Fast, robust total variation-based reconstruction of noisy, blurred images

IEEE Transactions on Image Processing

Volumn 7, Issue 6, 1998, Pages 813-824

  Vogel, Curtis R ^a   Oman, Mary E ^b  

^a MONTANA STATE UNIVERSITY   (United States)

^b IOWA STATE UNIVERSITY   (United States)

Author keywords

Cell centered finite differences; Conjugate gradient; Deconvolution; Fixed point iteration; Image reconstruction; Preconditioner; Regularization; Total variation

Indexed keywords

ALGORITHMS; FINITE DIFFERENCE METHOD; IMAGE ANALYSIS; IMAGE QUALITY; ITERATIVE METHODS;

CELL CENTERED FINITE DIFFERENCES; PRECONDITIONED CONJUGATE GRADIENT ITERATION;

IMAGE RECONSTRUCTION;

EID: 0032096686     PISSN: 10577149     EISSN: None     Source Type: Journal    
DOI: 10.1109/83.679423     Document Type: Article

References (34)

1. R. Acar and C. R. Vogel, "Analysis of total variation penalty methods," Inv. Prob., vol. 10, pp. 1217-1229, 1994.
2. O. Axelsson, Iterative Solution Methods. Cambridge, U.K.: Cambridge Univ. Press, 1994.
3. O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems. New York: Academic, 1984.
4. C. T. H. Baker, L. Fox, D. F. Meyer, and K. Wright, "Numerical solution of Fredholm integral equations of the first kind," Comput. J., vol. 7, pp. 141-148, 1964.
5. R. H. Chan, M. K. Ng, and R. J. Plemmons, "Generalization of Strang's preconditioner with applications to Toeplitz least squares problems," Numer. Lin. Alg. Applic., vol. 3, pp. 45-64, 1996.
6. R. H. Chan and M. K. Ng, "Conjugate gradient method for Toeplitz systems," SIAM Rev., vol. 38, pp. 427-482, Sept. 1996.
7. T. F. Chan, G. H. Golub, and P. Mulet, "A nonlinear primal-dual method for TV-based image restoration," in Proc. ICAOS'96, 12th Int. Conf. Analysis and Optimization of Systems: Images, Wavelets, and PDE's, Pans, France, June 26-28, 1996, pp. 241-252.
8. T. F. Chan, H. M. Zhou, and R. H. Chan, "Continuation method for total variation denoising problems," in Proc. SPIE 1995, Advanced Signal Processing Algorithms, F. T. Luk, Ed., San Diego, CA, vol. 2563.
9. A. Chambolle and P. L. Lions, "Image recovery via total variation minimization and related problems," Numer. Math., vol. 72, pp. 167-188, 1997.
10. P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, "Deterministic edge-preserving regularization in computed imaging," IEEE Trans. Image Proc., vol. 6, pp. 298-311, Feb. 1997.
11. J. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Englewood Cliffs, NJ: Prentice-Hall, 1983.
12. D. Dobson and F. Santosa, "An image enhancement technique for electrical impedance tomography," Inv. Prob., vol. 10, pp. 317-334, 1994.
13. D. Dobson and F. Santosa, "Recovery of blocky images from noisy and blurred data," SIAM J. Appl. Math, vol. 56, pp. 1181-1198, Aug. 1996.
14. D. Dobson and C. R. Vogel, "Convergence of an iterative method for total variation denoising," to appear in SIAM J. Numer. Anal., available from http://www.math.montana.edu/∼vogel.
15. A. Erhardt, G. Zinser, D. Komitowski, and J. Billie, "Reconstructing 3-d light-microscopic images by digital image processing," Appl. Opt., vol. 24, pp. 194-200, 1985.
16. R. E. Ewing and J. Shen, "A multigrid algorithm for the cell-centered finite difference scheme," in Proc. 6th Copper Mountain Conf. Multigrid Methods, NASA Conf. Pub. 3224, Apr. 1993.
17. D. Geman and G. Reynolds, "Constrained image restoration and the recovery of discontinuities," IEEE Trans. Pattern Anal. Machine Intell., vol. 14, pp. 367-383, 1992.
18. D. Geman and C. Yang, "Nonlinear image recovery with half-quadratic regularization," IEEE Trans. Image Processing, vol. 4, pp. 932-945, 1995.
19. E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Basel, Switzerland: Birkhäuser, 1984.
20. G. H. Golub and C. V. Loan, Matrix Computations, 2nd ed. Baltimore, MD: Johns Hopkins Univ. Press, 1989.
21. C. W. Groetsch, Inverse Problems in the Mathematical Sciences. Wiesbaden, Germany: Vieweg, 1993.
22. M. Hanke and P. C. Hansen, "Regularization methods for large-scale problems," Surv. Math. Ind., vol. 3, pp. 253-315, 1993.
23. M. Hanke and J. G. Nagy, "Restoration of atmospherically blurred images by symmetric indefinite conjugate gradient techniques," Inv. Prob., vol. 12, pp. 157-173, 1996.
24. K. Ito and K. Kunisch, "An active set strategy for image restoration based on the augmented Lagrangian formulation," Ctr. Res. Scient. Computat., North Carolina State Univ., 1994, preprint.
25. Y. Li and F. Santosa, "A computational algorithm for minimizing total variation in image restoration," IEEE Trans. Image Processing, vol. 5, pp. 987-995, 1996.
26. J. G. Nagy, R. J. Plemmons, and T. C. Torgersen, "Iterative image restoration using approximate inverse preconditioners," IEEE Trans. Image Processing, vol. 5, pp. 1151-1162, July 1996.
27. S. G. Nash and A. Sofer, Linear and Nonlinear Programming. New York: McGraw-Hill, 1996.
28. M. E. Oman, "Fast multigrid techniques in total variation based image reconstruction," in Proc. Copper Mountain Conf. Multigrid Methods, NASA, 1995.
29. L. I. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D, vol. 60, pp. 259-268, 1992.
30. A. N. Tikhonov, "Regularization of incorrectly posed problems," Soviet Math. Dokl., vol. 4, pp. 1624-1627, 1963.
31. C. R. Vogel, "A multigrid method for total variation based image denoising," in Progress in Systems and Control Theory: Computation and Control IV, K. L. Bowers and J. Lund, Eds. Basel, Switzerland: Birkhäuser, 1995.
32. C. R. Vogel and M. E. Oman, "Iterative methods for total variation denoising," SIAM J. Sci. Comput., vol. 17, no. 1, pp. 227-238, Jan. 1996.
33. C. R. Vogel and M. E. Oman, "Fast numerical methods for total variation minimization in image reconstruction," in Proc. SPIE 1995 Advanced Signal Processing Algorithms, San Diego, CA, vol. 2563.
34. A. Weiser and M. F. Wheeler, "On convergence of block-centered finite differences for elliptic problems," SIAM J. Numer. Anal., vol. 25, pp. 351-375, 1988.