Perfusion MRI can provide information about perfusion characteristics of the observed tissue, which makes it a widely applicable medical procedure. Measuring process of MRI is very time-consuming, and therefore, using classical reconstruction methods, we are often not able to obtain enough samples to accomplish the needed time and space resolution for perfusion analysis. That is why it is necessary to use compressed sensing, which allows reconstruction from under-sampled data by solving an optimization model. In this work, several models for reconstruction of an image sequence are verified on real and artificial data, along with multiple algorithms capable of solving these models. Among the optimization models used in this work are two L+S models with different regularization of the S component that are solved by Forward-Backward and Chambolle-Pock algorithm. The quality of reconstruction for various models was compared especially by their perfusion curves. In the last section, we explore possible modifications of the SASS model in order to increase quality of reconstruction and resistance to under sampling for the purpose of better adaptation for dynamic data.

This master’s thesis deals with the restoration of audio signals damaged by quantization. The theoretical part starts with a description of quantization and dequantization in general, few existing methods of dequantization of audio signals and theory of sparse representations of signals are also presented. The next part introduces algorithms suitable for dequantization, specifically Douglas–Rachford, Chambolle–Pock, SPADEQ and implementation of these algorithms in MATLAB application in the next chapter. In the last part of this thesis, testing of reconstructed signals using the algorithms takes place and results are evaluated by objective measures SDR, PEMO-Q, PEAQ and subjective listening test MUSHRA.

Speckle noise occurs in a wide range of images due to sampling and digital degradation. Understanding how noise can be present in images have led to multiple denoising techniques. Most of these denoising techniques assume equal noise distribution. When the noise present in the image is not uniform, the resulting denoised image becomes less than the highest standard or quality. For this research, we will be focusing on speckle noise. Unlike Gaussian noise, which affects single pixels on an image, speckle noise affects multiple pixels. Hence it is not possible to remove speckle noise with the traditional gaussian denoising model. We develope a more accurate speckle denoising model and its stable numerical methods. This model is based on the TV minimization and the associated non-linear PDE and Krissian $et$ $al$.'s speckle noise equation model. A realistic and efficient speckle noise equation model was introduced with an edge enhancing feature by adopting a non-convex functional. An effective numerical scheme was introduced and its stability was proved. Also, while working with TV minimization for non-linear PDE and Krissian $et$ $al$ we used a dual approach for faster computation. This work is based on Chambolle's approach for image denoising. The NLM algorithm takes advantage of the high degree of redundancy of any natural image. Also, the NLM algorithm is very accurate since all pixels contribute for denoising at any given pixel. However, due to non-local averaging, one major drawback is computational cost. For this research, we will discuss new denoising techniques based on NLM and total variation for images contaminated by speckle noise. We introduce blockwise and selective denoising methods based on NLM technique and Partial Differential Equations (PDEs) methods for total variation to enhance computational efficiency. Our PDE methods have shown to be very computational efficient and as mentioned before the NLM process is very accurate.

This dissertation is about the partial differential equation (PDE) based image denoising models. In particular, we are interested about speckle noise images. We provide the mathematical analysis of existing speckle denoising models and propose three new models based on total variation minimization methods. The first model is developed using a new speckle noise model and the solution of associated numerical scheme is proven to be stable. The second one is a speckle version of Chambolle algorithm and the convergence of the numerical solution was proved under certain assumptions. The final model is a nonlocal PDE based speckle denoising model derived by combining the excellent noise removal properties of the nonlocal means algorithm with the PDE models. We enhanced the computational efficiency of this model by adopting the Split Bregman method. Numerical results of all three models show that they compare favorably to the conventional models.