Wavelet Bases on the Sphere Obtained by Radial Projection

We present a method for constructing wavelet bases on the unit sphere S2 of R3, using the radial projection and an inner product associated to a convex polyhedron having the origin inside. The main advantage of this method is the avoidance of singularities and distortions around poles, which occur in other approaches. Also, we can obtain spherical wavelets with small support, a fact which is crucial in working with large amounts of data, since the algorithms deal with sparse matrices.

[1]  Ilona Weinreich,et al.  A Construction of C1-Wavelets on the Two-Dimensional Sphere☆ , 2001 .

[2]  Laurent Jacques,et al.  Wavelets on the sphere: implementation and approximations , 2002 .

[3]  Peter Schröder,et al.  Spherical wavelets: efficiently representing functions on the sphere , 1995, SIGGRAPH.

[4]  P. Vandergheynst,et al.  Wavelets on the 2-sphere: A group-theoretical approach , 1999 .

[5]  Daniela Rosca.  Piecewise Constant Wavelets on Triangulations Obtained by 1-3 Splitting , 2008, Int. J. Wavelets Multiresolution Inf. Process..

[6]  Georges-Pierre Bonneau.  Optimal triangular Haar bases for spherical data , 1999, Proceedings Visualization '99 (Cat. No.99CB37067).

[7]  Matthias Holschneider,et al.  Continuous wavelet transforms on the sphere , 1996 .

[8]  Michael S. Floater,et al.  Piecewise linear prewavelets on arbitrary triangulations , 1999, Numerische Mathematik.

[9]  Willi Freeden,et al.  Spaceborne gravitational field determination by means of locally supported wavelets , 2005 .

[10]  Daniela Rosca.  Haar Wavelets on Spherical Triangulations , 2005, Advances in Multiresolution for Geometric Modelling.

[11]  D. RoÅŸca.  Optimal Haar wavelets on spherical triangulations , 2004 .

[12]  Laurent Jacques,et al.  Stereographic wavelet frames on the sphere , 2005 .

[13]  Daniela Rosca.  Weighted Haar Wavelets on the Sphere , 2007, Int. J. Wavelets Multiresolution Inf. Process..

[14]  W. Dahmen,et al.  Multiresolution analysis and wavelets on S2 and S3 , 1995 .

[15]  Gregory M. Nielson,et al.  Haar wavelets over triangular domains with applications to multiresolution models for flow over a sphere , 1997 .