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 coefcients. 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 representa-tion 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. Denois-ing and compression algorithms based on footprints and directional wavelets show interesting improvement over traditional wavelet methods.