Multiresolution analysis on multidimensional dyadic grids

We propose a modified adaptive multiresolution scheme for representing d-dimensional signals which is based on cell-average discretization in dyadic grids. A dyadic grid is an hierarchy of meshes where a cell at a certain level is partitioned into two equal children at the next refined level by hyperplanes perpendicular to one of the coordinate axes which varies cyclically from level to level. Adaptivity is obtained by interrupting the refinement at the locations where appropriate scale (wavelet) coefficients are sufficiently small. One important aspect of such multiresolution representation is that we can use a binary tree data structure in all dimensions, that helps to compress data while still being able to navigate through it. Dyadic grids provide a more gradual refinement as compared with traditional multiresolution analyses that use, for instance, different quad-trees or oct-trees in 2D or 3D multiresolution applications. The cells may have different scales in different directions, this property can be explored to improve data compression of signals having anisotropic aspects.