Smoothness of interpolatory multivariate subdivision in Lie groups
نویسندگان
چکیده
Nonlinear subdivision schemes that operate on manifolds are of use whenever manifold valued data have to be processed in a multiscale fashion. This paper considers the case where the manifold is a Lie group and the nonlinear subdivision schemes are derived from linear interpolatory ones by the so-called log-exp analogy. The main result of the paper is that a multivariate interpolatory Lie group valued subdivision scheme derived from a linear scheme is at least as smooth as the linear scheme, where smoothness is understood in terms of Hölder exponents. subdivision, nonlinear subdivision, multivariate subdivision, Lie group, Lie group subdivison, Hölder exponents, smoothness equivalence
منابع مشابه
Smoothness of interpolatory multivariate subdivision schemes in Lie groups
Nonlinear subdivision schemes that operate on manifolds are of use whenever manifold valued data have to be processed in a multiscale fashion. This paper considers the case where the manifold is a Lie group and the nonlinear subdivision schemes are derived from linear interpolatory ones by the so-called log-exp analogy. The main result of the paper is that a multivariate interpolatory Lie group...
متن کاملApproximation order of interpolatory nonlinear subdivision schemes
Linear interpolatory subdivision schemes of C smoothness have approximation order at least r + 1. The present paper extends this result to nonlinear univariate schemes which are in proximity with linear schemes in a certain specific sense. The results apply to nonlinear subdivision schemes in Lie groups and in surfaces which are obtained from linear subdivision schemes. We indicate how to exten...
متن کاملSmoothness Equivalence Properties of Interpolatory Lie Group Subdivision Schemes
We prove that any interpolatory Lie group subdivision scheme based on combining a linear interpolatory subdivision scheme S with the log-exp adaption to Lie group valued data in [8] produces parametrized curves on the Lie group which are as smooth as the smoothness of S – no matter how smooth S is. We present both an extrinsic proof and an intrinsic proof. We discuss two variations of our main ...
متن کاملInterpolatory Wavelets for Manifold-valued Data
Geometric wavelet-like transforms for univariate and multivariate manifold-valued data can be constructed by means of nonlinear stationary subdivision rules which are intrinsic to the geometry under consideration. We show that in an appropriate vector bundle setting for a general class of interpolatory wavelet transforms, which applies to Riemannian geometry, Lie groups and other geometries, Hö...
متن کاملOptimal Interpolatory Subdivision Schemes in Multidimensional Spaces * Bin Han † and Rong-qing Jia ‡
We analyse the approximation and smoothness properties of fundamental and refinable functions that arise from interpolatory subdivision schemes in multidimensional spaces. In particular, we provide a general way for the construction of bivariate interpolatory refinement masks such that the corresponding fundamental and refinable functions attain the optimal approximation order and smoothness or...
متن کامل