Abstract
Let n, m be positive integers; we consider m×n real linear systems. We define regularized solutions of a linear system as the minimizers of an optimization problem. The objective function of this optimization problem can be seen as the Tikhonov functional when the p-norm is considered instead of the Euclidean norm. The cases p=1 and p=∞ are studied. This analysis is used to restore defocused synthetic images and real images with encouraging results.
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References
Barrodale, I., and Roberts, F. D. K., An Improved Algorithm for Discrete l1-Linear Approximation, SIAM Journal on Numerical Analysis, Vol. 10, pp. 839–848, 1973.
BjÖrck, Å., Numerical Methods for Least-Squares Problems, SIAM, Philadelphia, Pennsylvania, 1996.
Levy, S., and Fullagar, P. K., Reconstruction of a Sparse Spike Train from a Portion of Its Spectrum and Application to High-Resolution Deconvolution, Geophysics, Vol. 46, pp. 1235–1243, 1981.
Dax, A., On Regularized Least-Norm Problems, SIAM Journal on Optimization, Vol. 2, pp. 602–618, 1992.
Dax, A., A Row Relaxation Method for Large l1-Problems, Linear Algebra and Its Applications, Vol. 156, pp. 793–818, 1991.
Santosa, F., and Symes, W. W., Linear Inversion of Band-Limited Reflection Seismograms, SIAM Journal on Scientific and Statistical Computing, Vol. 7, pp. 1307–1330, 1986.
Alliney, S., A Property of the Minimum Vectors of a Regularizing Functional Defined by Means of the Absolute Norm, IEEE Transactions on Signal Processing, Vol. 45, pp. 913–917, 1997.
Taylor, H. L., Banks, S. C., and McCoy, J. F., Deconvolution with the l1-Norm, Geophysics, Vol. 44, pp. 39–52, 1979.
Claerbout, J. F., and Muir, F., Robust Modeling with Erratic Data, Geophysics, Vol. 38, pp. 826–844, 1973.
Ellis, R. G., Farquharson, C. G., and Oldenburg, D. W., Approximate Inverse Mapping of the COPROD2 Data, Journal of Geomagnetism and Geoelectricity, Vol. 45, pp. 1001–1012, 1993.
NÜrnberger, G., Approximation by Spline Functions, Springer Verlag, Berlin, Germany, 1989.
Osher, S., and Rudin, L. I., Feature-Oriented Image Enhancement Using Shock Filters, SIAM Journal on Numerical Analysis, Vol. 27, pp. 919–940, 1990.
Dobson, D. C., and Santosa, F., Recovery of Blocky Images from Noisy and Blurred Data, SIAM Journal on Applied Mathematics, Vol. 56, pp. 1181–1198, 1996.
Cohen, L. D., Auxiliary Variables and Two-Step Iterative Algorithms in Computer Vision Problems, Journal of Mathematical Imaging and Vision, Vol. 6, pp. 59–83, 1996.
Mumford, D., and Shah, J., Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems, Communications on Pure and Applied Mathematics, Vol. 42, pp. 577–685, 1989.
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Aluffi-Pentini, F., Castrignanò, T., Maponi, P. et al. Generalized Solution of Linear Systems and Image Restoration. Journal of Optimization Theory and Applications 103, 45–64 (1999). https://doi.org/10.1023/A:1021717215386
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DOI: https://doi.org/10.1023/A:1021717215386