Variational Approaches to Digital Image Zooming a Thesis Submitted to the Graduate School of the University of Minnesota by Todd Cameron Wittman in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

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. Unfortunatley, zooming is an ill-posed mathematical problem and the linear filters common in imaging software are often not adequate for the task. Other interpolation approaches include wavelets, PDEs, machine learning, and statistical filters, but variational methods offer computational advantages in the application and flexibility of the models. We discuss the theoretical and compuational 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 blurred grayscale image, the goal of quantized zooming is to produce a clean, high-resolution image taking on only a limited number of gray values. The graph cut method has proven successful for exact minimization of the