Linear Algebra and Optimization in Image Processing

This work, joint with various experts in image processing, involves the adaptation of techniques in linear algebra and optimization. Projects included using the singular value decomposition for classifying images [J9], applying function minimization methods to noise smoothing and edge reinforcement [J13] [J14], using multi-level iterative methods for function minimization [J15], and analyzing convergence of iterations used in image processing [J17]. An efficient algorithm for image compression was also developed, making use of linear algebra and discrete optimization techniques [J16], and several algorithms were studied for approximating two dimensional convolution operators by a product of convolutions with smaller support [J27].

Recent work has focussed on the solution of the ill-posed problems arising in deblurring. Various optimization criteria have been evaluated [C13], and J. G. Nagy and I have developed algorithms that are efficient when the point spread function (the blurring function) is spatially variant, as in the Hubble Space Telescope [C18],[J45]. We have also worked on computing and displaying confidence intervals for the reconstructed images [J61]. Armin Pruessner and I studied blind deconvolution, in which the blurring function as well as the true image is to be determined [J63],and the structure of the blurring matrix was exploited in [J67] and [J74].

An application to Ladar images was made in [J66].

Some of this work is summarized in a monograph [B1], written at the level of an advanced undergraduate or beginning graduate student, designed to motivate mathematics and computer science students to learn about computational methods.

[B1]

Per Christian Hansen, James G. Nagy, and Dianne P. O'Leary, Deblurring Images: Matrices, Spectra, and Filtering, SIAM Press, Philadelphia, 2006.

[C13]

Dianne P. O'Leary, ``Regularization of Ill-Posed Problems in Image Restoration," Proceedings of the Fifth SIAM Conference on Applied Linear Algebra, J.G. Lewis, ed., SIAM Press, Philadelphia, 1994, 102-105. [C18]] James G. Nagy and Dianne P. O'Leary, "Fast Iterative Image Restoration with a Spatially Varying PSF," in Advanced Signal Processing Algorithms, Architectures, and Implementations VII F. T. Luk, ed., SPIE, 1997, 388-399.

[J9]

Timothy J. O'Leary, Dianne P. O'Leary, Mary C. Habbersett, and Chester J. Herman, ``Classification of gynecologic flow cytometry data: a comparison of methods,'' J. of Analytical and Quantitative Cytology 3 (1981) 135-142.

[J13]

K. A. Narayanan, Dianne P. O'Leary, and Azriel Rosenfeld, ``Image smoothing and segmentation by cost minimization,'' IEEE Transactions on Systems, Man, and Cybernetics SMC-12 (1982) 91-96.

[J14]

K. A. Narayanan, Dianne P. O'Leary, and Azriel Rosenfeld, ``An optimization approach to edge reinforcement,'' IEEE Transactions on Systems, Man, and Cybernetics SMC-12 (1982) 551-553.

[J15]

K. A. Narayanan, Dianne P. O'Leary, and Azriel Rosenfeld, ``Multi-resolution relaxation,'' Pattern Recognition 16 (1983) 223-230.

[J16]

Dianne P. O'Leary and Shmuel Peleg, ``Digital image compression by outer product expansion,'' IEEE Transactions on Communications COM-31 (1983) 441-444.

[J17]

Dianne P. O'Leary and Shmuel Peleg, ``Analysis of relaxation processes: the two node, two label case,'' IEEE Transactions on Systems, Man, and Cybernetics SMC-13 (1983) 618-623.

[J27]

Dianne P. O'Leary, ``Some algorithms for approximating convolutions,'' Computer Vision, Graphics, and Image Processing 41 (1988) 333-345.

[J45]

James G. Nagy and Dianne P. O'Leary, ``Restoring Images Degraded by Spatially-Variant Blur," SIAM Journal on Scientific Computing, 19 (1998), 1063-1082.

[J61]

James G. Nagy and Dianne P. O'Leary, ``Image Restoration through Subimages and Confidence Images," Electronic Transactions on Numerical Analysis, 13 (2002) 22-37.

[J63]

Armin Pruessner and Dianne P. O'Leary, ``Blind Deconvolution Using a Regularized Structured Total Least Norm Approach," SIAM J. on Matrix Analysis and Applications, 24 (2003) 1018-1037.

[J66]

David E. Gilsinn, Geraldine S. Cheok, and Dianne P. O'Leary, ``Reconstructing Images of Bar Codes for Construction Site Object Recognition," Automation in Construction (Elsevier), 13 (2004) 21-35.

[J67]

Nicola Mastronardi, Phillip Lemmerling, Anoop Kalsi, Dianne O'Leary, and Sabine Van Huffel, ``Regularized structured total least squares algorithms for blind image deblurring," Linear Algebra and Its Applications, 391 (2004) 203-221

[J74]

Anoop Kalsi and Dianne P. O'Leary, ``Algorithms for Structured Total Least Squares Problems with Applications to Blind Image Deblurring," Journal of Research of the National Institute of Standards and Technology, 111, No. 2 (2006) pp. 113-119.