Constructing Wavelet Frames and Orthogonal Wavelet Bases on the Sphere
Daniela Roşca, Jean-Pierre Antoine · Signal Processing · 2010
A classical problem is to analyse a signal (function) by decomposing it into suitable building blocks, then approximate it by truncating the expansion. Well-known examples are Fourier transform and its localized version, the Short Time Fourier transform (sometimes called the Gabor transform), and the wavelet transform. In the best case, the elementary blocks form a basis in the space of signals, with the pleasant consequence that the expansion coefficients are uniquely defined. Unfortunately, this is not always possible and often one has to resort to frames. In image processing, in particular, two-dimensional wavelets are by now a standard tool in image processing, under the two concurrent approaches, the Discrete Wavelet Transform (DWT), based on the concept of multiresolution analysis, and the Continuous Wavelet Transform (CWT). While the former usually leads to wavelet bases, the CWT has to be discretized for numerical implementation and produces in general only frames. Nowadays, many situations yield data on spherical surfaces. For instance, in Earth and Space sciences (geography, geodesy, meteorology, astronomy, cosmology, etc), in crystallography (texture analysis of crystals), in medicine (some organs are regarded as sphere-like surfaces), or in computer graphics (modelling of closed surfaces as the graph of a function defined on the sphere). So one needs a suitable analysis tool for such data. In the spherical case, the Fourier transform amounts to an expansion in spherical harmonics, whose support is the whole sphere. Fourier analysis on the sphere is thus global and cumbersome. Therefore many different methods have been proposed to replace it with some sort of wavelet analysis. In addition, some data may live on more complicated manifolds, such as a two-sheeted hyperboloid, in cosmology for instance (an open expanding model of the universe). In optics also, in the catadioptric image processing, where a sensor overlooks a mirror with the shape of a hyperboloid or a paraboloid. Another example is a closed sphere-like surface, that is, a surface obtained from a sphere by a smooth deformation. Thus it would be useful to have a wavelet transform available on such manifolds as well. In this chapter, we will review the various aspects of the wavelet transform on the two-sphere, both continuous and discrete, with some emphasis on the construction of bases and frames. 4