49,349 research outputs found
Pants Decomposition of the Punctured Plane
A pants decomposition of an orientable surface S is a collection of simple
cycles that partition S into pants, i.e., surfaces of genus zero with three
boundary cycles. Given a set P of n points in the plane, we consider the
problem of computing a pants decomposition of the surface S which is the plane
minus P, of minimum total length. We give a polynomial-time approximation
scheme using Mitchell's guillotine rectilinear subdivisions. We give a
quartic-time algorithm to compute the shortest pants decomposition of S when
the cycles are restricted to be axis-aligned boxes, and a quadratic-time
algorithm when all the points lie on a line; both exact algorithms use dynamic
programming with Yao's speedup.Comment: 5 pages, 1 grayscale figur
Quantifying Homology Classes
We develop a method for measuring homology classes. This involves three
problems. First, we define the size of a homology class, using ideas from
relative homology. Second, we define an optimal basis of a homology group to be
the basis whose elements' size have the minimal sum. We provide a greedy
algorithm to compute the optimal basis and measure classes in it. The algorithm
runs in time, where is the size of the simplicial
complex and is the Betti number of the homology group. Third, we
discuss different ways of localizing homology classes and prove some hardness
results
Computing Optimal Morse Matchings
Morse matchings capture the essential structural information of discrete
Morse functions. We show that computing optimal Morse matchings is NP-hard and
give an integer programming formulation for the problem. Then we present
polyhedral results for the corresponding polytope and report on computational
results
On Computing the Translations Norm in the Epipolar Graph
This paper deals with the problem of recovering the unknown norm of relative
translations between cameras based on the knowledge of relative rotations and
translation directions. We provide theoretical conditions for the solvability
of such a problem, and we propose a two-stage method to solve it. First, a
cycle basis for the epipolar graph is computed, then all the scaling factors
are recovered simultaneously by solving a homogeneous linear system. We
demonstrate the accuracy of our solution by means of synthetic and real
experiments.Comment: Accepted at 3DV 201
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