We propose a shape matching method that produces dense correspondences tuned to a specific class of shapes and deformations. In a scenario where this class is represented by a small set of example shapes, the proposed method learns a shape descriptor capturing the variability of the deformations in the given class. The approach enables the wave kernel signature to extend the class of recognized deformations from near isometries to the deformations appearing in the example set by means of a random forest classifier. With the help of the introduced spatial regularization, the proposed method achieves significant improvements over the baseline approach and obtains stateof-the-art results while keeping short computation times.

Many algorithms for the computation of correspondences between deformable shapes rely on some variant of nearest neighbor matching in a descriptor space. Such are, for example, various point-wise correspondence recovery algorithms used as a post-processing stage in the functional correspondence framework. Such frequently used techniques implicitly make restrictive assumptions (e.g., nearisometry) on the considered shapes and in practice suffer from lack of accuracy and result in poor surjectivity. We propose an alternative recovery technique capable of guaranteeing a bijective correspondence and producing significantly higher accuracy and smoothness. Unlike other methods our approach does not depend on the assumption that the analyzed shapes are isometric. We derive the proposed method from the statistical framework of kernel density estimation and demonstrate its performance on several challenging deformable 3D shape matching datasets.

Figure 1: Qualitative examples on FAUST models (left), SHREC'16 (middle) and SCAPE (right). In the SHREC experiment, the green parts mark where no correspondence was found. Notice how those areas are close to the parts that are hidden in the other model. The missing matches (marked in black) in the SCAPE experiment are an artifact due to the multiscale approach. AbstractWe present a method to match three dimensional shapes under non-isometric deformations, topology changes and partiality. We formulate the problem as matching between a set of pair-wise and point-wise descriptors, imposing a continuity prior on the mapping, and propose a projected descent optimization procedure inspired by difference of convex functions (DC) programming.

We propose novel point descriptors for 3D shapes with the potential to match two shapes representing the same object undergoing natural deformations. These deformations are more general than the often assumed isometries, and we use labeled training data to learn optimal descriptors for such cases. Furthermore, instead of explicitly defining the descriptor, we introduce new Mercer kernels, for which we formally show that their corresponding feature space mapping is a generalization of either the Heat Kernel Signature (HKS) [3] or the Wave Kernel Signature (WKS) [1]. I.e. the proposed descriptors are guaranteed to be at least as precise as any Heat Kernel Signature or Wave Kernel Signature of any parameterisation.A point descriptor φ : P → R T takes points from a set of shapes P := ∪ i M i and maps them to a space R T . Ideally, the descriptors of points that are at corresponding locations on the shapes should have a small distance in the descriptor space. Points at distinct locations on the shapes should be mapped to distinct locations in the descriptor space (see Figure 1).Comparing points with a point descriptor: Ideally the point descriptor φ : P → R T should map corresponding points to nearby locations and noncorresponding points to distinct locations.In general one cannot assume that a given descriptor φ groups similar points as well as depicted in Fig. 1. The proposed method optimizes for the positive semi-definite matrix M = L T L inducing a pseudo distance in the descriptor space R T via d 2 M (x, y) = x − y, x − y M such that the point descriptors are grouped as good as possible. Optimizing for M is equivalent to looking for the best linear transformation L of the descriptor space with respect to the Euclidean distance, since d M (x, y) = L(x − y) . In Figure 2 we see that L projects the images of φ onto the dotted line resulting in the much better descriptor L • φ . As an optimization criterion for L we use LMNN [4] (see Figure 3). Figure 4: Qualitative comparison of descriptors: Distance maps between the descriptor at a reference point (indicated by a red arrow) and the descriptors computed on the shape after deformation. Colours range from blue (small distance) to red (large distance). Qualitatively, WKS and the proposed method do very well at indicating the right location while the cubic spline descriptor exhibits several local minima across the shape. Both test shapes are from the class michael (TOSCA), whereas the proposed descriptor and the spline descriptor were trained on the class david. The distances on the reference shape a generated by the proposed method.

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