Basis Selection for Wavelet Processing of Sparse Source Signals

An attractive property of wavelet bases is their ability to sparsely represent piecewise polynomial signals. The sparsity of a wavelet-domain representation depends on several factors such as the mother wavelet, the number of decomposition levels, and the structure of the original signal. We consider the problem of selecting an overcomplete or dyadic wavelet basis that can sparsely represent a sparse piecewise polynomial signal. Most existing applications that apply wavelet-domain processing techniques to signals that are inherently sparse have not considered the sparsity of underlying signal when selecting a wavelet basis. By accounting for the initial sparseness of a signal, the maximum wavelet filter length and number of decomposition levels can be computed. Selecting a wavelet basis that satisfies these maximum values guarantees that the resulting wavelet-domain representation will be at least as sparse as the original signal. This criteria for wavelet basis selection is of use in applications having sparse source signals.