Piecewise

About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid function.

Ergodic transformations from an interval into itself

Abstract: A class of piecewise continuous, piecewise C1 transformations on the interval J c R with finitely many discontinuities n are shown to have at most n invariant measures. 1. The way phenomena or processes evolve or change in time is often described by differential equations or difference equations. One of the sim- plest mathematical situations occurs when the phenomenon can be described by a single number and when this number can be estimated purely as a function of the previous number. That is, when the number xn+x can be written as xn+x = t(x") where t maps an interval J c R into itself. For x E J, let t°(x) denote x and t"+x(x) denote t(t"(x)) for n = 0,1.We will sayp E J is a periodic point with period n if p = t"(p) andp ¥= rk(p) for 1 1. In this paper we assume t is piecewise continuous and piecewise twice continuous differentiable. We also assume that a t ^ > 1 where Jx — j x E /, — t(x) exists \.

Piecewise smooth subdivision surfaces with normal control

Abstract: In this paper we introduce improved rules for Catmull-Clark and Loop subdivision that overcome several problems with the original schemes, namely, lack of smoothness at extraordinary boundary vertices and folds near concave corners. In addition, our approach to rule modification allows the generation of surfaces with prescribed normals, both on the boundary and in the interior, which considerably improves control of the shape of surfaces.

One-dimensional transport equations with discontinuous coefficients

Abstract: We consider one-dimensional linear transport equations with bounded but possibly discontinuous coefficient a. The Cauchy problem is studied from two different points of view. In the first case we assume that a is piecewise continuous. We give an existence result and a precise description of the solutions on the lines of discontinuity. In the second case, we assume that a satisfies a one-sided Lipschitz condition. We give existence, uniqueness and general stability results for backward Lipschitz solutions and forward measure solutions, by using a duality method. We prove that the flux associated to these measure solutions is a product by some canonical representative â of a. Key-words. Linear transport equations, discontinuous coefficients, weak stability, duality, product of a measure by a discontinuous function, nonnegative solutions. 1991 Mathematics Subject Classification. Primary 35F10, 35B35, 34A12. To appear in Nonlinear Analysis, TMA ∗Departement de Mathematiques et Applications, UMR CNRS 8553, Ecole Normale Superieure et CNRS, 45 rue d’Ulm, 75230 Paris Cedex 05, France, francois.bouchut@ens.fr †Mathematiques, Applications et Physique Mathematique d’Orleans, UMR CNRS 6628, Universite d’Orleans, 45067 Orleans Cedex 2, France, james@math.cnrs.fr

Trajectory Space: A Dual Representation for Nonrigid Structure from Motion

Abstract: Existing approaches to nonrigid structure from motion assume that the instantaneous 3D shape of a deforming object is a linear combination of basis shapes. These bases are object dependent and therefore have to be estimated anew for each video sequence. In contrast, we propose a dual approach to describe the evolving 3D structure in trajectory space by a linear combination of basis trajectories. We describe the dual relationship between the two approaches, showing that they both have equal power for representing 3D structure. We further show that the temporal smoothness in 3D trajectories alone can be used for recovering nonrigid structure from a moving camera. The principal advantage of expressing deforming 3D structure in trajectory space is that we can define an object independent basis. This results in a significant reduction in unknowns and corresponding stability in estimation. We propose the use of the Discrete Cosine Transform (DCT) as the object independent basis and empirically demonstrate that it approaches Principal Component Analysis (PCA) for natural motions. We report the performance of the proposed method, quantitatively using motion capture data, and qualitatively on several video sequences exhibiting nonrigid motions, including piecewise rigid motion, partially nonrigid motion (such as a facial expressions), and highly nonrigid motion (such as a person walking or dancing).