Multiscale Total Variation Methods for Integral Equation Models in Image Processing

图像处理中积分方程模型的多尺度全变分法

• 批准号: 0712827
• 负责人: Yuesheng Xu
• 金额: $ 35.89万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-15 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0712827&HistoricalAwards=false
• 关键词: Multiscale; Total; Variation; Methods; Integral

Models currently used in image processing are discrete. They are piecewise constant approximations of the true models - integral equation models. The discrete models impose bottleneck model errors which cannot be compensated from the numerical methods developed based upon them. To overcome this drawback, the PIs propose to directly use the integral equation models for image processing. The integral equation models will offer us much greater flexibility for the in-depth analysis of the corresponding images. An ideal method for image processing should be sensitive to geometric features of images and computationally efficient. Presently, the total variation method and the multiscale method are two main mathematical approaches for image processing. They are complementary to each other in their strength and weakness. The total variation method is sensitive to geometric features of images but it is computationally inefficient. The standard multiscale method is convenient for computation due to its multiscale structure but it is not very sensitive to the geometric features of images. Aiming at designing computationally efficient algorithms that are sensitive to geometric features of images, the PIs propose to develop multiscale total variation methods which combine the strengths from both of these two methods. The PIs study the following four mathematical problems related to image processing: (1) Multiscale approximation of images based on integral equation models; (2) Multiscale total variation regularization; (3) Missing data recovery with redundant systems; and (4) Design of application-driven wavelet and framelet filter banks. Image processing arises in a variety of scientific, medical and engineering applications. Specifically, applications in medical sciences and technologies range from computer tomography to diagnoses of diseases, applications in environmental sciences include natural resources and pollution control via satellite imaging, applications in art sciences have vision analysis and digital restorations of cracked ancient paintings in digitized fine art museums and applications in security identification include weapon, fingerprints and face identifications. In these applications, a key issue is restorating images from available data. This is an ill-posed problem. Solving this problem needs advanced mathematical models and efficient computational algorithms. The main objective of this proposal directly addresses this issue by proposing multiscale total variation methods for integral equation models in image processing. The projects in the proposal will enhance the integration of high level pure mathematics with the contemporary digital and computer technology. These projects will train graduate studetns in this important area to prepare them to face the mathematical and computational challenge in future scientifical and technological development. Moreover, th PIs will develop a multidisciplinary course for upper level undergraduate students based on research results of these projects.

图像处理中当前使用的模型是离散的。它们是真实模型的分段恒定近似值 - 积分方程模型。离散模型施加了瓶颈模型误差，这些误差无法从基于它们开发的数值方法中得到补偿。为了克服这一缺点，PI提议直接使用用于图像处理的积分方程模型。积分方程模型将为我们提供更大的灵活性，以对相应图像进行深入分析。图像处理的理想方法应该对图像的几何特征和计算效率敏感。目前，总变异方法和多尺度方法是图像处理的两种主要数学方法。他们的力量和劣势是彼此互补的。总变化方法对图像的几何特征敏感，但在计算上效率低下。由于其多尺度结构，标准的多尺度方法对于计算很方便，但对图像的几何特征并不十分敏感。旨在设计对图像几何特征敏感的计算有效算法，PIS提议开发多尺度的总变异方法，这些方法结合了这两种方法的强度。 PIS研究以下与图像处理有关的四个数学问题：（1）基于积分方程模型的图像的多尺度近似； （2）多尺度的总变化正规化； （3）缺少冗余系统的数据恢复； （4）设计由应用程序驱动的小波和帧过滤器库设计。 图像处理发生在各种科学，医学和工程应用中。具体而言，医学和技术的应用从计算机层析成像到诊断疾病，环境科学的应用包括自然资源和通过卫星成像进行污染控制，艺术科学中的应用具有视觉分析，并且在数字化美术博物馆中进行破解的古代绘画的数字修复以及在安全性美术博物馆中的应用，以及在安全标识中包括武器，武器，构成武器，构成武器，识别武器，并识别武器。 在这些应用程序中，关键问题是从可用数据中恢复图像。这是一个不适的问题。解决此问题需要高级数学模型和有效的计算算法。该提案的主要目的直接通过提出图像处理中积分方程模型的多尺度总变化方法来直接解决此问题。提案中的项目将增强高水平纯数学与当代数字和计算机技术的整合。这些项目将在这一重要领域培训研究生锻炼，以便他们准备面对未来科学和技术发展中的数学和计算挑战。此外，根据这些项目的研究结果，PIS将为上层本科生开发一门多学科课程。