A game-theoretic approach to deformable shape matching

Abstract

We consider the problem of minimum distortion intrinsic correspondence between deformable shapes, many useful formulations of which give rise to the NP-hard quadratic assignment problem (QAP). Previous attempts to use the spectral relaxation have had limited success due to the lack of sparsity of the obtained "fuzzy" solution. In this paper, we adopt the recently introduced alternative L 1 relaxation of the QAP based on the principles of game theory. We relate it to the Gromov and Lipschitz metrics between metric spaces and demonstrate on state-of-the-art benchmarks that the proposed approach is capable of finding very accurate sparse correspondences between deformable shapes.

Dynamic Optimal Path Selection for 3D Triangulation with Multiple Cameras

“…This means that the constraints expressed in equation (2) are actually guaranteed to be enforced. This approach, while novel within the scenario covered by this paper, has shown to be very successful in addressing a wide range of problems, including featurebased matching [19], medical images segmentation [20] and registration [21], rigid [22] and non-rigid [23] 3D object recognition. To give to the reader a better understanding of how the process works we illustrate a complete case in Fig.…”

Elastic Net Constraints for Shape Matching

“…Given the inherent difficulty to solve for a minimum distortion correspondence under general deformations, in a re- cent paper [14] we proposed to shift the focus to the search of a maximal group of matches having least distortion, regardless of its cardinality. To achieve this, we proposed to optimize over the probability simplex…”

“…Following [11,14], we first relax the correspondence from a discrete to a fuzzy notion by letting c : X × Y → [0, 1], effectively setting off the problem from its combinatorial nature and bringing it to a continuous optimization domain. Further, we adopt the so called Gromov-Wasserstein [11] family of metrics, which give rise to a relaxed notion of proximity between shapes:…”

“…This idea of a fuzzy map of assignments is not novel, and can be traced back, for instance, to the softassign method for graph matching [6]. In the specific case of non-rigid shapes, fuzzy schemes are typically adopted to relax the point-to-point mappings [11,14]. 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.…”

“…Notable attempts at relaxing the NP-hard QAP include graduated assignment [6], spectral relaxation [8] and the more recent game-theoretic approach [14]. Motivated by the observation that good accuracy often comes at the price of high sparsity, 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, namely the spectral and game-theoretic techniques, which we relate to their regularizer counterparts from regression analysis.…”