114,042 research outputs found
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
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
A Bayesian Approach to Manifold Topology Reconstruction
In this paper, we investigate the problem of statistical reconstruction of piecewise linear manifold topology. Given a noisy, probably undersampled point cloud from a one- or two-manifold, the algorithm reconstructs an approximated most likely mesh in a Bayesian sense from which the sample might have been taken. We incorporate statistical priors on the object geometry to improve the reconstruction quality if additional knowledge about the class of original shapes is available. The priors can be formulated analytically or learned from example geometry with known manifold tessellation. The statistical objective function is approximated by a linear programming / integer programming problem, for which a globally optimal solution is found. We apply the algorithm to a set of 2D and 3D reconstruction examples, demon-strating that a statistics-based manifold reconstruction is feasible, and still yields plausible results in situations where sampling conditions are violated
Computing the Similarity Between Moving Curves
In this paper we study similarity measures for moving curves which can, for
example, model changing coastlines or retreating glacier termini. Points on a
moving curve have two parameters, namely the position along the curve as well
as time. We therefore focus on similarity measures for surfaces, specifically
the Fr\'echet distance between surfaces. While the Fr\'echet distance between
surfaces is not even known to be computable, we show for variants arising in
the context of moving curves that they are polynomial-time solvable or
NP-complete depending on the restrictions imposed on how the moving curves are
matched. We achieve the polynomial-time solutions by a novel approach for
computing a surface in the so-called free-space diagram based on max-flow
min-cut duality
Approximating Dynamic Time Warping and Edit Distance for a Pair of Point Sequences
We give the first subquadratic-time approximation schemes for dynamic time
warping (DTW) and edit distance (ED) of several natural families of point
sequences in , for any fixed . In particular, our
algorithms compute -approximations of DTW and ED in time
near-linear for point sequences drawn from k-packed or k-bounded curves, and
subquadratic for backbone sequences. Roughly speaking, a curve is
-packed if the length of its intersection with any ball of radius
is at most , and a curve is -bounded if the sub-curve
between two curve points does not go too far from the two points compared to
the distance between the two points. In backbone sequences, consecutive points
are spaced at approximately equal distances apart, and no two points lie very
close together. Recent results suggest that a subquadratic algorithm for DTW or
ED is unlikely for an arbitrary pair of point sequences even for . Our
algorithms work by constructing a small set of rectangular regions that cover
the entries of the dynamic programming table commonly used for these distance
measures. The weights of entries inside each rectangle are roughly the same, so
we are able to use efficient procedures to approximately compute the cheapest
paths through these rectangles
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