3D Non-rigid Surface Matching and Registration Based on Holomorphic Differentials

Abstract

3D surface matching is fundamental for shape registration, deformable 3D non-rigid tracking, recognition and classification. In this paper we describe a novel approach for generating an efficient and optimal combined matching from multiple boundary-constrained conformal parameterizations for multiply connected domains (i.e., genus zero open surface with multiple boundaries), which always come from imperfect 3D data acquisition (holes, partial occlusions, change of pose and nonrigid deformation between scans). This optimality criterion is also used to assess how consistent each boundary is, and thus decide to enforce or relax boundary constraints across the two surfaces to be matched. The linear boundary-constrained conformal parameterization is based on the holomorphic differential forms, which map a surface with n boundaries conformally to a planar rectangle with (n − 2) horizontal slits, other two boundaries as constraints. The mapping is a diffeomorphism and intrinsic to the geometry, handles an open surface with arbitrary number of boundaries, and can be implemented as a linear system. Experimental results are given for real facial surface matching, deformable cloth nonrigid tracking, which demonstrate the efficiency of our method, especially for 3D non-rigid surfaces with significantly inconsistent boundaries.

Möbius voting for surface correspondence

“…For example, [Wang et al 2007] use point constraints predicted by spin images [Johnson and Hebert 1999] to establish a sparse set of correspondences and then use least-squares conformal mapping to create a cross-parameterization. In later work, [Zeng et al 2008] further elaborate this direction by cutting the surfaces into patches, given user-defined boundary correspondence, and combine several comformal mappings. Furthermore, they measure deformation error by an integral of the differences between conformal factors and curvatures over the domain of the map.…”

“…For example, Jin et al[2004] and Zeng et al[2008] define the distortion as the L 2 difference of the conformal factors. However, in our case the distance measure should not be a "regular" L p norm.…”

Dense non-rigid surface registration using high-order graph matching

“…In applications such as recognition of subtle facial expressions, there are localized, high-degree of freedom deformations. To tackle this problem, several approaches have been developed to obtain dense point correspondences by embedding the surfaces to a canonical domain which preserves the geodesics or angles [5,6,29,32,33]. Such embedding requires an initial set of feature correspondences or boundary conditions.…”

“…For example, in [2,26], only around 100 correspondences are found using geodesic information. The conformal mapping approach ( [33,29,32]) is more flexible. According to the uniformization theory [14], any 3D surface can be conformally mapped to a 2D domain.…”

Shape matching based on diffusion embedding and on mutual isometric consistency

“…Most of the previous attempts of shape matching can be broadly categorized as extrinsic or intrinsic approaches depending on how they analyze the properties of the underlying manifold. Intrinsic approaches are a natural choice for finding dense correspondences between articulated shapes, as they embed the shape in some canonical domain which preserves some important properties of the manifold, e.g., geodesics and angles [4,29,16,31,22,24,18,10].…”