Smoothness Equivalence Properties of General Manifold-Valued Data Subdivision Schemes

Based on a vector-bundle formulation, we introduce a new family of nonlinear subdivision schemes for manifold-valued data. Any such nonlinear subdivision scheme is based on an underlying linear subdivision scheme. We show that if the underlying linear subdivision scheme reproduces Πk, then the nonlinear scheme satisfies an order k proximity condition with the linear scheme. We also develop a new “proximity ⇒ smoothness” theorem, improving the one in [12]. Combining the two results, we can conclude that if the underlying linear scheme is C and stable, the nonlinear scheme is also C. The family of manifold-valued data subdivision scheme introduced in this paper includes a variant of the log-exp scheme, proposed in [10], as a special case, but not the original log-exp scheme when the underlying linear scheme is non-interpolatory. The original log-exp scheme uses the same tangent plane for both the odd and the even rules, while the variant uses two different, judiciously chosen, tangent planes. We also present computational experiments that indicate that the original smoothness equivalence conjecture posted in [10] is unlikely to be true. Our result also generalizes the recent results in [17, 16, 5, 6]. It uses only the intrinsic smoothness structure of the manifold and (hence) does not rely on any embedding or Lie group or symmetric space or Riemannian structure. In particular, concepts such as geodesics, log and exp maps, or projection from ambient space play no explicit role in the theorem. Also, the underlying linear scheme needs not be interpolatory. Acknowledgments. The work of this research was partially supported by the the National Science Foundation grant DMS 0512673. The main result in this paper was first announced, based on an extrinsic formulation, in the MAIA 2007 conference (Ålesund, Norway, August 22-26, 2007.) (The formulation was then revised into the current intrinsic setup.) The second named author would like to thank the organizers Michael Floater and Tom Lyche for their invitation, while the first named author would like to thank the hospitality of the Drexel University mathematics department during his postdoctoral visit at Drexel in 2007.