Statistical Analysis of Image Restoration and Its Applications in Magnetic Resonance Imaging

图像分析是一个具有广泛应用的跨学科研究领域。该项目的一个主要重点是当观察到的图像具有逐点噪声和空间模糊并且模糊机制未知时的图像去模糊，这通常被称为盲图像去模糊问题。在统计文献中，很少有现有的讨论可以找到图像去模糊。在应用数学和计算机科学文献中，现有的图像去模糊方法假设由点扩散函数描述的模糊机制或者是完全已知的，或者遵循具有一个或多个未知参数的参数模型，或者满足各种条件，其中一些条件在应用中过于限制而不能满足。这个计画提出了另一种方法来解决盲影像去模糊的问题，而不会对点扩散函数施加限制性的假设。其主要思想是从观测到的测试图像或具有简单图像结构的观测图像的部分估计点扩散函数，然后使用估计的点扩散函数恢复真实图像。如果测试图像或具有简单图像结构的观察图像的部分不可用，则研究者基于真实图像可以通过局部邻域中的简单结构图像很好地近似的观察，建议局部地使用上述想法。这种局部去模糊过程允许模糊机制随位置而变化。近年来，磁共振成像（MRI）被广泛用于显示活体组织的病理或其他生理变化。由于成像对象的运动等原因，所观察到的MRI图像通常包含各种污染，其中空间模糊和逐点噪声是最常见的。目前，人们通常使用现有的图像去模糊技术来逐切片地对成像对象的各个二维（2-D）切片的MRI图像进行去模糊。本课题将二维图像去模糊方法推广到三维图像，提出将不同层面的二维MRI图像作为三维图像的轮廓处理，直接对三维图像进行去模糊处理，这将大大提高MRI图像的恢复质量。本计画亦建议同时使用频域与空域的观测资料，并利用其优势，借由影响影像去模糊技术的进展，对社会产生更广泛的影响。拟议的去模糊程序应该大大提高恢复的MRI图像的质量，这应该有助于人们更好地了解人类和其他动物的活组织和神经活动，并更准确地诊断各种疾病。除了MRI和功能性MRI，图像去模糊在其他地方被广泛使用。例如，它用于手写体的机器识别，包括邮政地址、银行支票等的机器阅读。它还用于预处理卫星图像和医学、气象学、海洋学、军事、空间通信等领域的各种其他图像。该项目还将通过其教育活动促进科学和工程方面的人力资源开发。例如，研究员将提供关于图像分析的高级主题课程，来自各个部门的研究生将在科学研究的各个方面得到系统的培训。在本项目结束时将创建一个网页，包括所有计算机程序和研究结果，以便其他研究人员可以轻松下载和使用它们。博士学位研究员的学生目前正在写他的论文关于图像处理。这些教育活动对相关研究领域的普及和进一步发展具有重要影响。

Image analysis is an interdisciplinary research area with broad applications. One major focus of this project is on image deblurring when an observed image has both pointwise noise and spatial blur and when the blurring mechanism is unknown, which is often referred to as the blind image deblurring problem. In the statistical literature, little existing discussion can be found on image deblurring. In the applied mathematics and computer sciences literatures, existing image deblurring methods assume that the blurring mechanism, described by a point spread function, either is completely known, or follows a parametric model with one or more unknown parameters, or satisfies various conditions some of which are too restrictive to satisfy in applications. This project proposes alternative approaches to the blind image deblurring problem without imposing restrictive assumptions on the point spread function. The major idea is to estimate the point spread function from an observed test image or portions of an observed image with simple image structure, and then to restore true images using the estimated point spread function. If test images or portions of an observed image with simple image structure are not available, then the investigator suggests using the above idea locally, based on the observation that a true image can be well approximated by a simple structured one in a local neighborhood. Such local deblurring procedures allow the blurring mechanism to vary over location. In recent years, magnetic resonance imaging (MRI) is widely used for demonstrating pathological or other physiological alterations of living tissues. Due to movement of the imaged object and many other reasons, observed MRI images usually contain various contaminations, among which spatial blur and pointwise noise are most common. Right now, people usually use existing image deblurring techniques to deblur MRI images of individual 2-dimensional (2-D) slices of the imaged object slice by slice. This project suggests treating 2-D MRI images of different slices as profiles of a 3-D image and deblurring the 3-D image directly, after generalizing the proposed 2-D deblurring methods to 3-D cases, which would greatly improve the quality of the restored MRI images. This project also suggests using observed data in both frequency domain and spatial domain, and making use of their major strengths.This project would have broader impacts on the society through its impacts on the progress of image deblurring techniques. Proposed deblurring procedures should greatly improve the quality of restored MRI images, which should help people better understand living tissues and neural activities of humans and other animals and diagnose various diseases more accurately. Besides MRI and functional MRI, image deblurring is used widely elsewhere. For instance, it is used in machine recognition of handwriting, including machine reading of postal addresses, bank checks, and so forth. It is also used in preprocessing satellite images and various other images in medical sciences, meteorology, oceanography, military, space communication, etc. Therefore, this project could have a positive impact in all these areas. This project would also contribute to the development of human resources in science and engineering through its educational activities. For instance, the investigator will offer an advanced topics course on image analysis, from which graduate students from various departments will get systematic training in all aspects of scientific research. A web page will be created at the end of this project to include all computer programs and research results so that other researchers can easily download and use them. A Ph.D. student of the investigator is currently writing his thesis on image processing. All these educational activities should have a great impact on the popularity and further development of the related research areas.