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...Figure 8: Cumulative error distributions for BIM and our method based on the known ground truth for SCAPE dataset. this method from our comparisons. Second, we test our algorithm on the SCAPE dataset =-=[52]-=- consisting of 71 scans of a male scanned in different postures. In our experiment, we match the neutral posture to all 70 remaining postures using BIM and our approach. The cumulative error distribut...

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...n be exploited for matching. One solution is to replace the notion of distance. For example, by using diffusion distances, or variants thereof, one can reduce the sensitivity to topological artifacts =-=[17, 20, 21]-=- when the pieces of geometry that cause the problem are small in relation to the overall shape. Nonetheless, these invariants still break down in case of large artifacts (wide contacts, large holes, a...

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...linearized analysis. In order to understand the degrees of freedom of the local matching model, we can use the tool of modal analysis of such elastic models, first introduced by Pentland and Williams =-=[41]-=- to the field of computer graphics (the aim of their paper was actually to speed up the simulation of extrinsic elastic deformations of solid objects in threespace). Modal analysis represent the defor...

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...s paper, we exploit a different way to uniquely describe an isometry: fixing one point, a tangential direction, and the surface orientation is necessary and sufficient to specify an isometric mapping =-=[23]-=-. This provides the fewest possible degrees of freedom while still covering all cases including shapes with global intrinsic symmetries. In case of small, global error margins, statistical triangulati...

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... recognition and matching [27–29]. Alternatively, the geometry of shape S can be embedded into shape T using generalized multi-dimensional scaling, thereby computing a cross-parameterization directly =-=[14]-=-. Exact isometry results in a set of matching candidates with few degrees of freedom. Lipman and Funkhouser [10] have noticed that isometries are special cases of conformal maps, thereby having only t...

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...that was processed by fitting a template shape to the raw data. For all experiments in this section, we use rmin = 0.9R and rmax = 1.7R. We first use two frames of the samba sequence by Vlasic et al. =-=[51]-=-. These frames are locally very close to isometric, but globally have high non-isometric distortion due to the dress being connected to the legs. We therefore set ρ = 0 in the clustering step. Vlasic ...

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...en all pairs of correspondences quickly becomes prohibitively expensive and would limit the practical applicability of our algorithm. To remedy this, we construct a topology hierarchy on S similar to =-=[47]-=- as follows. We define level 0 of the hierarchy to be the original set of vertices and their connectivity–either the original triangle mesh or the k-nn graph for a point cloud. (In all our experiments...

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...along surfaces of objects such as humans, animals, plants, or cloths do not change a lot without serious injury or damage. This restriction leads to a strongly constrained search space. Lipman et al. =-=[10]-=- argue that isometries between topological disks are a special case of conformal mappings, thereby limiting the degrees of freedom to six (three pointto-point correspondences are sufficient). Ovsjanik...

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...sing purely geometric information, we outperform the other purely geometric methods tested. Second, we compare our method to BIM and Tevs et al.’s method using two models of the BU-3DFE face database =-=[53]-=-. The two models contain numerous small holes and outliers. Furthermore, the models have different topology because the mouth is closed in one model and open in the other one. For these reasons, we se...

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...n be exploited for matching. One solution is to replace the notion of distance. For example, by using diffusion distances, or variants thereof, one can reduce the sensitivity to topological artifacts =-=[17, 20, 21]-=- when the pieces of geometry that cause the problem are small in relation to the overall shape. Nonetheless, these invariants still break down in case of large artifacts (wide contacts, large holes, a...

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...ely related: They either include the preservation of the curvature tensor in the objective function to maintain the extrinsic shape, or apply Eq. 6 to the 3-manifold of the embedding Euclidean volume =-=[39, 40]-=-. All of these methods are designed for partial matching. The problem with both intrinsic and extrinsic elastic matching models is that the search space becomes very large, rendering any approach base...

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...uding shapes with global intrinsic symmetries. In case of small, global error margins, statistical triangulation algorithms can be applied that compute all correspondences from a few landmark matches =-=[15, 16]-=- or, similarly, by voting for several approximate solutions [10,11]. Depending on the geometry of the shape, errors can become amplified so that a bit more than only the minimal set of initial corresp...

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...cted in O(1) time. In this view, we can perform a relaxation: In order to also find approximate isometries, we can cast this problem as the task of finding approximate equivalence classes. Kim et al. =-=[18]-=- have used this idea in the context of globally consistent isometric mappings in order to efficiently find approximate isometries. In our paper, we demonstrate how our partial matching framework can b...

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...ng a large area of S. An example where the clustering step fails to identify a cluster of consistent mappings covering a large area of S is shown in Figure 12. The two frames of the flashkick dataset =-=[54]-=- contain a large topological change due to a merge of the subject’s pants in one of the models. Note that our algorithm correctly stops the growing of single partial mapping in this area, as shown in ...

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...he algorithm we require a continuous surface representation. We obtain a continuous surface representation as implicit moving-least-squares (MLS) manifolds using the robust method of Öztireli et al. =-=[46]-=-. Using this method, a discretized surface S is represented continuously as the zero level-set of an implicit function derived from the oriented vertices of S . Using a MLS representation allows a poi...

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...lly consistent approaches more robust w.r.t. topological changes, several approaches have been proposed to perform a band-limited analysis in the eigenspace of the manifold’s LaplaceBeltrami operator =-=[17,20,21,31]-=-. Unlike prior approaches based on embedding the intrinsic geometry of the shape directly [14, 27–29], these approaches successfully handle small topological errors. However, they break down 3 in the ...

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...w capture setup. We denote the feature descriptor at a point s ∈ S with a vector DS(s), and similarly for points on T . We identify features on S and T using the geodesic fingerprint descriptor DS(s) =-=[48]-=-, which compares how an isocontour of the geodesic distance from a point s ∈ S deviates in length from the isocontour of the same 2D Euclidean distance. We use 10 isocontours in all experiments, for g...

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...ely related: They either include the preservation of the curvature tensor in the objective function to maintain the extrinsic shape, or apply Eq. 6 to the 3-manifold of the embedding Euclidean volume =-=[39, 40]-=-. All of these methods are designed for partial matching. The problem with both intrinsic and extrinsic elastic matching models is that the search space becomes very large, rendering any approach base...

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... large parts of two surfaces that are similar to each other, where similarity is measured according to Eq. 2. In practice, the parts are found using generalized multidimensional scaling. Raviv et al. =-=[35]-=- use a similar technique to find partial intrinsic symmetries. 2.3 Local Approximate Isometry Another common way to relax the requirement of exact isometry towards approximate matching is to maintain ...

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... large search space for elastic matching, existing methods use additional constraints, such as, template models [40], temporal coherence [39], or fairly large sets of coherent feature correspondences =-=[37]-=-. Recently, Kim et al. [18] proposed a new paradigm for the elastic matching problem. First, the method computes multiple dense maps between two shapes by assuming a global near-isometric deformation ...

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...uding shapes with global intrinsic symmetries. In case of small, global error margins, statistical triangulation algorithms can be applied that compute all correspondences from a few landmark matches =-=[15, 16]-=- or, similarly, by voting for several approximate solutions [10,11]. Depending on the geometry of the shape, errors can become amplified so that a bit more than only the minimal set of initial corresp...

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...al planar parametrizations. Different techniques have been proposed to locally parameterize the intrinsic geometry of a surface to a plane using a local approximate isometry criterion. Schmidt et al. =-=[42]-=- used exponential maps to transport a local coordinate system along a surface for the purpose of texture mapping. More recently, Schmidt [43] used transported exponential maps to produce a parameteriz...

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...alyzed explicitly in literature. We do not aim to give an extensive review of surface registration methods; for this, we refer the reader to recent surveys such as van Kaick et al. [25] or Tam et al. =-=[26]-=-. We start by introducing some formal notations. Manifolds: We consider smooth, orientable 2-manifolds M ⊂ R3 embedded in three-dimensional space. In order to represent partial data (such as scans wit...

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...our approach would be compatible with this task given a reliable way to estimate scale change during metric propagation, possibly using shape extremities or other intrinsic features. Bronstein et al. =-=[34]-=- introduced a general framework to evaluate partial similarity using Pareto optimality. In case of partial intrinsic shape matching, this method aims to find large parts of two surfaces that are simil...

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...Laplace-Beltrami operator is not degenerate, thus showing that there are only two degrees of freedom in this special case. Additional cues, such as carefully selected constellations of local features =-=[12]-=-, can even reduce the complexity for many shapes to the point of leaving a trivial search space, eliminating all degrees of freedom. As approximate isometry makes the correspondence problem feasible w...

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...pological errors. However, they break down 3 in the presence of large artifacts, such as wide contacts or missing pieces. A notable exception is the heuristic region growing approach by Sharma et al. =-=[32]-=-. It tries to match points with similar spectral signatures using an expectation-maximization framework, which has been shown to perform well in the presence of large contacts. Despite good practical ...

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...15, 16] and follow-ups) in order to handle partial matching. For future work, to address this limitation, we plan to combine our approach with an approach that detects continuous intrinsic symmetries =-=[55]-=-, thereby reducing the search space for the initial feature matching and allowing us to efficiently enumerate all partial matches. Similarly, our framework could be extended to find partial intrinsic ...

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...ind approximate isometries. In our paper, we demonstrate how our partial matching framework can be adapted to perform the same task in a partial matching scenario. We perform agglomerative clustering =-=[24]-=- in the space of near-isometric mappings, which are concisely represented using the tuples introduce above. We validate our algorithm on standard benchmark data sets and raw scanner data, and compare ...

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...raditional method accordingly) but need to be determined along with the matching. This seems to re-introduce prohibitive complexity as we now have to choose from an exponential number of such subsets =-=[22]-=-. The most important contribution of this paper is to show that with an appropriate matching model this is not the case. The search space is not much larger than in the global problem and we give an e...

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...ficient and can be considered a more or less solved problem. The global approximate isometry criterion has also been employed to study matching partial overlaps of complete surfaces. Van Kaick et al. =-=[30]-=- use a pair of features to define a map that captures a local geodesic region between the two features, and show how this descriptor can be used for partial matching. Here, using two features instead ...

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...nian manifold S fixing one point s on S, a tangential direction ds in TsS, and a surface normal ns at s suffices to specify an isometric mapping [23]. For details on the smoothness criteria, refer to =-=[45]-=-. The following proof sketch aims to give the intuition behind this statement for 2-dimensional surfaces embedded in R3 by showing that we can transfer the local coordinate frame defined by ds and ns ...

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...setups for acquiring moving geometry in real-time [1–4] and there are even consumer devices on the market such as the Microsoft KinectTM. This leads to new applications such as virtual cinematography =-=[5]-=-, or the creation of data-driven generative shape models of deformable objects [6–9]. Finding correspondences among such data is a fundamental problem for all of these applications: Almost any further...

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...tion of the new match on T to reduce discretization errors for differently sampled surfaces. For robustness, we use all available oriented point matches inN1(u) and take the Riemannian center of mass =-=[49]-=- of the transported points, where all transported points have equal mass. The newly matched source point u is added to U only if it locally respects the stretch factor ν as given in Eq. 5. This is nec...

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... the performance of our algorithm against four existing methods, namely heat kernel maps (HKM) [11], blended intrinsic maps (BIM) [18], the method of Sharma et al. [32], and the method of Tevs et al. =-=[50]-=-. We select these methods for comparison because they represent the state of the art for matching between surfaces. More specifically, HKM is derived from the theoretical complexity of isometric mappi...

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...the consistent points separately, which leads to a rather complex algorithm. Further, the notion of a scale parameter is not canonically related to the original matching problem. Sahilloglu and Yemez =-=[33]-=- consider the case where one of the surfaces is complete and the other an incomplete, deformed part of that surface. Using a coarse sampling and matching strategy between shape extremities, they can d...

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...ing a local approximate isometry criterion. Schmidt et al. [42] used exponential maps to transport a local coordinate system along a surface for the purpose of texture mapping. More recently, Schmidt =-=[43]-=- used transported exponential maps to produce a parameterization of a local surface patch to the plane that has low metric distortion. The local surface patch is provided through user interaction as a...

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...a parameterization of a local surface patch to the plane that has low metric distortion. The local surface patch is provided through user interaction as an input stroke on the surface. Malvaer et al. =-=[44]-=- propose an alternative parameterization based on an extension of polar coordinates to surfaces. In our work, we crossparameterize local surface patches from S to T . This cross-parameterization is mo...