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

In the past few decades, mathematics based approaches have been widely adopted in various image restora-tion 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, implementa-tions 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