Spectral Mappings for Graph Wavelets

Graph wavelet filters can be used to construct dictionaries of atoms, and signals defined over graphs can be represented as linear combinations of these atoms. These representations can facilitate the development of various signal processing tasks, e.g. denoising or feature extraction. We present here a method to construct these filters, using mappings on simple prototype filters, that readily allows for spectral domain shaping. Two types of spectral maps will be developed in this work. The first type of maps sharpens (increase the frequency selectivity) of the prototype filter. The second type of maps is equivalent to upsampling in the spectral domain, and can be used to construct a'trous like transforms. The a'trous like transform allows for a tree-structured implementation, and is reminiscent of the classical a'trous (with-holes) wavelet transform. The resulting filters, constructed using these maps, are polynomial functions, and can therefore be implemented without the need for any approximation or eigendecomposition. Both non-redundant and redundant frames can be constructed with these filters. A variety of design examples will be presented to demonstrate the versatility of the proposed method. Applications of the filters to non-linear approximation and denoising will also be considered.