Optimal Intrinsic Descriptors for Non-Rigid Shape Analysis

Abstract

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.

Robust Nonrigid Registration by Convex Optimization

“…While this strategy has been successful for rigid matching [1], its application to nonrigid registration is complicated by the requirement that the descriptors themselves be invariant to deformations. Many intrinsic descriptors have been proposed [35,42,3,25,50]. The downside of their invariance to isometric deformations is their sensitivity to gross topological inconsistencies.…”

Dense Human Body Correspondences Using Convolutional Networks

“…A second popular objective is to match selected subsets of points on the two surfaces with similar feature descriptors [38,54,27,5]. However, finding descriptors that are both invariant to typical human and clothing deformations and also robust to topological changes remains a challenge.…”

4D Human Body Correspondences from Panoramic Depth Maps

“…Their techniques, however, still assume that input surfaces are nearly isometric and therefore cannot handle complex, articulated human motions. More recent approaches aim to design feature descriptors [51,28] matching points. Pottemann et al [36] use local geometric descriptors that can handle small motions.…”