Results 11  20
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619
Conformal Curvature Flows: From Phase Transitions to Active Vision
, 1995
"... In this paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new featurebased Riemannian metrics. This leads to a novel edgedetection paradigm in which the feature of interest may be consid ..."
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Cited by 118 (30 self)
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In this paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new featurebased Riemannian metrics. This leads to a novel edgedetection paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus an edgeseeking curve is attracted very naturally and efficiently to the desired feature. Comparison with the AllenCahn model clarifies some of the choices made in these models, and suggests inhomogeneous models which may in return be useful in phase transitions. We also consider some 3D active surface models based on these ideas. The justification of this model rests on the careful study of the viscosity solutions of evolution equations derived from a levelset approach. Key words: Active vision, antiphase boundary, visual tracking, edge detection, segmentation, gradient flows, Riemannian metrics, viscosity solutions, geometric heat equ...
Choosing Good Distance Metrics and Local Planners for Probabilistic Roadmap Methods
 In Proc. IEEE Int. Conf. Robot. Autom. (ICRA
, 1998
"... Abstract This paper presents a comparative evaluation of different distance metrics and local planners within the context of probabilistic roadmap methods for motion planning. Both Cspace andWorkspace distance metrics and local planners are considered. The study concentrates on cluttered threedim ..."
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Cited by 84 (22 self)
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Abstract This paper presents a comparative evaluation of different distance metrics and local planners within the context of probabilistic roadmap methods for motion planning. Both Cspace andWorkspace distance metrics and local planners are considered. The study concentrates on cluttered threedimensionalWorkspaces typical, e.g., of mechanical designs. Our results include recommendations for selecting appropriate combinationsof distance metrics and local planners for use in motion planning methods, particularly probabilistic roadmap methods. Wefind that each local planner makes some connections than none of the others do indicating that better connectedroadmaps will beconstructed using multiple local planners. We propose a new local planning method we call rotateats that outperforms the commonstraightline in Cspace method in crowded environments. 1
A Search Algorithm for Motion Planning with Six Degrees of Freedom
 ARTIFICIAL INTELLIGENCE
, 1987
"... The motion planning problem is of central importance to the fields of robotics, spatial planning, and automated design. In robotics we are interested in the automatic synthesis of robot motions, given highlevel specifications of tasks and geometric models of the robot and obstacles. The "Movers'" p ..."
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Cited by 76 (4 self)
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The motion planning problem is of central importance to the fields of robotics, spatial planning, and automated design. In robotics we are interested in the automatic synthesis of robot motions, given highlevel specifications of tasks and geometric models of the robot and obstacles. The "Movers'" problem is to find a continuous, collisionfree path for a moving object through an environment containing obstacles. We present an implemented algorithm for the classical formulation of the threedimensional Movers' problem: Given an arbitrary rigid polyhedral moving object P with three translational and three rotational degrees of freedom, find a continuous, collisionfree path taking P from some initial configuration to a desired goal configuration. This paper describes an implementation of a complete algorithm (at a given resolution)for the full six degree of freedom Movers' problem. The algorithm transforms the six degree of freedom planning problem into a point navigation problem in a sixdimensional configuration space (called Cspace). The Cspace obstacles, which characterize the physically unachievable configurations, are directly represented by sixdimensional manifolds whose boundaries are fivedimensional Csurfaces. By characterizing these surfaces and their intersections, collisionfree paths may be found by the
Seamless Texture Mapping of Subdivision Surfaces by Model Pelting and Texture Blending
"... Subdivision surfaces solve numerous problems related to the geometry of character and animation models. However, unlike on parametrised surfaces there is no natural choice of texture coordinates on subdivision surfaces. Existing algorithms for generating texture coordinates on nonparametrised surfa ..."
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Cited by 59 (0 self)
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Subdivision surfaces solve numerous problems related to the geometry of character and animation models. However, unlike on parametrised surfaces there is no natural choice of texture coordinates on subdivision surfaces. Existing algorithms for generating texture coordinates on nonparametrised surfaces often find solutions that are locally acceptable but globally are unsuitable for use by artists wishing to paint textures. In addition, for topological reasons there is not necessarily any choice of assignment of texture coordinates to control points that can satisfactorily be interpolated over the entire surface. We introduce a technique, pelting, for finding both optimal and intuitive texture mapping over almost all of an entire subdivision surface and then show how to combine multiple texture mappings together to produce a seamless result.
Smooth Interpolation of Orientations with Angular Velocity Constraints using Quaternions
, 1992
"... In this paper we present methods to smoothly interpolate orientations, given N rotational keyframes of an object along a trajectory. The methods allow the user to impose constraints on the rotational path, such as the angular velocity at the endpoints of the trajectory. We convert the rotations to q ..."
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Cited by 56 (2 self)
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In this paper we present methods to smoothly interpolate orientations, given N rotational keyframes of an object along a trajectory. The methods allow the user to impose constraints on the rotational path, such as the angular velocity at the endpoints of the trajectory. We convert the rotations to quaternions, and then spline in that nonEuclidean space. Analogous to the mathematical foundations of flatspace spline curves, we minimize the net "tangential acceleration" of the quaternion path. We replace the flatspace quantities with curvedspace quantities, and numerically solve the resulting equation with finite difference and optimization methods. 1 Introduction The problem of using spline curves to smoothly interpolate mathematical quantities in flat Euclidean spaces is a wellstudied problem in computer graphics [bartels et al 87], [kochanek&bartels 84]. Many quantities important to computer graphics, however, such as rotations, lie in nonEuclidean spaces. In 1985, a method to...
Shortest Paths For The ReedsShepp Car: A Worked Out Example Of The Use Of Geometric Techniques In Nonlinear Optimal Control.
, 1991
"... We illustrate the use of the techniques of modern geometric optimal control theory by studying the shortest paths for a model of a car that can move forwards and backwards. This problem was discussed in recent work by Reeds and Shepp who showed, by special methods, (a) that shortest path motion coul ..."
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Cited by 51 (5 self)
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We illustrate the use of the techniques of modern geometric optimal control theory by studying the shortest paths for a model of a car that can move forwards and backwards. This problem was discussed in recent work by Reeds and Shepp who showed, by special methods, (a) that shortest path motion could always be achieved by means of trajectories of a special kind, namely, concatenations of at most five pieces, each of which is either a straight line or a circle, and (b) that these concatenations can be classified into 48 threeparameter families. We show how these results fit in a much more general framework, and can be discovered and proved by applying in a systematic way the techniques of Optimal Control Theory. It turns out that the "classical" optimal control tools developed in the 1960's, such as the Pontryagin Maximum Principle and theorems on the existence of optimal trajectories, are helpful to go part of the way and get some information on the shortest paths, but do not suffice ...
Complementarity Modeling of Hybrid Systems
, 1998
"... A complementarity framework is described for the modeling of certain classes of mixed continuous /discrete dynamical systems. The use of such a framework is wellknown for mechanical systems with inequality constraints, but we give a more general formulation which applies for instance also to sys ..."
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Cited by 50 (11 self)
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A complementarity framework is described for the modeling of certain classes of mixed continuous /discrete dynamical systems. The use of such a framework is wellknown for mechanical systems with inequality constraints, but we give a more general formulation which applies for instance also to systems with relays in a feedback loop. The main theoretical results in the paper are concerned with uniqueness of smooth continuations; the solution of this problem requires the construction of a map from the continuous state to the discrete state. A crucial technical tool is the socalled linear complementarity problem (LCP); we introduce various generalizations of this problem. Specific results are obtained for Hamiltonian systems, passive systems, and linear systems.
From high energy physics to low level vision
 Presented in ONR workshop, UCLA
, 1996
"... Abstract. A geometric framework for image scale space, enhancement, and segmentation is presented. We consider intensity images as surfaces in the (x � I) space. The image is thereby a 2D surface in 3D space for gray level images, and a 2D surface in 5D for color images. The new formulation uni es m ..."
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Cited by 50 (21 self)
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Abstract. A geometric framework for image scale space, enhancement, and segmentation is presented. We consider intensity images as surfaces in the (x � I) space. The image is thereby a 2D surface in 3D space for gray level images, and a 2D surface in 5D for color images. The new formulation uni es many classical schemes and algorithms via a simple scaling of the intensity contrast, and results in new and e cient schemes. Extensions to multi dimensional signals become natural and lead to powerful denoising and scale space algorithms. Here, we demonstrate the proposed framework by applying it to denoise and improve graylevel and color images. 1 Introduction: A philosophical point of view In this paper we adopt an action potential that was recently introduced in physics and use it to produce a natural scale space for images as surfaces. It will lead us to the construction of image enhancement procedures for gray and color images. This model also integrates many existing segmentation and scale space procedures
How many zeros of a random polynomial are real
 Bull. Amer. Math. Soc. (N.S
, 1995
"... Abstract. We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve (1, t, ..."
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Cited by 46 (0 self)
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Abstract. We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve (1, t,..., t n) projected onto the surface of the unit sphere, divided by π. The probability density of the real zeros is proportional to how fast this curve is traced out. We then relax Kac’s assumptions by considering a variety of random sums, series, and distributions, and we also illustrate such ideas as integral geometry and the FubiniStudy metric. Contents 1.