Image restoration: Total variation, wavelet frames, and beyond

The variational techniques (e.g. the total variation based method) are well established and effective for image restoration, as well as many other applications, while the wavelet frame based approach is relatively new and came from a different school. This paper is designed to establish a connection between these two major approaches for image restoration. The main result of this paper shows that when spline wavelet frames of are used, a special model of a wavelet frame method, called the analysis based approach, can be viewed as a discrete approximation at a given resolution to variational methods. A convergence analysis as image resolution increases is given in terms of objective functionals and their approximate minimizers. This analysis goes beyond the establishment of the connections between these two approaches, since it leads to new understandings for both approaches. First, it provides geometric interpretations to the wavelet frame based approach as well as its solutions. On the other hand, for any given variational model, wavelet frame based approaches provide various and flexible discretizations which immediately lead to fast numerical algorithms for both wavelet frame based approaches and the corresponding variational model. Furthermore, the built-in multiresolution structure of wavelet frames can be utilized to adaptively choose proper differential operators in different regions of a given image according to the order of the singularity of the underlying solutions. This is important when multiple orders of differential operators are used in various models that generalize the total variation based method. These observations will enable us to design new methods according to the problems at hand, hence, lead to wider applications of both the variational and wavelet frame based approaches. Links of wavelet frame based approaches to some more general variational methods developed recently will also be discussed.