Index Mapping between Tensor-Product Wavelet Bases of Different Number of Variables, and Computing Multivariate Orthogonal Discrete Wavelet Transforms on Graphics Processing Units

An algorithm for computation of multivariate wavelet transforms on graphics processing units (GPUs) was proposed in [1]. This algorithm was based on the so-called

isometric conversion between dimension and resolution

(see [2] and the references therein) achieved by mapping the indices of orthonormal tensor-product wavelet bases of different number of variables and a tradeoff between the number of variables versus the resolution level, so that the resulting wavelet bases of different number of variables are with different resolution, but the overall dimension of the bases is the same.

In [1] we developed the algorithm only up to mapping of the indices of

blocks

of wavelet basis functions. This was sufficient to prove the consistency of the algorithm, but not enough for the

mapping of the individual basis functions

in the bases needed for a programming implementation of the algorithm. In the present paper we elaborate the full details of this ‘book-keeping’ construction by passing from block-matrix index mapping on to the detailed index mapping of the individual basis functions. We also consider some examples computed using the new detailed index mapping.