The structure of optimal parameters for image restoration problems

We study the qualitative properties of optimal regularisation parameters in variational models for image restoration. The parameters are solutions of bilevel optimisation problems with the image restoration problem as constraint. A general type of regulariser is considered, which encompasses total variation (TV), total generalized variation (TGV) and infimal-convolution total variation (ICTV). We prove that under certain conditions on the given data optimal parameters derived by bilevel optimisation problems exist. A crucial point in the existence proof turns out to be the boundedness of the optimal parameters away from 0 which we prove in this paper. The analysis is done on the original -- in image restoration typically non-smooth variational problem -- as well as on a smoothed approximation set in Hilbert space which is the one considered in numerical computations. For the smoothed bilevel problem we also prove that it Γ converges to the original problem as the smoothing vanishes. All analysis is done in function spaces rather than on the discretised learning problem.In Cambridge, this project has been supported by King Abdullah University of Science and Technology (KAUST) Award No. KUK-I1-007-43, EPSRC grants Nr. EP/J009539/1 “Sparse & Higher-order Image Restoration”, and Nr. EP/M00483X/1 “Efficient computational tools for inverse imaging problems”. In Quito, the project has been supported by the Escuela Politécnica Nacional de Quito under award PIS 12-14 and the MATHAmSud project SOCDE “Sparse Optimal Control of Differential Equations”. When in Quito, T. Valkonen was moreover supported by a Prometeo scholarship of the Senescyt (Ecuadorian Ministry of Science, Technology, Education, and Innovation).This is the final version of the article. It first appeared from Elsevier via http://dx.doi.org/10.1016/j.jmaa.2015.09.02

Statistical mechanics of image restoration and error-correcting codes

We develop a statistical-mechanical formulation for image restoration and error-correcting codes. These problems are shown to be equivalent to the Ising spin glass with ferromagnetic bias under random external fields. We prove that the quality of restoration/decoding is maximized at a specific set of parameter values determined by the source and channel properties. For image restoration in mean-field system a line of optimal performance is shown to exist in the parameter space. These results are illustrated by solving exactly the infinite-range model. The solutions enable us to determine how precisely one should estimate unknown parameters. Monte Carlo simulations are carried out to see how far the conclusions from the infinite-range model are applicable to the more realistic two-dimensional case in image restoration.Comment: 20 pages, 9 figures, ReVTe

Multilevel Approach For Signal Restoration Problems With Toeplitz Matrices

We present a multilevel method for discrete ill-posed problems arising from the discretization of Fredholm integral equations of the first kind. In this method, we use the Haar wavelet transform to define restriction and prolongation operators within a multigrid-type iteration. The choice of the Haar wavelet operator has the advantage of preserving matrix structure, such as Toeplitz, between grids, which can be exploited to obtain faster solvers on each level where an edge-preserving Tikhonov regularization is applied. Finally, we present results that indicate the promise of this approach for restoration of signals and images with edges