This thesis examines the concept of image self-similarity and provides solutions to various associated inverse problems such as resolution enhancement and missing fractal codes. In general, many real-world inverse problems are ill-posed, mainly because of the lack of existence of a unique solution. The procedure of providing acceptable unique solutions to such problems is known as regularization. The concept of image prior, which has been of crucial importance in image modelling and processing, has also been important in solving inverse problems since it algebraically translates to the regularization procedure. Indeed, much recent progress in imaging has been due to advances in the formulation and practice of regularization. This, coupled with progress in optimization and numerical analysis, has yielded much improvement in computational methods of solving inverse imaging problems. Historically, the idea of self-similarity was important in the development of fractal image coding. Here we show that the self-similarity properties of natural images may be used to construct image priors for the purpose of addressing certain inverse problems. Indeed, new trends in the area of non-local image processing have provided a rejuvenated appreciation of image self-similarity and opportunities to explore novel self-similarity-based priors. We first revisit the concept of fractal-based methods and address some open theoretical problems in the area. This includes formulating a necessary and sufficient condition for the contractivity of the block fractal transform operator. We shall also provide some more generalized formulations of fractal-based self-similarity constraints of an image. These formulations can be developed algebraically and also in terms of the set-based method of Projection Onto Convex Sets (POCS). We then revisit the traditional inverse problems of single frame image zooming and multi-frame resolution enhancement, also known as super-resolution. Some ideas will be borrowed from newly developed non-local denoising algorithms in order to formulate self-similarity priors. Understanding the role of scale and choice of examples/samples is also important in these proposed models. For this purpose, we perform an extensive series of numerical experiments and analyze the results. These ideas naturally lead to the method of self-examples, which relies on the regularity properties of natural images at different scales, as a means of solving the single-frame image zooming problem. Furthermore, we propose and investigate a multi-frame super-resolution counterpart which does not require explicit motion estimation among video sequences.

This thesis examines the concept of image self-similarity and provides solutions to various associated inverse problems such as resolution enhancement and missing fractal codes. In general, many real-world inverse problems are ill-posed, mainly because of the lack of existence of a unique solution. The procedure of providing acceptable unique solutions to such problems is known as regularization. The concept of image prior, which has been of crucial importance in image modelling and processing, has also been important in solving inverse problems since it algebraically translates to the regularization procedure. Indeed, much recent progress in imaging has been due to advances in the formulation and practice of regularization. This, coupled with progress in optimization and numerical analysis, has yielded much improvement in computational methods of solving inverse imaging problems. Historically, the idea of self-similarity was important in the development of fractal image coding. Here we show that the self-similarity properties of natural images may be used to construct image priors for the purpose of addressing certain inverse problems. Indeed, new trends in the area of non-local image processing have provided a rejuvenated appreciation of image self-similarity and opportunities to explore novel self-similarity-based priors. We first revisit the concept of fractal-based methods and address some open theoretical problems in the area. This includes formulating a necessary and sufficient condition for the contractivity of the block fractal transform operator. We shall also provide some more generalized formulations of fractal-based self-similarity constraints of an image. These formulations can be developed algebraically and also in terms of the set-based method of Projection Onto Convex Sets (POCS). We then revisit the traditional inverse problems of single frame image zooming and multi-frame resolution enhancement, also known as super-resolution. Some ideas will be borrowed from newly developed non-local denoising algorithms in order to formulate self-similarity priors. Understanding the role of scale and choice of examples/samples is also important in these proposed models. For this purpose, we perform an extensive series of numerical experiments and analyze the results. These ideas naturally lead to the method of self-examples, which relies on the regularity properties of natural images at different scales, as a means of solving the single-frame image zooming problem. Furthermore, we propose and investigate a multi-frame super-resolution counterpart which does not require explicit motion estimation among video sequences.

In this paper, we study ill-posedness concepts of nonlinear and linear inverse problems in a Hilbert space setting. We define local ill-posedness of a nonlinear operator equation $F(x) = y_0$ in a solution point $x_0$ and the interplay between the nonlinear problem and its linearization using the Frechet derivative $F\acent(x_0)$ . To find an appropriate ill-posedness concept for the linarized equation we define intrinsic ill-posedness for linear operator equations $Ax = y$ and compare this approach with the ill-posedness definitions due to Hadamard and Nashed.

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We revisit in L2-spaces the autoconvolution equation x ∗ x = y with solutions which are real-valued or complex-valued functions x(t) defined on a finite real interval, say t ∈ [0,1]. Such operator equations of quadratic type occur in physics of spectra, in optics and in stochastics, often as part of a more complex task. Because of their weak nonlinearity deautoconvolution problems are not seen as difficult and hence little attention is paid to them wrongly. In this paper, we will indicate on the example of autoconvolution a deficit in low order convergence rates for regularized solutions of nonlinear ill-posed operator equations F(x)=y with solutions x† in a Hilbert space setting. So for the real-valued version of the deautoconvolution problem, which is locally ill-posed everywhere, the classical convergence rate theory developed for the Tikhonov regularization of nonlinear ill-posed problems reaches its limits if standard source conditions using the range of F (x† )∗ fail. On the other hand, convergence rate results based on Hölder source conditions with small Hölder exponent and logarithmic source conditions or on the method of approximate source conditions are not applicable since qualified nonlinearity conditions are required which cannot be shown for the autoconvolution case according to current knowledge. We also discuss the complex-valued version of autoconvolution with full data on [0,2] and see that ill-posedness must be expected if unbounded amplitude functions are admissible. As a new detail, we present situations of local well-posedness if the domain of the autoconvolution operator is restricted to complex L2-functions with a fixed and uniformly bounded modulus function.

The focus of this paper is on conditional stability estimates for ill-posed inverse problems in partial differential equations. Conditional stability estimates have been obtained in the literature by a couple different methods. In this paper we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We are going to work out the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capability of our method is illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation.

The development of highly coherent X-ray sources offers new possibilities to image biological structures at different scales exploiting the refraction of X-rays. The coherence properties of the third-generation synchrotron radiation sources enables efficient implementations of phase contrast techniques. One of the first measurements of the intensity variations due to phase contrast has been reported in 1995 at the European Synchrotron Radiation Facility (ESRF). Phase imaging coupled to tomography acquisition allows threedimensional imaging with an increased sensitivity compared to absorption CT. This technique is particularly attractive to image samples with low absorption constituents. Phase contrast has many applications, ranging from material science, paleontology, bone research to medicine and biology. Several methods to achieve X-ray phase contrast have been proposed during the last years. In propagation based phase contrast, the measurements are made at different sample-to-detector distances. While the intensity data can be acquired and recorded, the phase information of the signal has to be "retrieved" from the modulus data only. Phase retrieval is thus an illposed nonlinear problem and regularization techniques including a priori knowledge are necessary to obtain stable solutions. Several phase recovery methods have been developed in recent years. These approaches generally formulate the phase retrieval problem as a linear one. Nonlinear treatments have not been much investigated. The main purpose of this work was to propose and evaluate new algorithms, in particularly taking into account the nonlinearity of the direct problem. In the first part of this work, we present a Landweber type nonlinear iterative scheme to solve the propagation based phase retrieval problem. This approach uses the analytic expression of the Fréchet derivative of the phase-intensity relationship and of its adjoint, which are presented in detail. We also study the effect of projection operators on the convergence properties of the method. In the second part of this thesis, we investigate the resolution of the linear inverse problem with an iterative thresholding algorithm in wavelet coordinates. In the following, the two former algorithms are combined and compared with another nonlinear approach based on sparsity regularization and a fixed point algorithm. The performance of theses algorithms are evaluated on simulated data for different noise levels. Finally the algorithms were adapted to process real data sets obtained in phase CT at the ESRF at Grenoble.