Results 1  10
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30
Computing geodesics and minimal surfaces via graph cuts
 in International Conference on Computer Vision
, 2003
"... Geodesic active contours and graph cuts are two standard image segmentation techniques. We introduce a new segmentation method combining some of their benefits. Our main intuition is that any cut on a graph embedded in some continuous space can be interpreted as a contour (in 2D) or a surface (in 3D ..."
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Cited by 250 (27 self)
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Geodesic active contours and graph cuts are two standard image segmentation techniques. We introduce a new segmentation method combining some of their benefits. Our main intuition is that any cut on a graph embedded in some continuous space can be interpreted as a contour (in 2D) or a surface (in 3D). We show how to build a grid graph and set its edge weights so that the cost of cuts is arbitrarily close to the length (area) of the corresponding contours (surfaces) for any anisotropic Riemannian metric. There are two interesting consequences of this technical result. First, graph cut algorithms can be used to find globally minimum geodesic contours (minimal surfaces in 3D) under arbitrary Riemannian metric for a given set of boundary conditions. Second, we show how to minimize metrication artifacts in existing graphcut based methods in vision. Theoretically speaking, our work provides an interesting link between several branches of mathematicsdifferential geometry, integral geometry, and combinatorial optimization. The main technical problem is solved using CauchyCrofton formula from integral geometry. 1.
Target enumeration via Euler characteristic integrals I: sensor fields
, 2007
"... We solve the problem of counting the total number of observable targets (e.g., persons, vehicles, landmarks) in a region using local counts performed by a dense field of sensors, each of which measures the number of targets nearby but not their identities nor any positional information. We formulat ..."
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Cited by 30 (11 self)
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We solve the problem of counting the total number of observable targets (e.g., persons, vehicles, landmarks) in a region using local counts performed by a dense field of sensors, each of which measures the number of targets nearby but not their identities nor any positional information. We formulate and solve several such problems based on the types of sensors and mobility of the targets. The main contribution of this paper is the adaptation of a topological integration theory — integration with respect to Euler characteristic — to yield complete solutions to these problems.
Coverage in heterogeneous sensor networks
 in Proc. WiOpt
, 2006
"... We study the problem of coverage in planar heterogeneous sensor networks. Coverage is a performance metric that quantifies how well a field of interest is monitored by the sensor deployment. To derive analytical expressions of coverage for heterogeneous sensor networks, we formulate the coverage pro ..."
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Cited by 25 (2 self)
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We study the problem of coverage in planar heterogeneous sensor networks. Coverage is a performance metric that quantifies how well a field of interest is monitored by the sensor deployment. To derive analytical expressions of coverage for heterogeneous sensor networks, we formulate the coverage problem as a set intersection problem, a problem studied in integral geometry. Compared to previous analytical results, our formulation allows us to consider a network model where sensors are deployed according to an arbitrary stochastic distribution; sensing areas of sensors need not follow the unit disk model but can have any arbitrary shape; sensors need not have an identical sensing capability. Furthermore, our formulation does not assume deployment of sensors over an infinite plane and, hence, our derivations do not suffer from the border effect problem arising in a bounded field of interest. We compare our theoretical results with the spatial Poisson approximation that is widely used in modeling coverage. By computing the KullbackLeibler and total variation distance between the probability density functions derived via our theoretical results, the Poisson approximation, and the simulation, we show that our formulas provide a more accurate representation of the coverage in sensor networks. Finally, we provide examples of calculating network parameters such as the network size and sensing range in order to achieve a desired degree of coverage.
Lipschitzian mappings and total mean curvature of polyhedral surfaces
 I, Trans. Amer. Math. Soc
, 1985
"... Abstract. For a smooth closed surface C in E3 the classical total mean curvature is defined by M(C) = ¿/(«i + k2) do(p), where kx, k2 are the principal curvatures at p on C. If C is a polyhedral surface, there is a well known discrete version given by M(C) = IE/,(w a,), where 1 ¡ represents edge ..."
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Cited by 21 (0 self)
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Abstract. For a smooth closed surface C in E3 the classical total mean curvature is defined by M(C) = ¿/(«i + k2) do(p), where kx, k2 are the principal curvatures at p on C. If C is a polyhedral surface, there is a well known discrete version given by M(C) = IE/,(w a,), where 1 ¡ represents edge length and a, the corresponding dihedral angle along the edge. In this article formulas involving differentials of total mean curvature (closely related to the differential formula of L. Schlàfli) are applied to several questions concerning Lipschitizian mappings of polyhedral surfaces. For example, the simplest formula T.I, da, = 0 may be used to show that the remarkable flexible polyhedral spheres of R. Connelly must flex with constant total mean curvature. Related differential formulas are instrumental in showing that if/: E2> E2 is a distanceincreasing function and KciE2, then Per(convK~)< Per(conv/[A:]). This article (part I) is mainly concerned with problems in E". In the sequel (part II) related questions in S " and H", as well as E", will be considered.
Generalized Rolle Theorem in R^n and C
, 1996
"... . The Rolle theorem for functions of one real variable asserts that the number of zeros of f on a real connected interval can be at most that of f 0 plus 1. The following inequality is a multidimensional generalization of the Rolle theorem: if ` [0; 1] ! R n , t 7! x(t), is a closed smooth spat ..."
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Cited by 8 (6 self)
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. The Rolle theorem for functions of one real variable asserts that the number of zeros of f on a real connected interval can be at most that of f 0 plus 1. The following inequality is a multidimensional generalization of the Rolle theorem: if ` [0; 1] ! R n , t 7! x(t), is a closed smooth spatial curve and L(`) is the length of its spherical projection on a unit sphere, then for the derived curve ` 0 [0; 1] ! R n , t 7! x(t), the following inequality holds: L(`) 6 L(` 0 ). For the analytic function F (z) defined in a neighborhood of a closed plane curve \Gamma ae C ' R 2 this inequality implies that e V \Gamma (F ) 6 e V \Gamma (F 0 ) + (\Gamma), where e V \Gamma (F ) is the total variation of the argument of F along \Gamma, and (\Gamma) is the integral absolute curvature of \Gamma. As an application of this inequality, we find an upper bound for the number of complex isolated zeros of quasipolynomials. We also establish a twosided inequality between the variat...
An index formula for simple graphs
, 2012
"... Abstract. We prove that any odd dimensional geometric graph G = (V, E) has zero curvature everywhere. To do so, we prove that for every injective function f on the vertex set V of a simple graph the index formula 1 [1 − 2 χ(S(x))/2 − χ(Bf (x))] = (if (x) + i−f (x))/2 = jf (x) holds, where if (x) is ..."
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Cited by 7 (7 self)
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Abstract. We prove that any odd dimensional geometric graph G = (V, E) has zero curvature everywhere. To do so, we prove that for every injective function f on the vertex set V of a simple graph the index formula 1 [1 − 2 χ(S(x))/2 − χ(Bf (x))] = (if (x) + i−f (x))/2 = jf (x) holds, where if (x) is a discrete analogue of the index of the gradient vector field ∇f and where Bf (x) is a graph defined by G and f. The PoincaréHopf formula ∑ x jf (x) = χ(G) allows so to express the Euler characteristic χ(G) of G in terms of smaller dimensional graphs defined by the unit sphere S(x) and the ”hypersurface graphs ” Bf (x). For odd dimensional geometric graphs, Bf (x) is a geometric graph of dimension dim(G)−2 and jf (x) = −χ(Bf (x))/2 = 0 implying χ(G) = 0 and zero curvature K(x) = 0 for all x. For even dimensional geometric graphs, the formula becomes jf (x) = 1−χ(Bf (x))/2 and allows with PoincaréHopf to write the Euler characteristic of G as a sum of the Euler characteristic of smaller dimensional graphs. The same integral geometric index formula also is valid for compact Riemannian manifolds M if f is a Morse function, S(x) is a sufficiently small geodesic sphere around x and Bf (x) = S(x) ∩ {y  f(y) = f(x)}. 1.
Kinematic formulas for finite lattices
 Annals of Combinatorics
, 1997
"... Abstract. In analogy to valuation characterizations and kinematic formulas of convex geometry, we develop a combinatorial theory of invariant valuations and kinematic formulas for finite lattices. Combinatorial kinematic formulas are shown to have application to some probabilistic questions, leadin ..."
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Cited by 4 (2 self)
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Abstract. In analogy to valuation characterizations and kinematic formulas of convex geometry, we develop a combinatorial theory of invariant valuations and kinematic formulas for finite lattices. Combinatorial kinematic formulas are shown to have application to some probabilistic questions, leading in turn to polynomial identities for Möbius functions and Whitney numbers.