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

Probabilistisches Verfahren zum Vergleichen von Formen

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.In dieser Arbeit wird ein probabilistischer Ansatz zum Vergleichen von Formen untersucht. Der Ansatz entspricht der intuitiven Vorstellung von "Formanpassung": zwei Formen werden als ähnlich empfunden wenn es eine Transformation aus der Menge der erlaubten Transformationen gibt, die die beiden Formen gut zur Deckung bringt. Dabei bedeutet "gute Deckung" dass große Teile einer Form sich in der räumlichen Nähe der anderen Form befinden. Die Formen, für die wir den Ansatz untersuchen, werden als endliche Mengen von Kurvensegmenten endlicher Länge dargestellt. Als Menge der erlaubten Transformationen betrachten wir Unterklassen der affinen Abbildungen: Translationen (Parallelverschiebungen), starre Bewegungen (Translationen und Drehungen), Ähnlichkeitsabbildungen (Translationen, Drehungen und Skalierungen), Homothetien (Translationen und Skalierungen), Scherungen und affine Abbildungen selbst. Die Grundidee des Algorithmus ist folgende: Nimm zufällige Punktproben aus jeder der Formen und zeichne eine "Stimme" für die Transformation auf, die die Probe der einen Form auf die Probe der anderen Form abbildet. Nach mehrfacher Wiederholung des Experiments zeichnet sich im Transformationsraum eine gewisse Verteilung der Stimmen aus. Die Maxima dieser Verteilung deuten "gute" Transformationen an, wobei eine gute Transformation eine ist, die große Teile der Formen zur Deckung bringt. In dieser Arbeit untersuchen wir das dem Algorithmus zu Grunde liegende Ähnlichkeitsmaß und bestimmen die notwendige Anzahl an Experimenten um eine gute Anpassung von Formen innerhalb einer vorgegebenen Approximationsschranke mit vorgegebener Erfolgswahrscheinlichkeit zu bestimmen. Der durchgeführte Analyse ist generisch und gilt für beliebige Transformationsklassen. Zusätzlich führen wir eine ausführliche Analyse für die oben genannten Transformationsklassen durch. Weiterhin zeigen wir, dass die vom Experiment induzierte Wahrscheinlichkeitsdichtefunktion im Transformationsraum genau der verallgemeinerten Radon-Transformation für Translationen und starre Bewegungen entspricht. Die theoretische Analyse betrachten wir als den Hauptbeitrag dieser Arbeit, denn sie führt zum besseren Verständnis einer Klasse von Heuristiken die unter dem Sammelnamen "Abstimmungsmethoden" bekannt ist

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