## Dr. James Nagy

Emory University

Dept of Math & Computer Science

Suite W401, 400 Dowman Drive

Atlanta, GA 30322

Phone: Office: (404) 727-5601, Fax: (404) 727-5611

Email: nagy@mathcs.emory.edu

James Nagy received a BS degree in 1986 and an MS degree in 1988 from Northern Illinois University, and a PhD in 1991 from North Carolina State University. He was a Postdoctoral Research Associate at the Institute for Mathematics and Its Applications, University of Minnesota from 1991--1992, and a member of the faculty in the Department of Mathematics at Southern Methodist University from 1992--1999. Since 1999 he has been at Emory University, where he is a professor in the Department of Mathematics and Computer Science. His research interests include numerical linear algebra (especially large scale structured linear systems), inverse problems, and image processing.

**Linear Algebra and Image Processing**

Many problems in image processing require using tools from linear algebra. In this talk we show how to use linear algebra to perform basic operations on images that arise in computer graphics, image compression, image enhancement, and other imaging applications. Many examples and MATLAB demonstrations will be used to illustrate the main concepts.

Level: Elementary

**Deblurring Images: Matrices, Spectra, and Filtering**

When we use a camera, we want the recorded image to be a faithful representation of the scene that we see, but every image is more or less blurry. In image deblurring, the goal is to recover the original, sharp image by using a mathematical model of the blurring process. The key issue is that some information on the lost details is indeed present in the blurred image, but this ``hidden" information can be recovered only if we know the details of the blurring process. In this talk we describe the deblurring algorithms and techniques collectively known as spectral filtering methods, in which the singular value decomposition (or a similar decomposition with spectral properties) is used to introduce the necessary regularization or filtering in the reconstructed image.

Level: Intermediate

**Iterative Methods for Ill-Posed in Image Processing**** **

Ill-posed problems arise in many image processing applications, including microscopy, medicine and astronomy. Iterative methods are typically recommended for these large scale problems, but they can be difficult to use in practice. For example, it may be difficult to determine an appropriate stopping criteria for fast algorithms, such as the conjugate gradient method; noise contaminates the iterates very quickly, so an imprecise stopping criteria can lead to poor reconstructions. Lanczos based hybrid methods have been proposed to slow the introduction of noise in the iterates. These methods require choosing a regularization parameter for a small subproblem at each iteration, but implementation on realistic problems can be difficult.

Preconditioning to accelerate convergence can introduce additional instability into the algorithms. In this talk we discuss the behavior of iterative methods on ill-posed problems in image processing, and describe techniques to address the implementation difficulties for realistic problems. Examples are used to illustrate the concepts and to test and compare algorithms.

Level: Advanced