Diffusion wavelet packets

Diffusion wavelets can be constructed on manifolds, graphs and allow an efficient multiscale representation of powers of the diffusion operator that generates them. In many applications it is necessary to have time–frequency bases that are more versatile than wavelets, for example for the analysis, denoising and compression of a signal. In the Euclidean setting, wavelet packets have been very successful in many applications, ranging from image denoising, 2- and 3-dimensional compression of data (e.g., images, seismic data, hyper-spectral data) and in discrimination tasks as well. Till now these tools for signal processing have been available mainly in Euclidean settings and in low dimensions. Building upon the recent construction of diffusion wavelets, we show how to construct diffusion wavelet packets, generalizing the classical construction of wavelet packets, and allowing the same algorithms existing in the Euclidean setting to be lifted to rather general geometric and anisotropic settings, in higher dimension, on manifolds, graphs and even more general spaces. We show that efficient algorithms exists for computations of diffusion wavelet packets and discuss some applications and examples.