Wavelet transform footprints: catching singularities for compression and denoising

Wavelets have been widely used for signal compression, image compression being a prime example, and for signal denoising. What makes wavelets such an attractive tool is their capability of representing both transient and stationary behaviors of a signal with few coefficients. We consider the problem of compressing and denoising a particular class of functions: piecewise polynomial signals. We show the limit of usual wavelet coders and present an alternative compression algorithm. The main innovation of the algorithm is that it tries to efficiently compress the significant coefficients of the wavelet decomposition rather then the zero coefficients as in usual coders. The proposed algorithm can potentially be extended to more general signals and represents an effective solution to problems like signal denoising and image compression.