Maximizing Sparsity of Wavelet Representations via Parameterized Lifting

Our goal is to determine the wavelet basis that represents a given signal as sparsely as possible. In a previous paper (Hurley et al., 2005), we proposed a novel, two-parameter method for designing a stable biorthogonal wavelet basis which maximizes the sparseness of a signal´s wavelet representation. We chose the Gini index as a measure of sparsity and sparsify a signal by lifting the wavelet basis with the parameters that maximize the Gini index of the resulting wavelet representation. In this paper we show an efficient manner of calculating the optimal parameters obtained by taking the derivative of the wavelet coefficients through the differentiation of the Gini Index. This allows us to find the parameters that yield the most sparse (in a Gini index sense) set of wavelet coefficients in a fast, effective manner.