Abstract
We construct a new volume preserving map from the unit ball B 3 to the regular 3D octahedron, both centered at the origin, and its inverse. This map will help us to construct refinable grids of the 3D ball, consisting in diameter bounded cells having the same volume. On this 3D uniform grid, we construct a multiresolution analysis and orthonormal wavelet bases of L 2 ( B 3 ) , consisting in piecewise constant functions with small local support.
Highlights
Spherical 3D signals occur in a wide range of fields, including computer graphics, and medical imaging (e.g., 3D reconstruction of medical images [1]), crystallography [2,3]
We need suitable efficient techniques for manipulating such signals, and one of the most efficient technique consists in using wavelets on the 3D ball
In this paper we propose to construct an orthonormal basis of wavelets with small support, defined on the 3D ball B3, starting from a multiresolution analysis
Summary
Spherical 3D signals occur in a wide range of fields, including computer graphics, and medical imaging (e.g., 3D reconstruction of medical images [1]), crystallography (texture analysis of crystals) [2,3]. Compared to the wavelets on the 3D ball constructed in [8,10], with localized support, our wavelets have local support, and this is very important when dealing with data consisting in big jumps on small portions, as shown in [11]. The author assumes that his wavelets are the first Haar wavelets on the 3D ball which are orthogonal and symmetric, even though we do not see any symmetry, neither in the cells, nor in the decomposition matrix His 8 × 8 decomposition matrices change in each step of the refinement, the entries depending on the volumes of the cells, which are, in our opinion, difficult to evaluate and for this reason they are not calculated explicitly in [7].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
