Journal article
Single Basepoint Subdivision Schemes for Manifold-valued Data: Time-Symmetry Without Space-Symmetry
Foundations of computational mathematics, v 13(5), pp 693-728
01 Oct 2013
Abstract
This paper establishes smoothness results for a class of nonlinear subdivision schemes, known as the single basepoint manifold-valued subdivision schemes, which shows up in the construction of wavelet-like transform for manifold-valued data. This class includes the (single basepoint) Log-Exp subdivision scheme as a special case. In these schemes, the exponential map is replaced by a so-called retraction map f from the tangent bundle of a manifold to the manifold. It is known that any choice of retraction map yields a C (2) scheme, provided the underlying linear scheme is C (2) (this is called "C (2) equivalence"). But when the underlying linear scheme is C (3), Navayazdani and Yu have shown that to guarantee C (3) equivalence, a certain tensor P (f) associated to f must vanish. They also show that P (f) vanishes when the underlying manifold is a symmetric space and f is the exponential map. Their analysis is based on certain "C (k) proximity conditions" which are known to be sufficient for C (k) equivalence.
In the present paper, a geometric interpretation of the tensor P (f) is given. Associated to the retraction map f is a torsion-free affine connection, which in turn defines an exponential map. The condition P (f) =0 is shown to be equivalent to the condition that f agrees with the exponential map of the connection up to the third order. In particular, when f is the exponential map of a connection, one recovers the original connection and P (f) vanishes. It then follows that the condition P (f) =0 is satisfied by a wider class of manifolds than was previously known. Under the additional assumption that the subdivision rule satisfies a time-symmetry, it is shown that the vanishing of P (f) implies that the C (4) proximity conditions hold, thus guaranteeing C (4) equivalence. Finally, the analysis in the paper shows that for k >= 5, the C (k) proximity conditions imply vanishing curvature. This suggests that vanishing curvature of the connection associated to f is likely to be a necessary condition for C (k) equivalence for k >= 5.
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Details
- Title
- Single Basepoint Subdivision Schemes for Manifold-valued Data: Time-Symmetry Without Space-Symmetry
- Creators
- Tom Duchamp - University of WashingtonGang Xie - East China University of Science and TechnologyThomas Yu - Drexel University
- Publication Details
- Foundations of computational mathematics, v 13(5), pp 693-728
- Publisher
- Springer Nature
- Number of pages
- 36
- Grant note
- 1115915 / Division Of Mathematical Sciences; National Science Foundation (NSF); NSF - Directorate for Mathematical & Physical Sciences (MPS) Louis and Bessie Stein family Fundamental Research Funds for the Central Universities DMS 0915068; DMS 1115915 / National Science Foundation; National Science Foundation (NSF) 11101146 / National Natural Science Foundation of China; National Natural Science Foundation of China (NSFC)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000324819700001
- Scopus ID
- 2-s2.0-84884702546
- Other Identifier
- 991021878114304721
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- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Computer Science, Theory & Methods
- Mathematics
- Mathematics, Applied