Image restoration: Total variation, wavelet frames, and beyond

Abstract

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.

Projected Iterative Soft-Thresholding Algorithm for Tight Frames in Compressed Sensing Magnetic Resonance Imaging

“…Then, we apply proximal algorithm [19,24] to solve the problem in (16) by incorporating the proximal mapping…”

“…So far, we have converted the original analysis model based CS-MRI problem into a much simpler form in (19) where the objective function is separable. However, the constraint   Range  α Ψ makes us hard to find an analytical solution of (19).…”

“…However, the constraint   Range  α Ψ makes us hard to find an analytical solution of (19). Observing that without this constraint, (19) degenerates to the proximal mapping in (11) and its closed form solution is…”

“…For these redundant systems, there exists significant difference between these two models as reported in [8,14,17,18]. Theoretically, CS theory [10] reveals that analysis models works under a wider range of dictionaries than the synthesis models, and it was proved in [19] that analysis models converge to some partial differential equation models with geometric interpretations. In practice, researchers in signal processing observed that analysis models reaches lower reconstructed error than synthesis models [8,14,17,18].…”

A Dictionary Learning Approach for Poisson Image Deblurring

“…Other widely-used approaches are wavelets and frames [12]- [14], which provide a multiscale and sparse representation of images, hence breaking the illposedness of the initial inverse problem. These methods better preserve textures than TV regularization and interesting connections between wavelet-based and TV-based restoration methods were established by Chambolle et al [15] and later by Steidl et al [16] and Cai et al [17]. In 2002, Malgouyres [18] and Candes [19] proposed a hybrid restoration method combining TV and wavelets.…”