Analysis of a wavelet frame based two-scale model for enhanced edges

Image restoration is a class of important tasks that emerges from a wide range of scientific disciplines. It has been noticed that most practical images can be modeled as a composition from a sparse singularity set (edges) where the image contents or their gradients change drastically, and cartoon chunks in which a high degree of regularity is dominant. Enhancing edges while promoting regularity elsewhere has been an important criterion for successful restoration in many image classes. In this article, we present a wavelet frame based image restoration model that captures potential edges and facilitates the restoration procedure by a dedicated treatment both of singularity and of cartoon. Moreover, its geometric robustness is enhanced by exploiting subtle inter-scale information available in the coarse image. To substantiate our intuition, we prove that this model converges to one variant of the celebrated Mumford-Shah model when adequate asymptotic specifications are given.