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.