Sparse & Higher Order Image Restoration

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In the modern society we encounter digital images in many different situations: from everyday life, where analogue cameras have long been replaced by digital ones, to their professional use in medicine, earth sciences, arts, and security applications. Examples of medical imaging tools are MRI (Magnetic Resonance Imaging), PET (Positron Emission Tomography), CT (computed tomography) for imaging the brain and inner organs like the human heart. These imaging tools usually produce noisy or incomplete image data. Hence, before they can be evaluated by doctors, they have to be processed. Keywords in this context are image denoising, image deblurring, image decomposition and image inpainting.One of the most successful image processing approaches are so-called partial differential equations (PDEs) and variational models. Given a noisy image, its processed (denoised) version is computed as a solution of a PDE or as a minimiser of a functional (variational model). Both of these processes are regularising the given image and herewith eliminate noise or fill missing parts in images. Favourable imaging approaches are doing so by eliminating high-frequency features (noise) while preserving or even enhancing low-frequency features (object boundaries, edges).In this project we propose to focus on one of the most effective while least understood classes in this context: methods that involve expressions of high, especially fourth, differential order. Higher-order methods by far outperform standard image restoration algorithms in terms of the high-quality visual results they produce. Bringing together the expertises from different fields of mathematics, among them applied PDEs, variational calculus, geometric measure theory and modern numerical analysis, we attempt to answer and complement some of the many open questions evolving around higher-order imaging models.The punchline of the project is a specific image processing task called image inpainting. Inpainting denotes the process of filling-in missing parts in an image using the information gained from the intact part of the image. It is essentially a type of interpolation and has applications, e.g., in the restoration of old photographs and paintings, text erasing (e.g., removal of dates in digital images or subtitles in a movie), or special effects like object disappearance. Adding additional geometrical constraints to this interpolation process, higher-order methods are able to address some of the shortcomings of standard inpainting methods like the ability to restore contents in very large gaps in an image.In order to have effective and reliable higher-order inpainting approaches it is inevitable to analyse their mathematical properties thoroughly. Questions to answer are: what kind of solutions do these approaches produce? What are the characteristic features (like regularity and sparseness) they promote in the resulting image? Which terms in the mathematical setup do we have to manipulate and how, to stir the interpolation process to our liking?Another issue is their numerical implementation. In fact, the unfortunate reason why these models are not accommodated in applied tasks and standard imaging software is that their solution with current numerical algorithms is still expensive and far away from real-time user interaction.This project addresses the development, analysis and efficient numerical implementation of imaging models using PDEs and variational formulations of high-differential order with sophisticated tools from modern applied mathematics.