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

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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.