7,626 research outputs found
Pattern Matching for sets of segments
In this paper we present algorithms for a number of problems in geometric
pattern matching where the input consist of a collections of segments in the
plane. Our work consists of two main parts. In the first, we address problems
and measures that relate to collections of orthogonal line segments in the
plane. Such collections arise naturally from problems in mapping buildings and
robot exploration.
We propose a new measure of segment similarity called a \emph{coverage
measure}, and present efficient algorithms for maximising this measure between
sets of axis-parallel segments under translations. Our algorithms run in time
O(n^3\polylog n) in the general case, and run in time O(n^2\polylog n) for
the case when all segments are horizontal. In addition, we show that when
restricted to translations that are only vertical, the Hausdorff distance
between two sets of horizontal segments can be computed in time roughly
O(n^{3/2}{\sl polylog}n). These algorithms form significant improvements over
the general algorithm of Chew et al. that takes time . In the
second part of this paper we address the problem of matching polygonal chains.
We study the well known \Frd, and present the first algorithm for computing the
\Frd under general translations. Our methods also yield algorithms for
computing a generalization of the \Fr distance, and we also present a simple
approximation algorithm for the \Frd that runs in time O(n^2\polylog n).Comment: To appear in the 12 ACM Symposium on Discrete Algorithms, Jan 200
Many-to-Many Graph Matching: a Continuous Relaxation Approach
Graphs provide an efficient tool for object representation in various
computer vision applications. Once graph-based representations are constructed,
an important question is how to compare graphs. This problem is often
formulated as a graph matching problem where one seeks a mapping between
vertices of two graphs which optimally aligns their structure. In the classical
formulation of graph matching, only one-to-one correspondences between vertices
are considered. However, in many applications, graphs cannot be matched
perfectly and it is more interesting to consider many-to-many correspondences
where clusters of vertices in one graph are matched to clusters of vertices in
the other graph. In this paper, we formulate the many-to-many graph matching
problem as a discrete optimization problem and propose an approximate algorithm
based on a continuous relaxation of the combinatorial problem. We compare our
method with other existing methods on several benchmark computer vision
datasets.Comment: 1
Computation of protein geometry and its applications: Packing and function prediction
This chapter discusses geometric models of biomolecules and geometric
constructs, including the union of ball model, the weigthed Voronoi diagram,
the weighted Delaunay triangulation, and the alpha shapes. These geometric
constructs enable fast and analytical computaton of shapes of biomoleculres
(including features such as voids and pockets) and metric properties (such as
area and volume). The algorithms of Delaunay triangulation, computation of
voids and pockets, as well volume/area computation are also described. In
addition, applications in packing analysis of protein structures and protein
function prediction are also discussed.Comment: 32 pages, 9 figure
Planar maps, circle patterns and 2d gravity
Via circle pattern techniques, random planar triangulations (with angle
variables) are mapped onto Delaunay triangulations in the complex plane. The
uniform measure on triangulations is mapped onto a conformally invariant
spatial point process. We show that this measure can be expressed as: (1) a sum
over 3-spanning-trees partitions of the edges of the Delaunay triangulations;
(2) the volume form of a K\"ahler metric over the space of Delaunay
triangulations, whose prepotential has a simple formulation in term of ideal
tessellations of the 3d hyperbolic space; (3) a discretized version (involving
finite difference complex derivative operators) of Polyakov's conformal
Fadeev-Popov determinant in 2d gravity; (4) a combination of Chern classes,
thus also establishing a link with topological 2d gravity.Comment: Misprints corrected and a couple of footnotes added. 42 pages, 17
figure
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