Tight framelets and fast framelet filter bank transforms on manifolds

• Published in: Applied and computational harmonic analysis Vol. 48; no. 1; pp. 64 - 95
• Main Authors: Wang, Yu Guang, Zhuang, Xiaosheng
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.01.2020
• Subjects: Affine system; Compact Riemannian manifold; Fast spherical harmonic transform; FFT; Filter bank; Laplace–Beltrami operator; Quadrature rule; Tight framelets; Unitary extension principle; Fast spherical harmonic transform; Laplace–Beltrami operator; 58C35; 57N99; 41A55; Compact Riemannian manifold; Quadrature rule; Unitary extension principle; 93C55; Tight framelets; 42C40; FFT; 94C15; Filter bank; 94A12; 42C15; 42B05; Affine system; 93C95
• ISSN: 1063-5203; 1096-603X
• DOI: 10.1016/j.acha.2018.02.001

Tight framelets on a smooth and compact Riemannian manifold M provide a tool of multiresolution analysis for data from geosciences, astrophysics, medical sciences, etc. This work investigates the construction, characterizations, and applications of tight framelets on such a manifold M. Characterizations of the tightness of a sequence of framelet systems for L2(M) in both the continuous and semi-discrete settings are provided. Tight framelets associated with framelet filter banks on M can then be easily designed and fast framelet filter bank transforms on M are shown to be realizable with nearly linear computational complexity. Explicit construction of tight framelets on the sphere S2 as well as numerical examples are given.