90,513 research outputs found
Numerical Computation of Weil-Peterson Geodesics in the Universal Teichm\"uller Space
We propose an optimization algorithm for computing geodesics on the universal
Teichm\"uller space T(1) in the Weil-Petersson () metric. Another
realization for T(1) is the space of planar shapes, modulo translation and
scale, and thus our algorithm addresses a fundamental problem in computer
vision: compute the distance between two given shapes. The identification of
smooth shapes with elements on T(1) allows us to represent a shape as a
diffeomorphism on . Then given two diffeomorphisms on (i.e., two
shapes we want connect with a flow), we formulate a discretized energy
and the resulting problem is a boundary-value minimization problem. We
numerically solve this problem, providing several examples of geodesic flow on
the space of shapes, and verifying mathematical properties of T(1). Our
algorithm is more general than the application here in the sense that it can be
used to compute geodesics on any other Riemannian manifold.Comment: 21 pages, 11 figure
Nonsmooth Control Barrier Functions for Obstacle Avoidance between Convex Regions
In this paper, we focus on non-conservative obstacle avoidance between robots
with control affine dynamics with strictly convex and polytopic shapes. The
core challenge for this obstacle avoidance problem is that the minimum distance
between strictly convex regions or polytopes is generally implicit and
non-smooth, such that distance constraints cannot be enforced directly in the
optimization problem. To handle this challenge, we employ non-smooth control
barrier functions to reformulate the avoidance problem in the dual space, with
the positivity of the minimum distance between robots equivalently expressed
using a quadratic program. Our approach is proven to guarantee system safety.
We theoretically analyze the smoothness properties of the minimum distance
quadratic program and its KKT conditions. We validate our approach by
demonstrating computationally-efficient obstacle avoidance for multi-agent
robotic systems with strictly convex and polytopic shapes. To our best
knowledge, this is the first time a real-time QP problem can be formulated for
general non-conservative avoidance between strictly convex shapes and
polytopes.Comment: 17 page
Smooth Interpolation of Curve Networks with Surface Normals
International audienceRecent surface acquisition technologies based on microsensors produce three-space tangential curve data which can be transformed into a network of space curves with surface normals. This paper addresses the problem of surfacing an arbitrary closed 3D curve network with given surface normals.Thanks to the normal vector input, the patch finding problem can be solved unambiguously and an initial piecewise smooth triangle mesh is computed. The input normals are propagated throughout the mesh and used to compute mean curvature vectors. We then introduce a new variational optimization method in which the standard bi-Laplacian is penalized by a term based on the mean curvature vectors. The intuition behind this original approach is to guide the standard Laplacian-based variational methods by the curvature information extracted from the input normals. The normal input increases shape fidelity and allows to achieve globally smooth and visually pleasing shapes
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