Probabilistic Matching of Planar Regions

We analyze a probabilistic algorithm for matching shapes modeled by planar regions under translations and rigid motions (rotation and translation). Given shapes $A$ and $B$, the algorithm computes a transformation $t$ such that with high probability the area of overlap of $t(A)$ and $B$ is close to maximal. In the case of polygons, we give a time bound that does not depend significantly on the number of vertices

Computation of the Hausdorff distance between sets of line segments in parallel

We show that the Hausdorff distance for two sets of non-intersecting line segments can be computed in parallel in $O(\log^2 n)$ time using O(n) processors in a CREW-PRAM computation model. We discuss how some parts of the sequential algorithm can be performed in parallel using previously known parallel algorithms; and identify the so-far unsolved part of the problem for the parallel computation, which is the following: Given two sets of $x$-monotone curve segments, red and blue, for each red segment find its extremal intersection points with the blue set, i.e. points with the minimal and maximal $x$-coordinate. Each segment set is assumed to be intersection free. For this intersection problem we describe a parallel algorithm which completes the Hausdorff distance computation within the stated time and processor bounds

Empty pentagons in point sets with collinearities

An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328k^2 points in the plane contains an empty pentagon or k collinear points. This is optimal up to a constant factor since the (k-1)x(k-1) grid contains no empty pentagon and no k collinear points. The previous best known bound was doubly exponential.Comment: 15 pages, 11 figure

Shape matching by random sampling

AbstractIn order to determine the similarity between two planar shapes, which is an important problem in computer vision and pattern recognition, it is necessary to first match the two shapes as well as possible. As sets of allowed transformation to match shapes we consider translations, rigid motions, and similarities. We present a generic probabilistic algorithm based on random sampling for matching shapes which are modelled by sets of curves. The algorithm is applicable to the three considered classes of transformations. We analyze which similarity measure is optimized by the algorithm and give rigorous bounds on the number of samples necessary to get a prespecified approximation to the optimal match within a prespecified probability

Probabilistic matching of sets of Polygonal curves

Analysis and comparison of geometric shapes are of importance in various application areas within computer science, e.g., pattern recognition and computer vision. The general situation in a shape matchin

Probabilistic Matching and Resemblance Evaluation of Shapes in Trademark Images

We present a novel matching and similarity evaluation method for planar geometric shapes represented by sets of polygonal curves. Given two shapes, the matching algorithm randomly generates a point sample from each shape and records a vote for a transformation which maps one sample to the other. The experiment is repeated many times. Clusters of votes in the transformation space indicate good candidate transformations for matching the two shapes. Unlike most voting schemes, though, the samples taken in one random experiment are extended as much as possible and the vote is weighted depending on the samples. The best clusters are those with a large total weight. The second part of the method is a resemblance evaluation of the two matched shapes. The definition of our resemblance function incorporates the proximity of line segments as well as the similarity of their slopes. The system is evaluated using the MPEG-7 shape silhouette database and a collection of 10 745 trade mark images. The experiments demonstrate a high performance of our algorithms for contour shapes as well as for trademark images