Nonseparable Laplacian pyramids with multiscale local polynomials for scattered data

This paper introduces a family of nonseparable multiscale decompositions for two-dimensional scattered data based on a sample grid dependent implementation of a Laplacian pyramid. This Laplacian pyramid for two-dimensional, irregular observations coincides with a slightly redundant lifting scheme for second generation wavelet decompositions. We can thus associate a frame of wavelet functions with the decomposition and investigate from there the smoothness of a reconstruction from processed decomposition coefficients. The filters that appear in the lifting or pyramid scheme are realized by local polynomial smoothing operations. The novel design of nonseparable multiscale local polynomials does not require a multiscale triangulation of the scattered data, which is a major benefit compared to existing second generation wavelets on scattered data. The proposed scheme has also a better numerical condition, its implementation is faster, and the algorithm is easily extendible to more sophisticated versions.