Sparse & Higher Order Image Restoration

在现代社会中，我们在许多不同的情况下都会遇到数字图像：从模拟相机早已被数字相机取代的日常生活，到它们在医学、地球科学、艺术和安全应用中的专业应用。医学成像工具的例子是MRI（磁共振成像），PET（正电子发射断层扫描），CT（计算机断层扫描）成像的大脑和内部器官，如人类心脏。这些成像工具通常产生噪声或不完整的图像数据。因此，在医生对它们进行评估之前，它们必须经过处理。这方面的关键词是图像去噪、图像去模糊、图像分解和图像补漆。最成功的图像处理方法之一是所谓的偏微分方程（PDEs）和变分模型。给定一个有噪声的图像，其处理（去噪）版本被计算为PDE的解或函数（变分模型）的最小值。这两种处理都是对给定图像进行正则化，从而消除噪声或填充图像中的缺失部分。有利的成像方法是消除高频特征（噪声），同时保留甚至增强低频特征（物体边界、边缘）。在这个项目中，我们建议关注在这种情况下最有效但最不容易理解的类之一：涉及高阶，特别是四阶微分阶表达式的方法。高阶方法在产生高质量的视觉结果方面远远优于标准图像恢复算法。汇集了来自不同数学领域的专业知识，其中包括应用偏微分方程，变分微积分，几何测量理论和现代数值分析，我们试图回答和补充围绕高阶成像模型发展的许多开放问题。这个项目的亮点是一个特定的图像处理任务，称为图像绘制。补图是利用从图像完整部分获得的信息，对图像中缺失部分进行填充的过程。它本质上是一种插值，并有应用，例如，在旧照片和绘画的恢复，文本擦除（例如，删除数字图像中的日期或电影中的字幕），或特殊效果，如物体消失。在这个插值过程中添加额外的几何约束，高阶方法能够解决一些标准绘制方法的缺点，比如能够恢复图像中非常大的空白中的内容。为了获得有效可靠的高阶涂装方法，必须对其数学性质进行深入分析。要回答的问题是：这些方法会产生什么样的解决方案？它们在生成的图像中促进了哪些特征（如规律性和稀疏性）？在数学设置中，我们必须操作哪些项，以及如何按照我们的喜好搅动插值过程？另一个问题是它们的数值执行。事实上，这些模型不适合应用任务和标准成像软件的不幸原因是，它们的解决方案使用当前的数值算法仍然昂贵且远离实时用户交互。该项目利用现代应用数学的复杂工具，利用偏微分方程和高阶变分公式，解决成像模型的开发、分析和有效的数值实现。

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society we encounter digital images in many different situations: from everyday life, where analogue cameras have long been replaced by digital ones, to their professional use in medicine, earth sciences, arts, and security applications.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.