Smoothing transforms for wavelet approximation of piecewise smooth functions

Multi-resolution analysis with high vanishing moment wavelets provides a framework to efficiently approximate smooth functions. However, it is a well-known fact that wavelet approximation usually cannot achieve the same order of approximation in the vicinity of discontinuous points of functions as that in the smooth regions. Ringing artefacts in the reconstructed functions inevitably appear around discontinuous points. To reduce these artefacts, the authors propose to locally smooth piecewise smooth functions at the discontinuous points, prior to applying the wavelet transform, via a smoothing transform. The numerical experiments for one- and two-dimensional signals show the effectiveness of the proposed strategy.