| The purpose of this thesis is to discuss digital image resolution enhancement by variational methods and the associated computational issues. Two problems related to the basic zooming problem are also studied: super-resolution and quantized deconvolution.; Digital zooming is important for mundane computing activities such as web browsing as well as sophisticated applications like satellite imagery and medical diagnosis. Unfortunately, zooming is an ill-posed mathematical problem and the linear filters common in imaging software are often not adequate for the task. We discuss the theoretical and computational issues surrounding variational zooming, focusing on the Total Variation (TV) and Mumford-Shah energies. The variational inpainting model is very flexible and the interpolated result can be improved with energy modifications, including locally adaptive fidelity weights, soft inpainting, and post-processing.; Super-resolution refers to the process of producing a single high-resolution image from a set of low-resolution images such as a video sequence. Variational inpainting extends naturally to the multiple-image case and is shown to be effective for video enhancement, barcode processing, and MR image reconstruction. We propose a soft inpainting model to handle local variation and motion within a video sequence.; Text and barcode images should appear as strictly binary-valued images, but due to blurring and downsampling the actual images takes on many gray values and may be unreadable by recognition systems. Given a bluffed grayscale image, the goal of quantized zooming is to produce a dean, high-resolution image taking on only a limited number of gray values. The graph cut method has proven successful for exact minimization of the quantized TV energy. We show the graph cut method is effective for denoising, segmentation, and inpainting, but deconvolution is an open problem in the literature. We propose an alternating minimization method for debluffing that combines graph cuts and numerical relaxation inspired by linear programming. For the zooming problem, me approach is improved by the addition of local gradient information. We provide numerical results for barcode imaging, text enhancement, and medical image segmentation. |
