Wavelets on the sphere: implementation and approximations

We continue the analysis of the continuous wavelet transform on the 2-sphere, introduced in a previous paper. After a brief review of the transform, we define and discuss the notion of directional spherical wavelet, i.e., wavelets on the sphere that are sensitive to directions. Then we present a calculation method for data given on a regular spherical grid g. This technique, which uses the FFT, is based on the invariance of g under discrete rotations around the z axis preserving the phi sampling. Next, a numerical criterion is given for controlling the scale interval where the spherical wavelet transform makes sense, and examples are given, both academic and realistic. In a second part, we establish conditions under which the reconstruction formula holds in strong L-p sense, for 1 less than or equal to p < infinity. This opens the door to techniques for approximating functions on the sphere, by use of an approximate identity, obtained by a suitable dilation of the mother wavelet. (C) 2002 Elsevier Science (USA). All rights reserved.