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

建议：DMS-0312223PI：Seongjai Kim[skim@ms.uky.edu]机构：肯塔基大学标题：ITR：基于PDE的图像恢复：高效数值算法和软件工程摘要由于图像处理(IP)领域需要更高水平的可靠性和效率，数学IP已成为一个重要的组成部分。特别是，利用最近功能强大的偏微分方程(PDE)工具的数学框架已被广泛研究以回答IP中的基本问题。事实证明，这种基于偏微分方程的方法不仅允许研究人员引入创新的数学模型，还允许他们分析和改进传统算法，这些算法大多是启发式开发的。为偏微分方程模型开发合适的数值技术是基于偏微分方程组的方法的另一个重要组成部分。该提案涉及基于偏微分方程的图像恢复的数值算法的发展及其在挑战性问题中的应用。要解决的模型包括用平均曲率和拉普拉斯平均曲率流表示运动的非线性偏微分方程组，p-调和映射，以及基于曲率的总变分最小化。主要目标是(A)开发可靠和有效的图像恢复的数值算法，(B)将这些算法应用于诸如医学图像和卫星图像等真实图像的去噪、图像增强和修复，以及(C)构建相关的软件包。新开发的数值算法有望对数学图像分析产生重大影响；由此产生的软件包必须适用于各种处理图像的有趣问题。现代数字时代的许多应用程序都是基于图像的，因此所取得的成就必须依赖于它们的质量。由于各种类型的噪声，例如自然噪声、传感器缺陷和传输问题，图像质量并不总是很好，因此自动消除噪声是很重要的。从历史上看，这样的图像恢复是最古老的问题之一，仍然是一个必要的处理步骤。在过去的二十年里，由于该领域对可靠性和效率的要求越来越高，基于偏微分方程的数学图像恢复已经成为一个重要的组成部分；它被广泛地研究以回答图像处理中的基本问题，并分析/改进传统的方法。该提案涉及数学图像恢复的可靠和有效的计算算法的开发，它们在真实图像中的应用，以及软件包的构建。研究的重点将放在计算方面，试图实现基于PDE模型的真正实用的算法。这些可靠、高效、实用的算法将被实现在软件包中，这些算法将适用于医学成像、安全控制、犯罪现场调查和环境监测等关键问题。所提出的方法和结果不仅可以为学术界的研究和教育带来好处，也可以为实践者的图像处理带来好处。特别是，这些软件包将为各种应用程序提供易于维护和修改的方便来源。研究人员建议探索数学框架和软件工程技术，以产生可靠和有效的数值算法和相关软件包，这些算法和相关软件包不仅可以支持研究和开发，而且可以支持课堂情况。

Proposal: DMS-0312223PI: Seongjai Kim [skim@ms.uky.edu]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.