Image Restoration: Wavelet Frame Shrinkage, Nonlinear Evolution PDEs, and Beyond

In the past few decades, mathematics based approaches have been widely adopted in various image restoration problems, among which the partial differential equation (PDE) based approach (e.g. the total variation model [56] and its generalizations, nonlinear diffusions [15, 52], etc.), and wavelet frame based approach are some of successful examples. These approaches were developed through different paths and generally provided understandings from different angles of the same problem. As shown in numerical simulations, implementations of wavelet frame based approach and PDE based approach quite often end up with solving a similar numerical problem with similar numerical behaviors, even though different approaches have advantages in different applications. Since wavelet frame based and PDE based approaches have all been modeling the same type of problems with success, it is natural to ask whether wavelet frame based approach is fundamentally connected with PDE based approach when we trace all the way back to their roots. A fundamental connection of a wavelet frame based approach with total variation model and its generalizations were established in [8]. This connection gives wavelet frame based approach a geometric explanation and, at the same time, it equips a PDE based approach with a time frequency analysis. It was shown in [8] that a special type of wavelet frame model using generic wavelet frame systems can be regarded as an approximation of a generic variational model (with the total variation model as a special case) in the discrete setting. A systematic convergence analysis, as the resolution of the image goes to the infinity, which is the key step to link the two approaches, is also given in [8]. Motivated by [8] and [47], this paper is to establish a fundamental connection between wavelet frame based approach and nonlinear evolution PDEs, provide interpretations and analytical studies of such connections, and propose new algorithms for image restoration based on the new understandings. Together with the results in [8], we now have a better picture of how wavelet frame based approach can be used to interpret general PDE based approach (e.g. the variational models or nonlinear evolution PDEs) and can be used as a new and useful tool in numerical analysis to discretize and solve various variational and PDE models. To be more precise on our contributions, we shall establish that: (1) The connections between wavelet frame shrinkage and nonlinear evolution PDEs provide new and inspiring interpretations of both approaches that enable us to derive new PDE models and (better) wavelet frame shrinkage algorithms for image restoration. (2) A generic nonlinear evolution PDEs (of parabolic or hyperbolic type) can be approximated by wavelet frame shrinkage with properly chosen wavelet frame systems and carefully designed shrinkage functions. (3) The main idea of this work is beyond the scope of image restoration. Our analysis and discussions indicate that wavelet frame shrinkage is a new way of solving PDEs in general, which will provide a new insight that will enrich the existing theory and applications of numerical PDEs, as well as, those of wavelet frames.