Predicting wavelet coefficients over edges using estimates based on nonlinear approximants

It is well-known that wavelet transforms provide sparse decompositions over many types of image regions but not over image singularities/edges that manifest themselves along curves. It is now widely accepted that, on 2D piecewise smooth signals, wavelet compression performance is dominated by coefficients over edges. Research in this area has focused on two tracks, each suffering from issues related to translation invariance. Methods that directly model high order coefficient dependencies over edges have to combat aliasing issues, and new transforms that have been designed lose their full strength if they are not used in a translation invariant fashion. In this paper we combine these approaches and use translation invariant, overcomplete representations to predict wavelet edge coefficients. By starting with the lowest frequency band of an l level wavelet decomposition, we reliably estimate missing higher frequency coefficients over piecewise smooth signals. Unlike existing techniques, our approach does not model edges directly but implicitly obtains boundaries by aggressively determining regions where the utilized translation invariant decomposition is sparse.