Narrow Results

Starting from the one-dimensional Haar wavelets on the interval [0,1], we construct spherical Haar wavelets which are orthogonal with respect to a given scalar product. This scalar product induces a norm which is equivalent to the usual ‖ · ‖₂ norm of L²(𝕊²). Thus, the Riesz stability in L²(𝕊²) is assured and we can use the algorithms of decomposition and reconstruction from the Haar wavelets in 2D. Another advantage is that we avoid the problems around the poles, which occur in other approaches. As example, we decompose a data set, showing the graphs of the approximations and details and thus the capability to detect the singularities (contours). The method described here can be also used for constructing spherical wavelets starting from wavelets on an interval.

We construct piecewise constant wavelets on a bounded planar triangulation, the refinement process consisting of dividing each triangle into three triangles having the same area. Thus, the wavelets depend on two parameters linked by a certain relation. We perform a compression and try to compare different norms of the compression error, when one wavelet coefficient is canceled. Finally, we show how this construction can be moved on to the two-dimensional sphere and sphere-like surfaces, avoiding the distortions around the poles, which occur in other approaches. As numerical example, we perform a compression of some spherical data and calculate some norms of the compression error for different compression rates. The main advantage is the orthogonality and sparsity of the decomposition and reconstruction matrices.

We give the most general expression of orthonormal piecewise constant wavelet bases, in the case when the refinement process consists in dividing a domain into four subdomains having the same area. We show that the classical Haar bases are optimal with respect to the ∞-norm of the reconstruction error, but not with respect to the 1-norm. The construction applies not only to planar domains, but also to surfaces of revolution, like sphere, cone, paraboloid, or two-sheeted hyperboloid.