Title: Optimal Triangular Haar Bases for Spherical Data (G.P. Bonneau)
Abstract:
Multiresolution analysis based on FWT (Fast Wavelet Transform) is now widely used in Scientific Visualization.
Spherical bi-orthogonal wavelets for spherical triangular grids where introduced in [2].
In order to improve on the orthogonality of the wavelets, the concept of nearly orthogonality, and two new
piecewise-constant (Haar) bases were introduced in [1]. In our paper, we extend the results of
[1]. First we give two one-parameter families of triangular Haar wavelet bases that are
nearly orthogonal in the sense of [1]. Then we introduce a measure of orthogonality. This
measure vanishes for orthogonal bases. Eventually, we show that we can find an optimal parameter of our
wavelet families, for which the measure of orthogonality is minimized. Several numerical and visual
examples for a spherical topographic data set illustrates our results.
[1] Nielson G.M., Jung I-H., Sung J., Haar-Wavelets over Triangular Domains with applications
to Multiresolution Models for Flow over a Sphere, IEEE Visualization '97, nov. 1997, pp. 143-150.
[2] Schroeder P., Sweldens W., Spherical Wavelets: Efficiently Representing Functions on the Sphere, SIGGRAPH'95,
aug. 1995, pp. 161-172.
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