- 2012 (2) (entfernen)
- Minimization and Parameter Estimation for Seminorm Regularization Models with I-Divergence Constraints (2012)
- This papers deals with the minimization of seminorms \(\|L\cdot\|\) on \(\mathbb R^n\) under the constraint of a bounded I-divergence \(D(b,H\cdot)\). The I-divergence is also known as Kullback-Leibler divergence and appears in many models in imaging science, in particular when dealing with Poisson data. Typically, \(H\) represents here, e.g., a linear blur operator and \(L\) is some discrete derivative operator. Our preference for the constrained approach over the corresponding penalized version is based on the fact that the I-divergence of data corrupted, e.g., by Poisson noise or multiplicative Gamma noise can be estimated by statistical methods. Our minimization technique rests upon relations between constrained and penalized convex problems and resembles the idea of Morozov's discrepancy principle. More precisely, we propose first-order primal-dual algorithms which reduce the problem to the solution of certain proximal minimization problems in each iteration step. The most interesting of these proximal minimization problems is an I-divergence constrained least squares problem. We solve this problem by connecting it to the corresponding I-divergence penalized least squares problem with an appropriately chosen regularization parameter. Therefore, our algorithm produces not only a sequence of vectors which converges to a minimizer of the constrained problem but also a sequence of parameters which convergences to a regularization parameter so that the penalized problem has the same solution as our constrained one. In other words, the solution of this penalized problem fulfills the I-divergence constraint. We provide the proofs which are necessary to understand our approach and demonstrate the performance of our algorithms for different image restoration examples.
- Anisotropic Smoothing and Image Restoration Facing Non-Gaussian Noise (2012)
- Image restoration and enhancement methods that respect important features such as edges play a fundamental role in digital image processing. In the last decades a large variety of methods have been proposed. Nevertheless, the correct restoration and preservation of, e.g., sharp corners, crossings or texture in images is still a challenge, in particular in the presence of severe distortions. Moreover, in the context of image denoising many methods are designed for the removal of additive Gaussian noise and their adaptation for other types of noise occurring in practice requires usually additional efforts. The aim of this thesis is to contribute to these topics and to develop and analyze new methods for restoring images corrupted by different types of noise: First, we present variational models and diffusion methods which are particularly well suited for the restoration of sharp corners and X junctions in images corrupted by strong additive Gaussian noise. For their deduction we present and analyze different tensor based methods for locally estimating orientations in images and show how to successfully incorporate the obtained information in the denoising process. The advantageous properties of the obtained methods are shown theoretically as well as by numerical experiments. Moreover, the potential of the proposed methods is demonstrated for applications beyond image denoising. Afterwards, we focus on variational methods for the restoration of images corrupted by Poisson and multiplicative Gamma noise. Here, different methods from the literature are compared and the surprising equivalence between a standard model for the removal of Poisson noise and a recently introduced approach for multiplicative Gamma noise is proven. Since this Poisson model has not been considered for multiplicative Gamma noise before, we investigate its properties further for more general regularizers including also nonlocal ones. Moreover, an efficient algorithm for solving the involved minimization problems is proposed, which can also handle an additional linear transformation of the data. The good performance of this algorithm is demonstrated experimentally and different examples with images corrupted by Poisson and multiplicative Gamma noise are presented. In the final part of this thesis new nonlocal filters for images corrupted by multiplicative noise are presented. These filters are deduced in a weighted maximum likelihood estimation framework and for the definition of the involved weights a new similarity measure for the comparison of data corrupted by multiplicative noise is applied. The advantageous properties of the new measure are demonstrated theoretically and by numerical examples. Besides, denoising results for images corrupted by multiplicative Gamma and Rayleigh noise show the very good performance of the new filters.