Orhonormal wavelet bases on the 3D ball via volume preserving map from the regular octahedron

We construct a new volume preserving map from the unit ball ${\mathbb{B}}^3$ 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 $L^2({\mathbb{B}}^3)$, consisting in piecewise constant functions with small local support.