Mathematical Sciences: Collaborative Research on Iterative Methods for Image Restoration

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Calvetti The investigator and her colleague undertake collaborative studies of numerical methods for image processing. The aim of the project is to develop and analyze new fast algorithms for computing an estimate of an original image from a degraded image that has been subjected to noise and blur. Depending on the point of view, two philosophically different approaches to image restoration result: i) in algebraic restoration one views the problem as a linear, possibly ill-conditioned, system of equations, and ii) in stochastic restoration one regards it as a problem of estimating a vector under random disturbances. The project develops numerical methods for both approaches. Either approach yields linear systems of equations that have a structure that can be used in the development and analysis of efficient algorithms for image restoration. The structure depends on the assumptions made on the image and the noise. The matrices in these systems are typically quite large; orders of a million are common. The computational work, even with efficient methods, is therefore quite demanding. Because many applications require real-time image restoration, the development of algorithms for parallel computers is important. The proposed research focuses on the development of iterative methods that lend themselves well to implementation on multiprocessors. The central problem of image restoration is the estimation of the original image, given a version of the image degraded by noise and blur. When the image is of a star observed from earth, the blur may be due to the atmosphere and the noise can stem from transmission of the signal. Several different approaches to image restoration are available. They all give rise to large systems of equations; a million equations are common. The system of equations has a structure, which depends on the assumptions made on the image and on the statistical properties of the noise. The purpose of the project is to develop ne w algorithms for the restoration of images. These algorithms are designed so that they use the structure of the system of equations in order to reduce the computational work. They are moreover suitable for use on parallel computers. The latter is important due to the large amount of computations required.