig. 1A). While several approaches have been proposed to address\nthis problem of incorrect correspondences, their applicability is largely limited to problems\nwhere the deformation is local, and the initial alignment is approximately correct.\n Another line of related work is the work on deformable template matching in the com-\nputer vision community. In the 3D case, this framework is used for detection of articulated\nobject models in images [13, 22, 19]. The algorithms assume the decomposition of the\nobject into a relatively small number of parts is known, and that a detector for each object\npart is available. Template matching approaches have also been applied to deformable 2D\nobjects, where very efficient solutions exist [9, 11]. However, these methods do not extend\neasily to the case of 3D surfaces.\n\n\n3 The Correlated Correspondence Algorithm\n\nThe input to the algorithm is a set of two meshes (surfaces tessellated into polygons).\nThe model mesh X = (V X , EX ) is a complete model of the object, in a particular pose.\nV X = (x1, . . . , xN ) denotes the mesh points, while EX is the set of links between adjacent\npoints on the mesh surface. The data mesh Z = (V Z , EZ ) is either a complete model or a\npartial view of the object in a different configuration. Each data mesh point zk is associated\nwith a correspondence variable ck, specifying the corresponding model mesh point. The\ntask of registration is one of estimating the set of all correspondences C and a non-rigid\ntransformation which aligns the corresponding points.\n\n3.1 Probabilistic Model\n\nWe formulate the registration problem as one of finding an embedding of the data mesh\nZ into the model mesh X, which is encoded as an assignment to all correspondence vari-\nables C = (c1, . . . , cK ). The main idea behind our approach is to preserve the consis-\ntency of the embedding by explicitly correlating the assignments to the correspondence\nvariables. We define a joint distribution over the correspondence variables c1, . . . , cK , rep-\nresented as a Markov network. For each pair of adjacent data mesh points zk, zl, we want\nto define a probabilistic potential (ck, cl) that constrains this pair of correspondences to\nreasonable and consistent. This gives rise to a joint probability distribution of the form\np(C) = 1 (c (c\n Z k k ) k,l k , cl) which contains only single and pairwise potentials.\nPerforming probabilistic inference to find the most likely joint assignment to the entire set\nof correspondence variables C should yield a good and consistent registration.\n\nDeformation Potentials. We want our model to encode a preference for embeddings\nof mesh Z into mesh X, which minimize the amount of deformation induced by the\nembedding. In order to quantify the amount of deformation , applied to the model, we\n\n\f\nwill follow the ideas of Hahnel et al. [12] and treat the links in the set EX as springs, which\nresist stretching and twisting at their endpoints. Stretching is easily quantified by looking at\nchanges in the link length induced by the transformation . Link twisting, however, is ill-\nspecified by looking only at the Cartesian coordinates of the points alone. Following [12],\nwe attach an imaginary local coordinate system to each point on the model. This local\ncoordinate system allows us to quantify the \"twist\" of a point xj relative to a neighbor xi.\nA non-rigid transformation defines, for each point xi, a translation of its coordinates and\na rotation of its local coordinate system.\n To evaluate the deformation penalty, we parameterize each link in the model in terms\nof its length and its direction relative to its endpoints (see Fig. 1B). Specifically, we define\nli,j to be the distance between xi and xj; dij is a unit vector denoting the direction of\nthe point xj in the coordinate system of xi (and vice versa). We use ei,j to denote the set\nof edge parameters (li,j, dij, dji). It is now straightforward to specify the penalty for\nmodel deformations. Let be a transformation, and let ~\n ei,j denote the triple of parameters\nassociated with the link between xi and xj after applying . Our model penalizes twisting\nand stretching, using a separate zero-mean Gaussian noise model for each:\n\n P (~\n ei,j | ei,j) = P (~li,j | li,j) P ( ~\n dij | dij) P ( ~\n dji | dji) (1)\n\nIn the absence of prior information, we assume that all links are equally likely to deform.\n In order to quantify the deformation induced by an embedding C, we need to include\na potential d(ck, cl) for each link eZ EZ . Every probability \n k,l d(ck = i, cl = j)\ncorresponds to the deformation penalty incurred by deforming model link ei,j to generate\nlink eZ and is defined in (1). We do not restrict ourselves to the set of links in EX , since\n k,l\nthe original mesh tessellation is sparse and local. Any two points in X are allowed to\nimplicitly define a link.\n Unfortunately, we cannot directly estimate the quantity P (eZ | e\n k,l i,j ), since the link pa-\nrameters eZ depend on knowing the nonrigid transformation, which is not given as part\n k,l\nof the input. The key issue is estimating the (unknown) relative rotation of the link end-\npoints. In effect, this rotation is an additional latent variable, which must also be part of the\nprobabilistic model. To remain within the realm of discrete Markov networks, allowing the\napplication of standard probabilistic inference algorithms, we discretize the space of the\npossible rotations, and fold it into the domains of the correspondence variables. For each\npossible value of the correspondence variable ck = i we select a small set of candidate\nrotations, consistent with local geometry. We do this by aligning local patches around the\npoints xi and zk using rigid ICP. We extend the domain of each correspondence variables\nck, where each value encodes a matching point and a particular rotation from the precom-\nputed set for that point. Now the edge parameters eZ are fully determined and so is the\n k,l\nprobabilistic potential.\n\nGeodesic Distances. Our proposed approach raises the question as to what constitutes\nthe best constraint between neighboring correspondence variables. The literature on scan\nregistration -- for rigid and non-rigid models alike -- relies on the preserving Euclidean\ndistance. While Euclidean distance is meaningful for rigid objects, it is very sensitive to de-\nformations, especially those induced by moving parts. For example, in Fig. 1C, we see that\nthe two legs in one configuration of our puppet are fairly close together, allowing the algo-\nrithm to map two adjacent points in the data mesh to the two separate legs, with minimal\ndeformation penalty. In the complementary situation, especially when object symmetries\nare present, two distant yet similar points in one scan might get mapped to the same region\nin the other. For example, in the same figure, we see that points in both an arm and a leg in\nthe data mesh get mapped to a single leg in the model mesh.\n We therefore want to enforce constraints preserving distance along the mesh surface\n(geodesic distance). Our probabilistic framework easily incorporate such constraints as\ncorrelations between pairs of correspondence variables. We encode a nearness preservation\n\n\f\nFigure 2: A) Automatic interpolation between two scans of an arm and a wooden puppet. B) Regis-\ntration results on two scans of the same man sitting and standing up (select points were displayed)\nC) Registration results on scans of a larger man and a smaller woman. The algorithm is robust to\nsmall changes in object scale.\n\n\nconstraint which prevents adjacent points in mesh Z to be mapped to distant points in X\nin the geodesic distance sense. For adjacent points zk, zl in the data mesh, we define the\nfollowing potential:\n\n 0 dist\n Geodesic (xi, xj ) > \n n(ck = i, cl = j) = (2)\n 1 otherwise\n\nwhere is the data mesh resolution and is some constant, chosen to be 3.5.\n The farness preservation potentials encode the complementary constraint. For every\npair of points zk, zl whose geodesic distance is more than 5 on the data mesh, we have a\npotential:\n 0 dist\n Geodesic(xi, xj ) < \n f (ck = i, cl = j) = (3)\n 1 otherwise\n\nwhere is also a constant, chosen to be 2 in our implementation. The intuition behind this\nconstraint is fairly clear: if zk, zl are far apart on the data mesh, then their corresponding\npoints must be far apart on the model mesh.\n\nLocal Surface Signatures. Finally, we encode a set of potentials that correspond to\nthe preservation of local surface properties between the model mesh and data mesh. The\nuse of local surface signatures is important, because it helps to guide the optimization in\nthe exponential space of assignments. We use spin images [14] compressed with prin-\ncipal component analysis to produce a low-dimensional signature sx of the local surface\ngeometry around a point x. When data and model points correspond, we expect their lo-\ncal signatures to be similar. We introduce a potential whose values s(ck) = i enforce a\nzero-mean Gaussian penalty for discrepancies between sx and s .\n i zk\n\n\n3.2 Optimization\n\nIn the previous section, we defined a Markov network, which encodes a joint probability\ndistribution over the correspondence variables as a product of single and pairwise poten-\ntials. Our goal is to find a joint assignment to these variables that maximizes this proba-\nbility. This problem is one of standard probabilistic inference over the Markov network.\nHowever, the Markov network is quite large, and contains a large number of loops, so that\nexact inference is computationally infeasible. We therefore apply an approximate inference\nmethod known as loopy belief propagation (LBP)[21], which has been shown to work in a\nwide variety of applications. Running LBP until convergence results in a set of probabilis-\ntic assignments to the different correspondence variables, which are locally consistent. We\nthen simply extract the most likely assignment for each variable to obtain a correspondence.\n One remaining complication arises from the form of our farness preservation constraints.\nIn general, most pairs of points in the mesh are not close, so that the total number of\nsuch potentials grows as O(M 2), where M is the number of points in the data mesh.\nHowever, rather than introducing all these potentials into the Markov net from the start, we\n\n\f\nintroduce them as needed. First, we run LBP without any farness preservation potentials.\nIf the solution violates a set of farness preservation constraints, we add it and rerun BP. In\npractice, this approach adds a very small number of such constraints.\n\n\n4 Experimental Results\n\nBasic Registration. We applied our registration algorithm to three different datasets,\ncontaining meshes of a human arm, wooden puppet and the CAESAR dataset of whole\nhuman bodies [1], all acquired by a 3D range scanner. The meshes were not complete\nsurfaces, but several techniques exist for filling the holes (e.g., [10]).\n We ran the Correlated Correspondence algorithm using the same probabilistic model and\nthe same parameters on all data sets. We use a coarse-to-fine strategy, using the result of a\ncoarse sub-sampling of the mesh surface to constrain the correspondences at a finer-grained\nlevel. The resulting set of correspondences were used as markers to initialize the non-rigid\nICP algorithm of Hahnel et al. [12].\n The Correlated Correspondence algorithm successfully aligned all mesh pairs in our hu-\nman arm data set containing 7 arms. In the puppet data set we registered one of the meshes\nto the remaining 6 puppets. The algorithm correctly registered 4 out of 6 data meshes to the\nmodel mesh. In the two remaining cases, the algorithm produced a registration where the\ntorso was flipped, so that the front was mapped to the back. This problem arises from am-\nbiguities induced by the puppet symmetry, whose front and back are almost identical. Im-\nportantly, our probabilistic model assigns a higher likelihood score to the correct solution,\nso that the incorrect registration is a consequence of local maxima in the LBP algorithm.\n This fact allows us to address the issue in an unsupervised way simply by running loopy\nBP several times, with different initialization. For details on the unsupervised initialization\nscheme we used, please refer to our technical report [2]. We ran the modified algorithm\nto register one puppet mesh to the remaining 6 meshes in the dataset, obtaining the correct\nregistration in all cases. In particular, as shown in Fig. 1A, we successfully deal with the\ncase on which the straightforward nonrigid ICP algorithm failed. The modified algorithm\nwas applied to the CAESAR dataset and produced very good registration for challenging\ncases exhibiting both articulated motion and deformation (Fig. 2B), or exhibiting deforma-\ntion and a (small) change in object scale (Fig. 2C).\n Overall, the algorithm performed robustly, producing a close-to-optimal registrations\neven for pairs of meshes that involve large deformations, articulated motion or both. The\nregistration is accomplished in an unsupervised way, without any prior knowledge about\nobject shape, dynamics, or alignment.\n\nPartial view completion. The Correlated Correspondence algorithm allows us to register\na data mesh containing only a partial scan of an object to a known complete surface model\nof the object, which serves as a template. We can then transform the template mesh to the\npartial scan, a process which leaves undisturbed the links that are not involved in the partial\nmesh. The result is a mesh that matches the data on the observed points, while completing\nthe unknown portion of the surface using the template.\n We take a partial mesh, which is missing the entire back part of the puppet in a particular\npose. The resulting partial model is displayed in Fig. 3B-1; for comparison, the correct\ncomplete model in this configuration (which was not available to the algorithm), is shown in\nFig. 3B-2. We register the partial mesh to models of the object in a different pose (Fig. 3B-\n3), and compare the completions we obtain (Fig. 3B-4), to the ground truth represented in\nFig. 3B-2. The result demonstrates a largely correct reconstruction of the complete surface\ngeometry from the partial scan and the deformed template. We report additional shape\ncompletion results in [2].\n\nInterpolation. Current research [20] shows that if a nonrigid transformation between\nthe poses is available, believable animation can be produced by linear interpolation be-\n\n\f\nFigure 3: A) The results produced by the CC algorithm were used for unsupervised recovery of\narticulated models. 15 puppet parts and 4 arm parts, as well as the articulated object skeletons, were\nrecovered. B) Partial view completion results. The missing parts of the surface were estimated by\nregistering the partial view to a complete model of the object in a different configuration.\n\n\ntween the model mesh and the transformed model mesh. The interpolation is performed\nin the space of local link parameters (li,j, dij, dji), We demonstrate that transforma-\ntion estimates produced by our algorithm can be used to automatically generate believable\nanimation sequences between fairly different poses, as shown in Fig. 2A.\n\nRecovering Articulated Models. Articulated object models have a number of appli-\ncations in animation and motion capture, and there has been work on recovering them\nautomatically from 3D data [7, 3]. We show that our unsupervised registration capability\ncan greatly assist articulated model recovery. In particular, the algorithm in [3] requires\nan estimate of the correspondences between a template mesh and the remaining meshes in\nthe dataset. We supplied it with registration computed with the Correlated Correspondence\nalgorithm. As a result we managed to recover in a completely unsupervised way all 15\nrigid parts of the puppet, as well as the joints between them (Fig. 3A). We demonstrate\nsuccessful articulation recovery even for objects which are not purely rigid, as is the case\nwith the human arm (see Fig. 3A).\n\n\n5 Conclusion\n\nThe contribution of this paper is an algorithm for unsupervised registration of non-rigid 3D\nsurfaces in significantly different configurations. Our results show that the algorithm can\ndeal with articulated objects subject to large joint movements, as well as with non-rigid sur-\nface deformations. The algorithm was not provided with markers or other cues regarding\ncorrespondence, and makes no assumptions about object shape, dynamics, or alignment.\nWe show the quality and the utility of the registration results we obtain by using them as a\nstarting point for compelling computer graphics applications: partial view completion, in-\nterpolation between scans, and recovery of articulated object models. Importantly, all these\nresults were generated in a completely unsupervised manner from a set of input meshes.\n The main limitation of our approach is the fact that it makes the assumption of (approx-\nimate) preservation of geodesic distance. Although this assumption is desirable in many\ncases, it is not always warranted. In some cases, the mesh topology may change drastically,\nfor example, when an arm touches the body. We can try to extend our approach to handle\nthese cases by trying to detect when they arise, and eliminating the associated constraints.\nHowever, even this solution is likely to fail on some cases. A second limitation of our ap-\nproach is that it assumes that the data mesh is a subset of the model mesh. If the data mesh\ncontains clutter, our algorithm will attempt to embed the clutter into the model. We feel that\nthe general nonrigid registration problem becomes underspecified when significant clutter\nand occlusion are present simultaneously. In this case, additional assumptions about the\nsurfaces will be needed.\n Despite the fact that our algorithm performs quite well, there are limitations to what\ncan be accurately inferred about the object from just two scans. Given more scans of the\n\n\f\nsame object, we can try to learn the deformation penalty associated with different links,\nand bootstrap the algorithm. Such an extension would be a step toward the goal of learning\nmodels of object shape and dynamics from raw data.\n\nAcknowledgments. This work has been supported by the ONR Young Investigator (PECASE) grant\nN00014-99-1-0464, and ONR Grant N00014-00-1-0637 under the DoD MURI program.\n\n\nReferences\n\n [1] B Allen, B Curless, and Z Popovic. The space of human body shapes:reconstruction and pa-\n rameterization from range scans. In Proc. SIGGRAPH, 2003.\n\n [2] D. Anguelov, D.Koller, P. Srinivasan, S.Thrun, H. Pang, and J.Davis. The correlated correspon-\n dence algorithm for unsupervised registration of nonrigid surfaces. In TR-SAIL-2004-100, at\n http://robotics.stanford.edu/drago/cc/tr100.pdf, 2004.\n [3] D. Anguelov, D. Koller, H. Pang, P. Srinivasan, and S. Thrun. Recovering articulated object\n models from 3d range data. In Proc. UAI, 2004.\n\n [4] P. Besl and N. McKay. A method for registration of 3d shapes. 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NIPS, 2002.\n\n\f\n", "award": [], "sourceid": 2601, "authors": [{"given_name": "Dragomir", "family_name": "Anguelov", "institution": null}, {"given_name": "Praveen", "family_name": "Srinivasan", "institution": null}, {"given_name": "Hoi-cheung", "family_name": "Pang", "institution": null}, {"given_name": "Daphne", "family_name": "Koller", "institution": null}, {"given_name": "Sebastian", "family_name": "Thrun", "institution": null}, {"given_name": "James", "family_name": "Davis", "institution": null}]}