Geometrically consistent elastic matching of 3D shapes: A linear programming solution

Abstract

We propose a novel method for computing a geometrically consistent and spatially dense matching between two 3D shapes. Rather than mapping points to points we match infinitesimal surface patches while preserving the geometric structures. In this spirit we consider matchings as diffeomorphisms between the objects' surfaces which are by definition geometrically consistent. Based on the observation that such diffeomorphisms can be represented as closed and continuous surfaces in the product space of the two shapes we are led to a minimal surface problem in this product space. The proposed discrete formulation describes the search space with linear constraints. Computationally, our approach leads to a binary linear program whose relaxed version can be solved efficiently in a globally optimal manner. As cost function for matching, we consider a thin shell energy, measuring the physical energy necessary to deform one shape into the other. Experimental results demonstrate that the proposed LP relaxation allows to compute highquality matchings which reliably put into correspondence articulated 3D shapes. Moreover a quantitative evaluation shows improvements over existing works.

Optical techniques for 3D surface reconstruction in computer-assisted laparoscopic surgery

“…e.g. (Bronstein et al, 2011;Funkhouser and Shilane, 2006;Gelfand et al, 2005;Lipman and Funkhouser, 2009;Wang et al, 2010;Windheuser et al, 2011;Zeng et al, 2010;Zhang et al, 2008), the only fully-automatic non-rigid approaches to intra-operative registration of range data in abdominal procedures have been applied in open surgery (dos and do not provide real-time performance. To avoid the computational demands of repeating the registration process over time, an alternative registration approach involves continuously updating an initially performed registration via tissue tracking using the endoscopic image information acquired during surgery.…”

Relaxations for Minimizing Metric Distortion and Elastic Energies for 3D Shape Matching

“…In this section we discuss the matching energies that have been used to find correspondences among shapes in [23] and [30] respectively. 2.1.…”

“…This leads us to the introduction of the elastic net penalty function [33] into shape matching problems. Differently, our second approach [30] takes a physically motivated view on the problem and minimizes a functional that encodes the physical deformation energy [15,31] necessary to deform one shape into the other. The formulation we give in Section 4 is based on finding an optimal surface of codimension 2 in the product of the two shape surfaces.…”

Elastic Net Constraints for Shape Matching

“…While methods based on uniformization theory are made attractive by the low dimensionality of the embedding domain, they do not behave well with different kinds of deformations (e.g., topological changes), and are subject to global inconsistencies in the final mapping. More recently, Windheuser et al [16] gave a linear programming relaxation to the matching problem; the method notably allows to obtain continuous correspondences, but it is sensitive to topological changes and, as noted by the authors, its GPU implementation takes about 2 hours per matching.…”