The main idea of the thesis is to develop new connections between the theory of real interpolation and applications. Near and exact minimizers for E–, K– and L–functionals of the theory of real interpolation are very important in applications connected to regularization of inverse problems such as image processing. The problem which appears is how to characterize and construct these minimizers. These exact minimizers referred to as optimal decompositions in the thesis, have certain extremal properties that we completely express and characterize in terms of duality. Our characterization generalizes known characterization for a particular Banach couple. The characterization presented in the thesis also makes it possible to understand the geometrical meaning of optimal decomposition for some important particular cases and gives a possibility to construct them. One of the most famous models in image processing is the total variation regularization published by Rudin, Osher and Fatemi. We propose a new fast algorithm to find the exact minimizer for this model. Optimal decompositions mentioned have some connections to optimization problems which are also pointed out. The thesis is based on results that have been presented in international conferences and have been published in five papers.
In Paper 1, we characterize optimal decomposition for the E–, K– and L_{p0,p1} –functional. We also present a geometrical interpretation of optimal decomposition for the L_{p,1}–functional for the couple (ℓp, X) on R^{n}. The characterization presented is useful in the sense that it gives insights into the construction of these minimizers.
The characterization mentioned in Paper 1 is based on optimal decomposition for infimal convolution. The operation of infimal convolution is a very important and non–trivial tool in functional analysis and is also very well–known within the context of convex analysis. The L–, K– and E–functionals can be regarded as an infimal convolution of two well–defined functions. Unfortunately tools from convex analysis can not be applied in a straightforward way in this context of couples of spaces. The most important requirement that an infimal convolution would satisfy for a decomposition to be optimal is subdifferentiability. In Paper 2, we have used an approach based on the famous Attouch–Brezis theorem to prove subdifferentiability of infimal convolution on Banach couples.
In Paper 3, we apply result from Paper 1 to the well–known Rudin–Osher–Fatemi (ROF) image denoising model on a general finite directed graph. We define the space BV of functions of bounded variation on the graph and show that the unit ball of its dual space can be described as the image of the unit ball of the space `¥ on the graph by a divergence operator. Based on this result, we propose a new fast algorithm to find the exact minimizer for the ROF model. Proof of convergence of the algorithm is presented and its performance on image denoising test examples is illustrated.
In Paper 4, we present some extensions of results presented in Paper 1 and Paper 2. First we extend the results from Banach couples to Banach triples. Then we prove that our approach can apply when complex spaces are considered instead of real spaces. Finally we compare the performance of the algorithm that was proposed in Paper 3 with the Split Bregman algorithm which is one of the benchmark algorithms known for the ROF model. We find out that in most cases both algorithms behave in a similar way and that in some cases our algorithm decreases the error faster with the number of iterations.
In Paper 5, we point out some connections between optimal decompositions mentioned in the thesis and optimization problems. We apply the approach used in Paper 2 to two well–known optimization problems, namely convex and linear programming to investigate connections with standard results in the framework of these problems. It is shown that we can derive proofs for duality theorems for these problems under the assumptions of our approach.