Multiscale Total Variation Methods for Integral Equation Models in Image Processing

目前在图像处理中使用的模型是离散的。它们是真实模型——积分方程模型的分段常数近似。离散模型带来的瓶颈模型误差无法通过基于离散模型的数值方法得到补偿。为了克服这一缺点，pi建议直接使用积分方程模型进行图像处理。积分方程模型将为我们对相应图像的深入分析提供更大的灵活性。理想的图像处理方法应该对图像的几何特征敏感，计算效率高。目前，全变分法和多尺度法是图像处理的两种主要数学方法。他们的长处和短处是相辅相成的。全变分法对图像的几何特征敏感，但计算效率低。标准多尺度方法由于其多尺度结构，便于计算，但对图像的几何特征不太敏感。为了设计对图像几何特征敏感的计算效率高的算法，pi提出了将这两种方法的优点结合起来的多尺度全变分方法。pi研究了以下四个与图像处理相关的数学问题：(1)基于积分方程模型的图像多尺度逼近；(2)多尺度总变差正则化；(3)采用冗余系统恢复丢失数据；(4)应用驱动的小波和小框架滤波器组设计。图像处理出现在各种科学、医学和工程应用中。具体来说，医学科学和技术的应用范围从计算机断层扫描到疾病诊断，环境科学的应用包括通过卫星成像进行自然资源和污染控制，艺术科学的应用包括视觉分析和数字化美术博物馆中破碎的古代绘画的数字修复，安全识别的应用包括武器，指纹和面部识别。在这些应用程序中，一个关键问题是从可用数据中恢复图像。这是一个不适定的问题。解决这一问题需要先进的数学模型和高效的计算算法。本文的主要目的是通过提出图像处理中积分方程模型的多尺度全变分方法来直接解决这一问题。提案中的项目将促进高水平纯数学与当代数字和计算机技术的结合。这些项目将培养这一重要领域的研究生，使他们准备好面对未来科学技术发展中的数学和计算挑战。此外，pi将根据这些项目的研究成果，为高年级本科生开发多学科课程。

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.