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 graph-cut based methods in vision. Theoretically speaking, our work provides an interesting link between several branches of mathematics-differential geometry, integral geometry, and combinatorial optimization. The main technical problem is solved using Cauchy-Crofton formula from integral geometry. 1.

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.

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 Kullback-Leibler 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.

We show that every upper semicontinuous and equi-affine invariant valuation on the space of d-dimensional convex bodies is a linear combination of affine surface area, volume and the Euler characteristic.

. 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 two-sided inequality between the variat...

We consider the problem of traveling the contour of the set of all points that are within distance 1 of a connected planar curve arrangement P, forming an embedding of the graph G. We show that if the overall length of P is L, there is a closed roundtrip that visits all points of the contour and has length no longer than 2L + 2ß. This result carries over in a more general setting: if R is a compact convex shape with interior points and boundary length `, we can travel the boundary of the Minkowski sum P \Phi R on a closed roundtrip no longer than 2L + `. 1 Introduction When planning the motion of a robot, we are confronted with the problem of moving an object while avoiding collision with a set of obstacles. For the case of translational motion, this means considering Minkowski sums of the form P \Phi R, where P is the set of positions of a reference point and R is the shape of the robot itself. P is a set of feasible positions if and only if P \Phi R does not contain any part of an ...

Abstract. Following the pioneering work of Schapira, we consider topological Radon-type integral transforms on constructible Z-valued functions using the Euler characteristic as a measure. Contributions include: (1) application of the Schapira inversion formula to target localization and classification problems in sensor networks; (2) extension and application of the inversion formula to weighted Radon transforms; and (3) pseudo-inversion formulae for inverting annuli (sets of Euler measure zero).Inversion of Euler transforms 3 1.

Abstract. We solve the problem of counting the total number of observable targets (e.g., persons, vehicles, etc.) in a region based on local counts performed by a network of sensors, each of which measures the number of targets nearby but not their identities nor any positional information. We formulate 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. 1. Introduction. The

In earlier work, we investigated the dynamics of shape when rectangles are split into two. Further exploration, into the more general issues of Markovian sequences of rectangular shapes, has identified four particularly appealing problems. These problems, which lead to interesting invariant distributions on [0,1], have motivating links with the classical works of Blaschke, Crofton, D.G. Kendall, Renyi and Sulanke.

Abstract. This article surveys the Euler calculus — an integral calculus based on Euler characteristic — and its applications to data, sensing, networks, and imaging. 1.