Stability of Manifold-Valued Subdivision Schemes and Multiscale Transformations

作者：Grohs Philipp^*

来源：Constructive Approximation, 2010, 32(3): 569-596.

DOI：10.1007/s00365-010-9085-8

• Lipschitz
• Interpolatory wavelet transform
• Manifold-valued data
• Nonlinear subdivision

Linear subdivision schemes can be adapted in various ways so as to operate in nonlinear geometries such as Lie groups or Riemannian manifolds. It is well known that along with a linear subdivision scheme a multiscale transformation is defined. Such transformations can also be defined in a nonlinear setting. We show the stability of such nonlinear multiscale transforms. To do this we introduce a new kind of proximity condition which bounds the difference of the differential of a nonlinear subdivision scheme and a linear one. It turns out that-unlike the generic nonlinear case and modulo some minor technical assumptions-in the manifold-valued setting, convergence implies stability of the nonlinear subdivision scheme and associated nonlinear multiscale transformations.

• 出版日期2010