Formenvergleich in höheren Dimensionen

Cover and Contents 1 Introduction 1.1 Overview 1.2 Credits 2 Preliminaries 2.1 Representation of Shapes 2.2 Distance Measures 2.3 Miscellaneous 3 Hausdorff Distance Under Translations 3.1 Overview 3.2 Basic Properties of \delta;->H 3.3 Matching Points to Sites 3.4 Matching Two Sets of Sites 3.5 Approximate Algorithms 4 Matching Special Shape Classes Under Translations 4.1 Matching Terrains 4.2 Matching Convex Polyhedra 5 Matching Curves with respect to the Fréchet Distance 5.1 Basic Properties of the Fréchet Distance 5.2 Polygonal Curves Under Translations 5.3 Polygonal Curves Under Affine Transformations 5.4 Variants 6 Matching a Polygonal Curve in a Graph of Curves 6.1 Problem Statement 6.2 Algorithm 6.3 Variants Bibliography Index A Zusammenfassung B LebenslaufThe comparison of geometric shapes is a task which naturally arises in many applications, such as in computer vision, computer aided design, robotics, medical imaging, etc. Usually geometric shapes are represented by a number of simple objects (sites) that either describe the boundary of the shape, or the whole shape itself. Sites are often chosen to be linear objects, such as line segments, triangles, or simplices in general, since linear objects are easier to handle in algorithms. But sometimes also patches of algebraic curves or surfaces, such as circular arcs or portions of spheres or cylinders are of interest. In order to compare two shapes we need to have a notion of similarity or dissimilarity, which arises from the desired application. There is a large variety of different similarity measures. Popular similarity notions are, for example, the Hausdorff distance, the area of symmetric difference, or especially for curves the turn-angle distance, or the Fréchet distance. The application usually supplies a distance measure, and furthermore a set of allowed transformations, and the task is to find a transformation that, when applied to the first object, minimizes the distance to the second one. Typical transformation classes are translations, rotations, and rigid motions (which are combinations of translations and rotations). The contribution of this thesis consists of several algorithms for matching simplicial shapes in dimensions d >= 2. The shapes are either represented as sets of simplicial objects or as polygonal curves with a given parametrization. The considered distance measures are mainly the Hausdorff distance and the Fréchet distance. In the literature most matching algorithms either attack two-dimensional problems, or consider finite point sets in higher dimensions. In the first half of this thesis we present results for the Hausdorff distance in d >= 2 dimensions under translations, for a rather general notion of simplicial shapes, as well as for some special shape classes which allow to speed up the computations. In the second half of this thesis we investigate the Fréchet distance for polygonal curves. The Fréchet distance is a natural distance measure for curves, but has not been investigated much in the literature. We present the first algorithms to optimize the Fréchet distance under various transformation classes for polygonal curves in arbitrary dimensions. In the last chapter we consider a partial matching variant in which a geometric graph and another curve are given, and we show how to find a polygonal path in the graph which minimizes the Fréchet distance to the curve.Das Vergleichen zweier geometrischer Formen ist eine Aufgabe, die aus vielerlei Anwendungen natürlich hervorgeht. Einige Anwendungen sind Computer Vision, Computer Graphik, Computer Aided Design, Robotics, medizinische Bilderverarbeitung, etc. Normalerweise werden geometrische Formen aus einfacheren Objekten zusammengesetzt, die entweder den Rand der Form oder die ganze Form ansich beschreiben. Oft verwendet werden lineare Objekte wie Strecken, Dreicke, oder Simplizes in höheren Dimensionen. Um zwei Formen zu vergleichen braucht man zunächst einen Ähnlichkeits- oder Abstandsbegriff zwischen zwei Formen, der in der Regel aus der jeweiligen Anwendung hervorgeht. Naturgemäß gibt es eine große Vielfalt solcher Abstandsmaße; eines der natürlichsten ist der Hausdorff-Abstand. Weiterhin gibt die Anwendung in der Regel eine Menge von Transformationen vor, und möchte eine Transformation finden, die, angewandt auf die erste Form, den Abstand zur zweiten Form minimiert. Diese Aufgabe wird als Matching bezeichnet. Oft verwendete Transformationsklassen sind zum Beispiel Translationen, Rotationen und starre Bewegungen (Kombinationen von Translationen und Rotationen). Diese Arbeit beschäftigt sich mit dem Matching von geometrischen Formen in Dimensionen d >= 2, die aus stückweise linearen Objekten bestehen. Die Formen sind entweder als Mengen solcher Objekte, oder als Polygonzüge, die als parametrisierte Kurven aufgefaßt werden, beschrieben. Als Abstandsmaße werden hauptsächlich der Hausdorff-Abstand und der Fréchet-Abstand betrachtet. Bisherige Ergebnisse für das Matching von Formen behandeln in der Regel entweder zweidimensionale Formen, oder Punktmengen in höheren Dimensionen. Die erste Hälfte dieser Dissertation präsentiert Ergebnisse für den Hausdorff- Abstand in d >= 2 Dimensionen unter Translationen für einen allgemein gehaltenen Formenbegriff, sowie für einige spezielle Klassen geometrischer Formen, die eine schnellere Berechnung erlauben. Die zweite Hälfte der Dissertation beschäftigt sich mit dem Matching von parametrisierten Kurven bezüglich des Fréchet-Abstandes. Obwohl der Fréchet-Abstand ein natürliches Abstandsmaß für Kurven darstellt, gibt es bisher diesbezüglich wenig Ergebnisse in der Literatur. Für parametrisierte Kurven in d >= 2 Dimensionen wird in dieser Dissertation ein Matching-Algorithmus vorgestellt, der unter Translationen und relativ allgemein gehaltenen Teilmengen der affinen Abbildungen den Fréchet-Abstand minimiert. Als letztes Ergebnis wird eine weitere Matching-Variante bezüglich des Fréchet-Abstandes vorgestellt, in der eine Teilkurve in in einem eingebetteten planaren Graphens gefunden werden soll, die den Fréchet-Abstand zu einer gegebenen Kurve minimiert

Applying an edit distance to the matching of tree ring sequences in dendrochronology

AbstractIn dendrochronology wood samples are dated according to the tree rings they contain. The dating process consists of comparing the sequence of tree ring widths in the sample to a dated master sequence. Assuming that a tree forms exactly one ring per year a simple sliding algorithm solves this matching task.But sometimes a tree produces no ring or even two rings in a year. If a sample sequence contains this kind of inconsistencies it cannot be dated correctly by the simple sliding algorithm. We therefore introduce a O(α2mn+α4(m+n)) algorithm for dating such a sample sequence against an error-free master sequence, where n and m are the lengths of the sequences. Our algorithm takes into account that the sample might contain up to α missing or double rings and suggests possible positions for these kind of inconsistencies. This is done by employing an edit distance as the distance measure

Fast Frechet Distance Between Curves With Long Edges

Computing the Fr\'echet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fr\'echet distance computations become easier. Let $P$ and $Q$ be two polygonal curves in $\mathbb{R}^d$ with $n$ and $m$ vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fr\'echet distance between them: (1) a linear-time algorithm for deciding the Fr\'echet distance between two curves, (2) an algorithm that computes the Fr\'echet distance in $O((n+m)\log (n+m))$ time, (3) a linear-time $\sqrt{d}$-approximation algorithm, and (4) a data structure that supports $O(m\log^2 n)$-time decision queries, where $m$ is the number of vertices of the query curve and $n$ the number of vertices of the preprocessed curve

Inferring movement patterns from geometric similarity

Spatial movement data nowadays is becoming ubiquitously available, including data of animals, vehicles and people. This data allows us to analyze the underlying movement. In particular, it allows us to infer movement patterns, such as recurring places and routes. Many methods to do so rely on the notion of similarity of places or routes. Here we briefly survey how research on this has developed in the past 15 years and outline challenges for future work

On the Reconstruction of Geodesic Subspaces of RN\mathbb{R}^N

We consider the topological and geometric reconstruction of a geodesic subspace of $\mathbb{R}^N$ both from the \v{C}ech and Vietoris-Rips filtrations on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique leverages the intrinsic length metric induced by the geodesics on the subspace. We consider the distortion and convexity radius as our sampling parameters for a successful reconstruction. For a geodesic subspace with finite distortion and positive convexity radius, we guarantee a correct computation of its homotopy and homology groups from the sample. For geodesic subspaces of $\mathbb{R}^2$, we also devise an algorithm to output a homotopy equivalent geometric complex that has a very small Hausdorff distance to the unknown shape of interest