Recent research in generative models came up with a promising approach to modelling the prior proba- bility of natural images. The architecture of these prior models is based on deep neural networks. Although these priors were primarily designed for generating new natural-like images, its potential use is much broader. One of the possible applications is to use these models for solving the inverse problems in low-level vision (i.e., image reconstruction). This usage is mainly possible because the architecture of these models allows computing the derivative of the prior probability with respect to the input image. The main objective of this thesis is to evaluate the usage of these prior models in image reconstruction. This thesis proposes a novel model-based optimization method to two image reconstruction problems - image denoising and single-image super-resolution (SISR). The proposed method uses optimization algorithms for finding the maximum-a- posteriori probability, which is defined using the above mentioned prior models. The experimental results demonstrate that the proposed approach achieves reconstruction performance competitive with the current state-of-the-art methods, especially regarding SISR.

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.