IMAGE RESTORATIONS: A WAVELET FRAME BASED MODEL FOR PIECEWISE SMOOTH FUNCTIONS AND BEYOND

Abstract. In this paper, we propose a new wavelet frame based image restoration model that explicitly treats images as piecewise smooth functions. It estimates both the image to be restored and its singularity set. It can well protect singularities, which are important image features, and provide enough regularization in smooth regions at the same time. This model penalizes the ℓ2-norm of the wavelet frame coefficients away from the singularity set, while penalizes the ℓ1-norm of the coefficients on the singularity set. This model explicitly models images as piecewise smooth functions with a general smoothness regularization and characterizes rather general singularity set, which includes both jump discontinuities and jumps after certain orders of differentiations. As we know, all types of singularities are important image features and need to be recovered. Furthermore, the singularity set can be robustly estimated by wavelet frame transform during the image recovery procedure, which makes our model easy to solve numerically; hence, the model is insensitive to the estimation of the singularity set. The proposed model is in discrete setting and is a wavelet frame based approach. To further understand the piecewise smooth nature of the obtained solutions, we connect it to a variational model on the space of piecewise smooth functions and prove rigorously that the discrete mode