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We construct a new volume preserving map from the unit ball $\mathbb B^3$ tothe regular 3D octahedron, both centered at the origin, and its inverse. Thismap will help us to construct refinable grids of the 3D ball, consisting indiameter bounded cells having the same volume. On this 3D uniform grid, weconstruct a multiresolution analysis and orthonormal wavelet bases of$L^2(\mathbb B^3)$, consisting in piecewise constant functions with small localsupport.

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