Abstract
This paper is concerned with improvement in optical image quality by image restoration. Image restoration is an ill-posed inverse problem which involves the removal or minimization of degradations caused by noise and blur in an image, resulting from, in this case, imaging through a medium. Our work here concerns the use of the underlying Toeplitz structure of such problems, and associated techniques for accelerating the convergence of iterative image restoration computations. Denoising methods, including total variation minimization, followed by segmentation-based preconditioning methods for minimum residual conjugate gradient iterations, are investigated. Regularization is accomplished by segmenting the image into (smooth) segments and varying the preconditioners across the segments. By taking advantage of the Toeplitz structure, our algorithms can be implemented with computational complexity of onlyO (ln 2 logn), wheren 2 is the number of pixels in the image andl is the number of segments used. Also, parallelization is straightforward. Numerical tests are reported for atmospheric imaging problems, including the case of spatially varying blur.
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References
R. Acard andC. Vogel,Analysis of Bounded Variation Penalty Methods, Inverse Problems10, (1994) 1217–1229.
O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, 1994.
A. Björck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA, 1996.
J. Brown,Overview of Topical Issues on Inverse Problems in Astronomy, Inverse Problems11, (1995) 635–638.
K. Castleman, Digital Image Processing, Prentice-Hall, NJ, 1996.
P. J. Davis, Circulant Matrices, Wiley, New York, 1979.
D. Dobson andF. Santosa,Recovery of Blocky Images from Noisy and Blurred Data, SIAM. J. Appl. Math.56, (1996) 1181–1198.
G. Golub andC. Van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, MD, 2nd Edition, 1989.
M. Hanke, Conjugate Gradient Type Methods for Ill-posed Problems, Pitman Research Notes in Mathematics, Longman Scientific & Technical, Harlow, Essex, 1995.
M. Hanke andP. Hansen,Regularization Methods for Large-scale Problems, Surveys Math. Indust.3, (1993) 253–315.
M. Hanke andJ. Nagy,Restoration of Atmospherically Blurred Images by Symmetric Indefinite Conjugate Gradient Techniques, Inverse Problems12, (1996) 157–173.
M. Hanke, J. Nagy and R. Plemmons,Preconditioned Iterative Regularization for Ill-posed Problems, Numerical Linear Algebra and Scientific Computing, Eds. L. Reichel, A. Ruttan, R. Varga, Walter de Gruyter Press, Berlin 141–163.
P. Hansen andD. P. O’Leary,The use of the L-Curve in the Regularization of Discrete Ill-posed Problems, SIAM J. Sci. Comp.14, (1993), 1487–1503.
J. W. Hardy,Adaptive-Optics, Scientific American270, no. 6 (1994) 60–65.
R. Lagendijk andJ. Biedmond, Iterative Identification and Restoration of Images, Kluwer Press, Boston, 1991.
J. Nagy and D. P. O’Leary,Restoring Images Degraded by Space Variant Blur, Preprint, February 1995, to appear in SIAM J. Sci. Comput.
J. Nagy, R. Plemmons andT. Torgersen,Iterative Image Restoration using Approximate Inverse Preconditioning, IEEE Trans. on Image Processing5, (1996) 1151–1162.
S. Osher andL. Rudin,Feature-oriented Image Enhancement Using Shock Filters, SIAM J. Numer. Anal.27, (1990) 919–940.
M. Paige andM. A. Saunders,Solution of Sparse Indefinite Systems of Linear Equations, SIAM J. Numer. Anal.12 (1975) 617–629.
S. Reaves,Optimal Space-Varying Regularization in Iterative Image Restoration, IEEE Trans. on Image Processing3, (1994) 319–324.
L. Rudin andS. Osher,Total Variation Based Image Restoration with Free Local Constraints, Proc. IEEE International Conf. on Image Processing, II, (1994) 31–35.
L. Rudin, S. Osher andE. Fatemi,Nonlinear Total Variation Based Noise Removal Algorithms, Physica D.60, (1992) 259–268.
C. Vogel andM. Oman,Iterative Methods for Total Variation Denoising, SIAM J. Sci. Comput.17, (1996) 227–238.
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Research supported in part by a National Science Foundation Postdoctoral Research Fellowship.
Research sponsored by the U.S. Air Force Office of Scientific Research under grant F49620-97-1-1039.
Research sponsored by the U.S. Air Force Office of Scientific Research under grant F49620-97-1-0139, and by the National Science Foundation under grant CCR-96-23356.
Research sponsored by the National Science Foundation under grant CCR-96-23356.
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Nagy, J.G., Pauca, V.P., Plemmons, R.J. et al. Degradation reduction in optics imagery using Toeplitz structure. Calcolo 33, 269–288 (1996). https://doi.org/10.1007/BF02576005
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DOI: https://doi.org/10.1007/BF02576005