PDE-based Image Restoration: Efficient Numerical Algorithms and Software Engineering

基于偏微分方程的图像恢复：高效的数值算法和软件工程

• 批准号: 0312223
• 负责人: Seongjai Kim
• 金额: $ 11.05万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-09-15 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0312223&HistoricalAwards=false
• 关键词: PDE; based; Image; Restoration; Efficient

Proposal: DMS-0312223PI: Seongjai Kim [[e-mail]]Institution: University of KentuckyTitle: ITR: PDE-based Image Restoration: Efficient Numerical Algorithms and Software EngineeringABSTRACTAs the field of image processing (IP) requires higher levels of reliability and efficiency, mathematical IP has become an important component. In particular, mathematical frameworks employing recent powerful tools of partial differential equations (PDEs) have been extensively studied to answer fundamental questions in IP. Such PDE-based methods turn out to allow researchers not only to introduce innovative mathematical models but also to analyze and improve traditional algorithms most of which have been developed heuristically. Developing appropriate numerical techniques for the PDE models is another important component for the PDE-based approaches. The proposal is concerned with the development of numerical algorithms for PDE-based image restoration and their applications to challenging problems. The models to be solved include nonlinear PDEs representing motion by mean curvature and Laplacian mean curvature flows, p-harmonic maps, and curvature-based total-variation minimization. The main goals are (a) to develop reliable and efficient numerical algorithms for image restoration, (b) to apply those algorithms for noise removal, image enhancement, and inpainting for real-life images such as medical imagery and satellite images, and (c) to construct related software packages. The newly developed numerical algorithms are expected to deliver large impact on mathematical image analysis; the resulting software packages must be applicable to various interesting problems dealing with images.Many applications in the modern digital age are based on images and therefore the resulting achievements must rely on their quality. Since images are not always in a good quality due to various types of noise, e.g., natural noise, defects in the sensors, and transmission problems, it is important to eliminate the noise automatically. Such image restoration is historically one of the oldest concerns and still a necessary processing step. As the field requires higher levels of reliability and efficiency for the last two decades, mathematical (PDE-based) image restoration has become an important component; it has been extensively studied to answer fundamental questions in image processing and to analyze/improve traditional methods. The proposal is concerned with the development of reliable and efficient computational algorithms for mathematical image restoration, their applications to real-life images, and the construction of software packages. The research emphasis will be on the computational aspects, trying to achieve truly practical algorithms based on the PDE models. Such reliable, efficient, practical algorithms will be implemented for software packages, which will be applicable for critically important problems including medical imaging, security control, crime scene investigation, and environmental watch. The proposed approaches and results can surely deliver benefits not only to research and education in academia but also to practitioners' image processing in industry. In particular, the software packages will provide convenient sources that are easy to maintain and modify for various applications. The investigators propose to explore mathematical frameworks and software engineering techniques to produce reliable and efficient numerical algorithms and related software packages which can support not only research-and-development but also classroom situations.

Proposal: DMS-0312223PI: Seongjai Kim [[e-mail]]Institution: University of KentuckyTitle: ITR: PDE-based Image Restoration: Efficient Numerical Algorithms and Software EngineeringABSTRACTAs the field of image processing (IP) requires higher levels of reliability and efficiency, mathematical IP has become an important component. 特别是，已经对使用最新的部分微分方程工具（PDE）的数学框架进行了广泛的研究，以回答IP中的基本问题。 这种基于PDE的方法事实证明，研究人员不仅可以引入创新的数学模型，还可以分析和改善传统算法大多数是通过启发式发展的。 为PDE模型开发适当的数值技术是基于PDE的方法的另一个重要组成部分。 该提案涉及开发基于PDE的图像恢复的数值算法及其在具有挑战性问题上的应用。 要解决的模型包括代表运动曲率和拉普拉斯平均曲率流动，p谐波图和基于曲率的总差异最小化的非线性PDE。 主要目标是（a）为图像恢复开发可靠，高效的数值算法，（b）将这些算法应用于降噪，图像增强和对现实生活中的图像（例如医学图像和卫星图像）以及（c）构造相关软件包的现实图像。 新开发的数值算法有望对数学图像分析产生巨大影响。由此产生的软件包必须适用于与图像相关的各种有趣的问题。现代数字时代的许多应用程序基于图像，因此所得的成就必须依赖于其质量。 由于由于各种类型的噪声，例如自然噪声，传感器中的缺陷以及传输问题，图像并不总是质量良好，因此自动消除噪声很重要。 从历史上看，这种图像恢复是最古老的问题之一，并且仍然是必要的处理步骤。 由于该领域在过去二十年中需要更高的可靠性和效率水平，因此数学（基于PDE）的图像恢复已成为重要组成部分。已经对它进行了广泛的研究，以回答图像处理中的基本问题并分析/改善传统方法。 该提案涉及开发用于数学图像恢复的可靠，高效的计算算法，其应用于现实生活图像以及软件包的构建。 研究重点将放在计算方面，试图基于PDE模型实现真正实用的算法。 将针对软件包实施这种可靠，高效，实用的算法，该算法将适用于非常重要的问题，包括医学成像，安全控制，犯罪现场调查和环境观察。 拟议的方法和结果肯定可以为学术界的研究和教育带来好处，而且还可以为行业中的从业者的形象处理带来好处。 特别是，软件包将提供方便的来源，这些来源易于维护和修改各种应用程序。调查人员建议探索数学框架和软件工程技术，以生成可靠，高效的数值算法和相关软件包，不仅可以支持研究和发展，还可以支持课堂情况。