Robust nonlinear wavelet transform based on median-interpolation

It is well known that wavelet transforms can be derived from stationary linear refinement subdivision schemes. We discuss a special nonlinear refinement subdivision scheme-median-interpolation. It is a nonlinear cousin of Deslauriers-Dubuc (1992) interpolation and of average-interpolation. The refinement scheme is based on constructing polynomials which interpolate median functionals of the underlying object. The refinement scheme can be deployed in a multiresolution fashion to construct nonlinear pyramid schemes and associated forward and inverse transforms. We discuss the basic properties of this transform and its possible use in wavelet de-noising schemes for badly non-Gaussian data. Analytic and computational results are presented to show that the nonlinear pyramid has a very different performance compared to traditional wavelets when coping with non-Gaussian data.