Image Reconstruction Using PDE, Variational and Regularization Methods

Abstract

Image restoration and enhancement are two important requirements in the field of image processing. In this study three anisotropic non-linear diffusion filters are proposed for image restoration and enhancement and one filter for image inpainting. The orientation, type and extent of filtering are controlled by the decision mechanism based on the underlying image features. The first process is a conditionally anisotropic diffusion for deblurring and denoising images. This process is a time-dependent curvature based model and the steady state is attained at a faster rate, using the explicit time-marching scheme. The filter switches between isotropic and anisotropic behavior based on the local image features. Two other non-linear curvature based diffusion processes are devised, one for image enhancement and the other one for image inpainting. The diffusion process in these filters is driven by the Gauss curvature of the level curves of the image. Therefore, these methods are capable of preserving structures even with non-zero mean curvature values like curvy edges and corners. To be precise, the second process couples a hyperbolic shock filter together with a Gauss curvature driven diffusion term to enhance images. And the third one inpaints the intended domain based on the Gauss curvature. Finally, a fourth-order shock coupled diffusion filter is proposed for image enhancement. This is an anisotropic model that converges at a faster rate and preserves planar approximation while enhancing images. In this study a thorough theoretical and experimental analysis is carried out for each and every diffusion process introduced as a part of this thesis work. A variety of applications are presented for denoising and deblurring gray-level and color images. The required mathematical preliminaries are presented in the introduction of the thesis. We conclude the thesis highlighting some of the future enhancements that could be possibly taken forward for further research.