The Large Deviation Principle for Coarse-Grained Processes

The large deviation principle is proved for a class of $L^2$-valued processes that arise from the coarse-graining of a random field. Coarse-grained processes of this kind form the basis of the analysis of local mean-field models in statistical mechanics by exploiting the long-range nature of the interaction function defining such models. In particular, the large deviation principle is used in a companion paper to derive the variational principles that characterize equilibrium macrostates in statistical models of two-dimensional and quasi-geostrophic turbulence. Such macrostates correspond to large-scale, long-lived flow structures, the description of which is the goal of the statistical equilibrium theory of turbulence. The large deviation bounds for the coarse-grained process under consideration are shown to hold with respect to the strong $L^2$ topology, while the associated rate function is proved to have compact level sets with respect to the weak topology. This compactness property is nevertheless sufficient to establish the existence of equilibrium macrostates for both the microcanonical and canonical ensembles.Comment: 19 page

BM3D Frames and Variational Image Deblurring

A family of the Block Matching 3-D (BM3D) algorithms for various imaging problems has been recently proposed within the framework of nonlocal patch-wise image modeling [1], [2]. In this paper we construct analysis and synthesis frames, formalizing the BM3D image modeling and use these frames to develop novel iterative deblurring algorithms. We consider two different formulations of the deblurring problem: one given by minimization of the single objective function and another based on the Nash equilibrium balance of two objective functions. The latter results in an algorithm where the denoising and deblurring operations are decoupled. The convergence of the developed algorithms is proved. Simulation experiments show that the decoupled algorithm derived from the Nash equilibrium formulation demonstrates the best numerical and visual results and shows superiority with respect to the state of the art in the field, confirming a valuable potential of BM3D-frames as an advanced image modeling tool.Comment: Submitted to IEEE Transactions on Image Processing on May 18, 2011. implementation of the proposed algorithm is available as part of the BM3D package at http://www.cs.tut.fi/~foi/GCF-BM3

A transformation method for constrained-function minimization

A direct method for constrained-function minimization is discussed. The method involves the construction of an appropriate function mapping all of one finite dimensional space onto the region defined by the constraints. Functions which produce such a transformation are constructed for a variety of constraint regions including, for example, those arising from linear and quadratic inequalities and equalities. In addition, the computational performance of this method is studied in the situation where the Davidon-Fletcher-Powell algorithm is used to solve the resulting unconstrained problem. Good performance is demonstrated for 19 test problems by achieving rapid convergence to a solution from several widely separated starting points