Kronecker Products in Image Restoration

May 30, 2003

It's hard to imagine many fields---such as astronomy, geophysics, medicine, and microscopy---functioning without the use of images for analytic, diagnostic, and other purposes. Unfortunately, there's no such thing as a perfect imaging device. As a result, pictures of acceptable quality can often be obtained only with postprocessing techniques. Certain uncontrollable conditions, for example, can lead to an image (of a block of text) no better than the one shown here.

The postprocessing technique of image restoration involves developing a mathematical model and efficient computational methods to describe and remove distortions in an image. A class of methods used for this purpose, called filtering, makes extensive use of tools from linear algebra. Though it is common to discuss Toeplitz and circulant matrices in this context, my aim in the talk I've been invited to give this summer at the SIAM Conference on Applied Linear Algebra is to describe how Kronecker products arise in image restoration, and to show how the Kronecker product structure can be exploited to obtain very efficient computational filtering methods. In addition to examples from various applications, the important message hidden in the above image will be revealed only to those attending the SIAM Conference on Applied Linear Algebra!---James G. Nagy, Emory University.


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