Locally supported rational spline wavelets on a sphere

Abstract

Abstract. In this paper we construct certain continuous piecewise rational wavelets on arbitrary spherical triangulations, giving explicit expressions of these wavelets. Our wavelets have small support, a fact which is very important in working with large amounts of data, since the algorithms for decomposition, compression and reconstruction deal with sparse matrices. We also give a quasi-interpolant associated to a given triangulation and study the approximation error. Some numerical examples are given to illustrate the efficiency of our wavelets.

Constructing Wavelet Frames and Orthogonal Wavelet Bases on the Sphere

“…One would then get orthogonal wavelet bases in each patch, but there remains the problem of connection of one patch with the next one, using transition functions (the concatenation of all the local bases may also be considered as a dictionary). Notice the same problem of combining local orthogonal wavelet bases has been encountered, and solved, in the wavelet construction based on radial projection from a convex polyhedron (Roşca, 2005), described briefly in Section 3.2(3). A final example of orthogonal wavelet basis is that of the wavelet transform on graphs .…”

“…The idea of the method, due to one of us (Roşca, 2005;2007a;b), is to obtain wavelets on S 2 first by moving planar wavelets to wavelets defined on the faces of Γ and then projecting these radially onto S 2 . This proceeds as follows.…”

The Continuous Wavelet Transform on Conic Sections

“…36 This approach has become a common tool in computer graphics. We may also mention C 1 wavelets constructed by a factorization of the refinement matrices; 39 or wavelets obtained by radial projection from a polyhedron inscribed in the sphere, typically locally supported spline wavelets on spherical triangulations 32,34 (actually this construction extends to sphere-like surfaces, i.e., continuous deformations of a sphere 33 ). It is interesting to note that the authors of Refs.…”

Wavelets on the Two-Sphere and Other Conic Sections

“…Discrete wavelets on the sphere have also been designed, using an S 2 multiresolution analysis. For instance, Haar wavelets on a triangulation of S 2 and refined with the lifting scheme [32]; C 1 wavelets constructed by a factorization of the refinement matrices [35]; or wavelets obtained by radial projection from a polyhedron incribed in the sphere (typically locally supported spline wavelets on spherical triangulations) [29,30]. References to the (vast) literature on discrete spherical wavelets may be found in [29,35] for earlier work and in [21] for recent work.…”