Elastic Geometric Shape Matching

Abstract

In computational geometry, geometric shape matching (GSM) problems are among the classical and well-studied geometric optimization problems. In a conventional GSM problem, the pattern P and the model Q, both from a class S of geometric shapes, are given, along with a suitable distance measure d. The task is to compute a single transformation t from an admissible transformation class T acting on S that minimizes the distance between the transformed pattern and the model according to the distance measure d. Problems of this flavor have many applications such as traffic sign recognition, character recognition, human-computer-interaction, etc., in different scientific fields such as robotics, computer aided medicine and drug design. Yet in cases where local distortions or complex deformations occur, a single transformation from a simple transformation class is not enough to match the pattern well to the model. A more flexible approach is needed. Elastic geometric shape matching (EGSM) is a generalization of the conventional GSM and was designed with the intention of ensuring a both globally consistent and locally precise mapping in cases where the classical GSM approach is too restrictive. In an EGSM problem, one is given a pattern P and a model Q along with a graph G. The pattern P is partitioned into subshapes and instead of a single transformation, a so-called transformation ensemble is computed. A transformation ensemble consists of a set of transformations, one for each subshape of P, that are individually applied to the subshapes of P with the goal to minimize the distance of the transformed pattern to the model according to a suitable distance measure. Additionally, some of the transformations are enforced to be similar with respect to a suitable similarity measure defined for the transformation class at hand. In doing so, the consistency and “continuity” of the ensemble, and consequently, also of the transformed pattern, is ensured. The graph G is called neighborhood graph, and encodes which transformations need to be similar. There is a vast number of variations of EGSM problems, depending on how the different options for, eg., S, T , the distance measures, and structure of G, are chosen to match the application at hand. In particular, just slight changes in the problem setup may result in the need of completely different strategies to compute a solution. In this thesis, we analyze the computational complexity of several EGSM problem variants under translations under the L1-, the L2- and under polygonal norms for different distance measures and graph classes. Additionally, we present exact and approximative algorithms to solve the considered problem variants.