Variational and PDE Models, and their Computation for Image Inpainting

DMS奖摘要奖#：0202565PI：沈，建宏学院：明尼苏达大学，双城项目：应用数学项目经理：凯瑟琳·马夫里普利斯题目：变分和偏微分方程模型，及其图像绘制计算这个项目旨在发现和发展最基本和最关键的数学原理和框架，用于高度多样化的数字修复应用。这些框架将使我们能够进一步构建许多普遍适用的修复模型，并设计它们的高效和健壮的计算算法。我们建议的方法是使用几种高级数学工具来进行修复的建模和计算，其中包括贝叶斯决策理论、非线性偏微分方程(如平均曲率运动、非线性传输和扩散)、有界变分空间中的变分方法和自由边界问题、各种最先进的调和分析工具，如小波和多分辨分析，以及许多数值分析和计算偏微分方程的有效方案。该项目的突出之处还在于，我们首次提出将视觉上重要的曲线、曲面和图像几何图形集成到传统的统计模型和动态过程中。数字修复是开发一种自动过程，以智能地恢复和补充丢失的、不可用的或故意伪装的图像信息。这种信息丢失在各种重要领域无处不在，包括计算机视觉、网络(特别是无线)通信、稳健的图像编码和传输(例如来自哈勃太空望远镜)、从二维医学图像重建三维体积器官、伪装战场上的敌人武器和人员，以及数字化美术馆中破解的古画的数字修复。这个项目将为图像修复开发一个数学框架。除了对上述众多重要领域产生广泛影响外，该项目还将加强高水平纯数学与当代数字、计算机和人工智能技术的融合，并为数学建模、分析和计算创造大量机会。它还将帮助首席研究员开发新的课程，并在这个蓬勃发展的新应用数学领域培训新的研究生。日期：2002年5月22日

DMS Award AbstractAward #: 0202565PI: Shen, JianhongInstitution: University of Minnesota, Twin CitiesProgram: Applied MathematicsProgram Manager: Catherine MavriplisTitle: Variational and Partial Differential Equation Models, and their Computation for Image InpaintingThis project is intended to discover and develop the most fundamental and crucial mathematical principles and frameworks for highly diversified applications of digital inpainting. These frameworks will allow us to further construct many universally applicable inpainting models, and design their efficient and robust computational algorithms. Our proposed approach is to employ several high level mathematical tools for the modeling and computation of inpainting, which include the Bayesian decision theory, nonlinear partial differential equations (e.g. mean curvature motions, nonlinear transport and diffusion), variational methods in the space of Bounded Variations and for free boundary problems, a variety of state-of-the-art tools from harmonics analysis such as wavelets and multiresolution analysis, and many efficient schemes in numerical analysis and computational partial differential equations. The project is also highlighted by the fact that we are proposing for the first time to integrate visually important curve, surface, and image geometry into the traditionally statistical models and dynamic processes. Digital inpainting is to develop an automatic process to intelligently recover and complete the missing, unavailable, or purposely disguised image information. Such loss of information occurs ubiquitously in a variety of important fields including computer vision, network (especially wireless) communication, robust image coding and transmission (from the Hubble Space Telescope for example), three-dimensional volumetric organ reconstruction from two-dimensional medical images, disguise of enemy weapons and personnel in the battlefields, and the digital restoration of cracked ancient paintings in digitized fine art museums. This project will develop a mathematical framework for image inpainting. Besides the broad impact on the numerous important fields mentioned above, the project will also strengthen the integration of high level pure mathematics into the contemporary digital, computer, and artificial intelligence technology, and in return, create numerous opportunities for mathematical modeling, analysis, and computation. It will also help the principal investigator develop new curricula and train new graduate students in this booming fresh field of applied mathematics.Date: May 22, 2002