Splines and Wavelets on Geophysically Relevant Manifolds

Abstract

6. Radon transform on the group of rotations SO(3).3. Generalized variational splines on compact Riemannian manifolds 3.1 Generalized interpolating variational splines. 3.2. Approximation by pointwise interpolation and approximation. 3.3. A sampling theorem and a cubature formula. 4. Bandlimited and localized Parseval frames on homogeneous manifolds 5. Applications to the Radon transform on S d 5.1. Approximate inversion of the spherical Radon transform using generalized splines. 5.2. A sampling theorem for the spherical Radon transform of bandlimited functions on S d . 5.3. Exact formulas for Fourier coefficients of a bandlimited function f on S d from a finite number of samples of Rf .6. Applications to the Radon transform on SO(3) 6.1. Approximate inversion of the Radon transform on SO(3) using generalized splines.6.2. A sampling theorem for the Radon transform of bandlimited functions on SO(3).6.3. Exact formulas for Fourier coefficients of a bandlimited function f on SO(3) from a finite number of samples of Rf . Conclusion.References.

Simple proposal for radial 3D needlets

“…so that the "energy" of the function f is fully conserved in the collection of β i 's; we refer for instance to [24,43] and the references therein for more details and discussions. In words, a tight frame can be basically seen as a (possible redundant) basis; indeed we recall that tight frames enjoy the same reconstruction property as standard orthonormal systems, e.g.…”