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Directional wavelets and wavelet footprints for compression and denoising

In recent years, wavelet based algorithms have been successful in different signal processing tasks. The wavelet transform is a powerful tool because it manages to represent both transient and stationary behaviours of a signal with few transform coefficients. In this paper we present new expansions and algorithms which improve wavelet algorithms. First we focus on one dimensional piecewise smooth signals and propose a new representation of these signals in terms of elements which we call footprints. Then we consider two dimensional signals and present a new directional wavelet transform, which keeps the simplicity of the standard separable wavelet transform but allows for more directionalities. Denoising and compression algorithms based on footprints and directional wavelets show interesting improvement over traditional wavelet methods.