Elastic Geometric Shape Matching for Point Sets under Translations
In geometric shape matching problems one is given a pattern P, a model Q, a distance measure d (which formalizes the intuitive notion of similarity of such shapes), and a class of geometric transformations applicable to the pattern P. The task is to find a transformation t in the given class that minimizes the distance of the transformed pattern t(P) to the model Q (as measured by d) in order to compute the similarity of the given shapes.
In many applications, among them medical-image analysis, industrial design, robotics or computer vision, where local distortions and complex deformations can occur, this setting is too restrictive, since only the single transformation t is used to align the entire pattern with the model. Almost all known strategies that deal with non-rigid deformations apply heuristics (based on local descent, relaxed LP formulations, simulated annealing, or alike). The quality of the solution found by these heuristics can usually not be related to the quality of a global optimum.
Elastic geometric shape matching tries to remedy this situation by computing a whole set of transformations T. Each transformation \(t\in T\) is applied to a subpattern of P. The objective of the optimization problem becomes twofold: Minimize the distance of the (union of the) transformed subpatterns to the model while also maximizing the similarity of the transformations in the ensemble T. This modeling aims at strategies that compute provably optimal solutions, or alternatively approximative results of a guaranteed quality.
We consider variations of a simple elastic geometric shape matching problem in the plane where each subshape is just a single point. We show that this problem already is NP-hard for the directed Hausdorff- or bottleneck distance under arbitrary translations. We complement our result with efficient algorithms to compute transformation ensembles under both distances for variants of the problem where only translations in a prescribed, fixed direction are allowed.
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