Probabilistic Matching of Planar Shapes
In this thesis we study a probabilistic approach for the shape matching problem. The studied approach is based on an intuitive deﬁnition of the shape matching task: Given two shapes A and B ﬁnd that transformation within the class of allowable transformations which maps B to A in a best possible way. A mapping is considered to be good if large parts of the two shapes coincide within some tolerance distance delta.
We assume that the shapes are modeled by ﬁnite sets of rectifiable curves in the plane. As possible classes of transformations we consider sub-classes of affine transformations: translations, rigid motions (translations and rotations), similarity maps (translation, rotation, and scaling), homotheties (translation and scaling), shear transformations, and affine maps.
The major idea of the probabilistic algorithm is to take random samples of points from both shapes and give a "vote" for that transformation matching one sample with the other. If that experiment is repeated frequently, we obtain by the votes a certain probability distribution in the space of transformations. Maxima of this distribution indicate which transformations give the best match between the two ﬁgures. The matching step of the algorithm is, therefore, a voting scheme.
In this thesis we analyze the similarity measure underlying the algorithm and give rigorous bounds on the runtime (number of experiments) necessary to obtain the optimal match within a certain approximation factor with a prespecified probability. We perform a generic analysis of the algorithm for arbitrary transformation classes, as well as an in-depth analysis for diﬀerent sub-classes of affine transformations. It is also shown that the density function of the vote distribution is exactly the normalized generalized Radon transform in the cases of translations and rigid motions.
We consider the theoretical analysis as the major contribution of this thesis, since it leads to a better understanding of this kind of heuristic techniques.