Discrete finite variation: a new measure of smoothness for the design of wavelet basis

A new method for measuring and designing smooth wavelet basis which dispenses with the need for having a large number of zero moments of the wavelet is given. The method is based on minimizing the “discrete finite variation”, and is a measure of the local “roughness” of a sampled version of the scaling function giving rise to “visually smooth” wavelet basis. Smooth wavelet basis are deemed to be important for several applications and in particularly for image compression where the goal is to limit spurious artifacts due to non-smooth basis functions in the presence of quantization of the individual subbands. The definition of smoothness introduced here gives rise to new algorithms for designing smooth wavelet basis with only one vanishing moment leaving free parameters, otherwise used for setting moments to zero, for optimization.