Image restoration: A wavelet frame based model for piecewise smooth functions and beyond

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 ℓ2ℓ2-norm of the wavelet frame coefficients away from the singularity set, while penalizes the ℓ1ℓ1-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 model converges to the variational model as image resolution goes to infinity. Also, we show that the approximate solutions of the discrete model can be regarded as an approximation of those of the variational model. Through these theoretical analysis, we manage to connect the proposed discrete wavelet frame based model with the variational model. Such connection not only enables us to acquire deeper understandings of the discrete model, but also leads us to the discovery of a variational model new to the literature, which is more general and works better than the Mumford–Shah model [1] for image restoration problems. Although the focus of the paper is to propose the new model and provide theoretical studies, we still conduct numerical simulations to support our claims and theoretical findings. Our numerical studies show that the proposed model is the right one for image restorations, when the underlying solutions are piecewise smooth. Generally speaking, this model combines the merits of the PDE based approach [1], [2], [3], [4], [5], [6] and [7] and the wavelet frame based approach [8], [9] and [10].