Sparse & Higher Order Image Restoration

在现代社会中，我们在许多不同的情况下遇到数字图像：从日常生活中，模拟相机早已被数字相机取代，到医学，地球科学，艺术和安全应用中的专业用途。医学成像工具的示例是MRI（磁共振成像）、PET（正电子发射断层扫描）、CT（计算机断层扫描），用于对大脑和内部器官（如人类心脏）进行成像。这些成像工具通常产生有噪声或不完整的图像数据。因此，在医生对其进行评估之前，必须对其进行处理。图像去噪、图像去模糊、图像分解和图像修复是图像处理中最成功的方法之一，其中偏微分方程（PDE）和变分模型是图像处理中最常用的方法之一。给定噪声图像，其处理（去噪）版本被计算为PDE的解或泛函（变分模型）的最小值。这两个过程都是对给定图像进行正则化，从而消除图像中的噪声或填充图像中的缺失部分。最常用的成像方法是通过消除高频特征（噪声），同时保留甚至增强低频特征（物体边界，边缘）来实现的。在这个项目中，我们建议专注于最有效但最不了解的类之一：涉及高，特别是四阶微分表达式的方法。高阶方法在它们产生的高质量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，变分法，几何测度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性问题。该项目的重点是一个特定的图像处理任务，称为图像修复。修复是指利用图像完整部分的信息来填补图像中缺失部分的过程。它本质上是一种插值，并具有应用，例如，在旧照片和绘画的修复中，文本擦除（例如，去除数字图像中的日期或电影中的字幕），或者像对象消失这样的特殊效果。高阶修复方法通过在插值过程中增加额外的几何约束，能够解决标准修复方法的一些缺点，例如无法恢复图像中非常大的间隙中的内容。为了获得有效可靠的高阶修复方法，不可避免地要彻底分析它们的数学特性。需要回答的问题是：这些方法产生了什么样的解决方案？它们在生成的图像中促进了哪些特征（如规则性和稀疏性）？我们必须操纵数学设置中的哪些项，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实现。事实上，不幸的是，为什么这些模型不适应于应用任务和标准成像软件是，他们的解决方案与当前的数值算法仍然是昂贵的，远离实时用户interaction.This项目地址的发展，分析和有效的数值实现成像模型使用偏微分方程和变分制剂的高微分阶复杂的工具，从现代应用数学。

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society we encounter digital images in many different situations: from everyday life, where analogue cameras have long been replaced by digital ones, to their professional use in medicine, earth sciences, arts, and security applications.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