Nonlinear pyramid transforms based on median-interpolation

We introduce a nonlinear refinement subdivision scheme based on median-inte rpolation. The scheme constructs a polynomial interpolating adjacent block medians of an underlying object. The interpolating polynomial is then used to impute block medians at the next ner triadic scale. Perhaps surprisingly , expressions for the refinement operator can be obtained in closed-form fo r the scheme interpolating by polynomials of degree D = 2. Despite the nonl inearity of this scheme, convergence and regularity can be established usin g techniques reminiscent of those developed in analysis of linear refinemen t schemes. The refinement scheme can be deployed in multiresolution fashion to constru ct a nonlinear pyramid and an associated forward and inverse transform. In this paper we discuss the basic properties of these transforms and their po ssible use in removing badly non-Gaussian noise. Analytic and computational results are presented to show that in the presence of highly non-Gaussian noise, the coefficients of the nonlinear transform have much better propert ies than traditional wavelet coefficients.