Splitting Methods in Image Processing

It is often necessary to restore digital images which are affected by noise (denoising), blur (deblurring), or missing data (inpainting). We focus here on variational methods, i.e., the restored image is the minimizer of an energy functional. The first part of this thesis deals with the algorithmic framework of how to compute such a minimizer. It turns out that operator splitting methods are very useful in image processing to derive fast algorithms. The idea is that, in general, the functional we want to minimize has an additive structure and we treat its summands separately in each iteration of the algorithm which yields subproblems that are easier to solve. In our applications, these are typically projections onto simple sets, fast shrinkage operations, and linear systems of equations with a nice structure. The two operator splitting methods we focus on here are the forward-backward splitting algorithm and the Douglas-Rachford splitting algorithm. We show based on older results that the recently proposed alternating split Bregman algorithm is equivalent to the Douglas-Rachford splitting method applied to the dual problem, or, equivalently, to the alternating direction method of multipliers. Moreover, it is illustrated how this algorithm allows us to decouple functionals which are sums of more than two terms. In the second part, we apply the above techniques to existing and new image restoration models. For the Rudin-Osher-Fatemi model, which is well suited to remove Gaussian noise, the following topics are considered: we avoid the staircasing effect by using an additional gradient fitting term or by combining first- and second-order derivatives via an infimal-convolution functional. For a special setting based on Parseval frames, a strong connection between the forward-backward splitting algorithm, the alternating split Bregman method and iterated frame shrinkage is shown. Furthermore, the good performance of the alternating split Bregman algorithm compared to the popular multistep methods is illustrated. A special emphasis lies here on the choice of the step-length parameter. Turning to a corresponding model for removing Poisson noise, we show the advantages of the alternating split Bregman algorithm in the decoupling of more complicated functionals. For the inpainting problem, we improve an existing wavelet-based method by incorporating anisotropic regularization techniques to better restore boundaries in an image. The resulting algorithm is characterized as a forward-backward splitting method. Finally, we consider the denoising of a more general form of images, namely, tensor-valued images where a matrix is assigned to each pixel. This type of data arises in many important applications such as diffusion-tensor MRI

Forschungsbericht Universität Mannheim 2010 / 2011

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Approximation related to quotient functionals

AbstractWe examine the best approximation of componentwise positive vectors or positive continuous functions f by linear combinations fˆ=∑jαjφj of given vectors or functions φj with respect to functionals Qp, 1≤p≤∞, involving quotients max{f/fˆ,fˆ/f} rather than differences |f−fˆ|. We verify the existence of a best approximating function under mild conditions on {φj}j=1n. For discrete data, we compute a best approximating function with respect to Qp, p=1,2,∞ by second order cone programming. Special attention is paid to the Q∞ functional in both the discrete and the continuous setting. Based on the computation of the subdifferential of our convex functional Q∞ we give an equivalent characterization of the best approximation by using its extremal set. Then we apply this characterization to prove the uniqueness of the best Q∞ approximation for Chebyshev sets {φj}j=1n