Smoothness Equivalence Properties of General Manifold-Valued Data Subdivision Schemes

Abstract

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 $\Pi_k$, then the nonlinear scheme satisfies an order k proximity condition with the linear scheme. We also develop a new “proximity $\Rightarrow$ smoothness” theorem, improving the one in [J. Wallner, Constr. Approx., 24 (2006), pp. 289–318]. Combining the two results, we can conclude that if the underlying linear scheme is $C^k$ and stable, then the nonlinear scheme is also $C^k$. The family of manifold-valued data subdivision scheme introduced in this paper includes a variant of the log-exp scheme, proposed in [I. U. Rahman et al., Multiscale Model. Simul., 4 (2005), pp. 1201–1232], as a special case, but not the original log-exp scheme when the underlying linear scheme is noninterpolatory. 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 [I. U. Rahman et al., Multiscale Model. Simul., 4 (2005), pp. 1201–1232] is unlikely to be true. Our result also generalizes the recent results in [G. Xie and T. P.-Y. Yu, SIAM J. Numer. Anal., 45 (2007), pp. 1200–1225], [G. Xie and T. P.-Y. Yu, IMA J. Numer. Anal., to appear], [P. Grohs, IMA J. Numer. Anal., to appear], [P. Grohs, Smoothness Equivalence Properties of Univariate Subdivision Schemes and Their Projection Analogues, manuscript, 2007]. 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 need not be interpolatory.