Dense Point‐to‐Point Correspondences Between Genus‐Zero Shapes

Abstract

We describe a novel approach that addresses the problem of establishing correspondences between non‐rigidly deformed shapes by performing the registration over the unit sphere. In a pre‐processing step, each shape is conformally parametrized over the sphere, centered to remove Möbius inversion ambiguity, and authalically evolved to expand regions that are excessively compressed by the conformal parametrization. Then, for each pair of shapes, we perform fast SO(3) correlation to find the optimal rotational alignment and refine the registration using optical flow. We evaluate our approach on the TOSCA dataset, demonstrating that our approach compares favorably to state‐of‐the‐art methods.

Posture-Invariant Three Dimensional Human Hand Statistical Shape Model

“…Ma et al [38] introduced the non-rigid point set registration method by taking the advantages of Gaussian mixture models, which was able to robustly preserve both global and local structures during matching. The dense point-topoint correspondence algorithm proposed by Lee and Kazhdan [39] is a novel non-rigid registration algorithm for watertight objects. By mapping objects to unit Möbius spheres using a conformal map, the alignment between meshes can be established by performing Fast Fourier Transformation (FFT), and the registration is refined based on the optical flow.…”

“…using non-rigid ICP registration algorithms. In this paper, we adopted the algorithm developed by Lee and Kazhdan [39] to register the template to each mesh in for establishing a map as | → . The algorithm converts the correspondence problem between two meshes into the registration problem over two depicted spheres using the following steps: 1) Computing the conformal parameterization of the mesh over the sphere.…”

“…Eventually, the surface after utilizing the conformalized mean curvature flow is converged onto the sphere; 2) Authalic evolution. As the conformal parameterization of the mesh over a sphere may lead to the extreme compressive areas on the sphere, an authalic evolution approach is adopted [39] to decompress those areas; 3) Compute the Heat Kernel Signature (HKS) [52] of each vertex in the source and the target meshes, where the scan and the template can be represented as and , respectively; 4) Rotational alignment.…”

“…In this step, the correspondence between two mesh is established by finding the rotational * that maximizes the correlation between and as [39]:…”

“…where the inner-product 〈•〉 2 is obtained by integrating the point-wise product of the HKS of vertices over the sphere [39]. ( ) describes the rotation result of by the rotation .…”