Elastic Net Constraints for Shape Matching

Abstract

We consider a parametrized relaxation of the widely adopted quadratic assignment problem (QAP)

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

“…quasiisometric deformations, topological changes, displacement noise and changes in scale (we refer to [23] for a detailed numerical breakdown). Differently from most existing methods, the approach presented in this Section is quite general and not restricted to the quasi-isometric case.…”

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

“…Even though originating from distinct motivations, the first two methods share a convenient interpretation as partitioning problems in the space of potential assignments. In Section 3.3 we provide a different view on the problem, as presented in [23,22], by using the language of regression analysis.…”

“…A detailed explanation of our approach on the computation of the unique minimumdistance projection Π α in an efficient manner is given in [23]. Also note that, for practical purposes, we adopt a more efficient alternative to the standard projected gradient descent (3.5), namely its acceleration via vector extrapolation techniques [22].…”

“…The two approaches share the common perspective of minimizing a distortion criterion, derived from the metric information which the shapes to be matched are endowed with. In one case (Section 3), following [23], we consider a notion of pairwise metric distortion that directly captures to what extent two shapes can be isometrically put in correspondence. Motivated by the observation that good correspondences often come at the price of high sparsity (in terms of number of matched points), whereas large cardinality tends to bring distorted matches into the correspondence, we attempt to control the accuracy/sparsity trade-off by introducing a weighting parameter on the combination of two effective relaxations [12,21], which we relate to their regularizer counterparts from regression analysis.…”