Geodesic Distance Descriptors

Abstract

The Gromov-Hausdorff (GH) distance is traditionally used for measuring distances between metric spaces. It was adapted for non-rigid shape comparison and matching of isometric surfaces, and is defined as the minimal distortion of embedding one surface into the other, while the optimal correspondence can be described as the map that minimizes this distortion. Solving such a minimization is a hard combinatorial problem that requires pre-computation and storing of all pairwise geodesic distances for the matched surfaces. A popular way for compact representation of functions on surfaces is by projecting them into the leading eigenfunctions of the Laplace-Beltrami Operator (LBO). When truncated, The basis of the LBO is known to be the optimal for representing functions with bounded gradient in a min-max sense. Methods such as Spectral-GMDS exploit this idea to simplify and efficiently approximate a minimization related to the GH distance by operating in the truncated spectral domain, and obtain state of the art results for matching of nearly isometric shapes. However, when considering only a specific set of functions on the surface, such as geodesic distances, an optimized basis could be considered as an even better alternative. Moreover, current simplifications of approximating the GH distance introduce errors due to low rank approximations and relaxations of the permutation matrices.Here, we define the geodesic distance basis, which is optimal for compact approximation of geodesic distances, in terms of Frobenius norm. We use the suggested basis to extract the Geodesic Distance Descriptor (GDD), which encodes the geodesic distances information as a linear combination of the basis functions. We then show how these ideas can be used to efficiently and accurately approximate the metric spaces matching problem with almost no loss of information. We incorporate recent methods for efficient approximation of the proposed basis and descriptor without actually computing and storing all geodesic distances. These observations are used to construct a very simple and efficient procedure for shape correspondence. Experimental results show that the GDD improves both accuracy and efficiency of state of the art shape matching procedures.

Anisotropic Spectral Manifold Wavelet Descriptor

“…[COO15] introduced a multi‐scale signature based on the topological structure of the distribution of geodesic distances centered at a given point. [SK17] defined a set of geodesic distance bases and used them to build the Geodesic Distance Descriptor (GDD), which encoded the geodesic distance information as a linear combination of the basis functions. However, the above‐stated geodesic distance based methods suffer the problem that they are strong sensitive to shape topological noises.…”

Multidimensional Scaling

“…By setting the weights in (36) to (38) we get an equivalent LS-MDS problem to (35). The IRLS algorithm works by solving a series of LS-MDS problems with iteratively updated weights:…”

“…A spectral version of GMDS is developed in [34], following the spectral MDS algorithm [21]. Another form of non-Euclidean embedding is given in [35], who suggested to decompose the distance matrix D into XX . Since D is symmetric, such a decomposition is always possible, based on the spectral decomposition of D…”

“…These canonical forms can be aligned with a rigid transformation, and a threshold on the alignment mismatch rules weather two shapes belong to the same class or not. This idea has been carried out using multiple versions of the MDS algorithms [8,21,23,24,35]. Alternatively, the generalized MDS scheme (44) [33] or its spectral version [34] can compare two surfaces directly, without passing through an intermediate embedding space.…”

Functional Maps Representation On Product Manifolds

“…One of the key innovations of this framework is allowing bringing a new set of algebraic methods into the domain of shape correspondence. Several follow‐up works tried to improve the framework by employing sparsity‐based priors [PBB*13], manifold optimization [KBB*13, KGB16], non‐orthogonal [KBBV15] or localized [CSBK17, MRCB18] bases, coupled optimization over the forward and inverse maps [ERGB16, EBC17, HO17], combination of functional maps with metric‐based approaches [ADK16, SK17] and kernelization [WGBS18]. Recent works of [NO17, NMR*18] considered functional algebra (function point‐wise multiplications together with addition).…”