Block-iterative algorithms

International Transactions in Operational Research

Volumn 16, Issue 4, 2009, Pages 427-463

  Byrne, Charles ^a  

^a University of Massachusetts Lowell   (United States)

Author keywords

Image; Optimization; Reconstruction

Indexed keywords

EID: 70350673979     PISSN: 09696016     EISSN: 14753995     Source Type: Journal    
DOI: 10.1111/j.1475-3995.2008.00683.x     Document Type: Article

References (50)

1. Agmon, S., 1954. The relaxation method for linear inequalities. Canadian Journal of Mathematics 6, 382–392.
2. Baillon, J.-B., Haddad, G., 1977. Quelques propriétés des operateurs angle-bornes et n-cycliquement monotones. Israel Journal of Mathematics 26, 137–150.
3. Bauschke, H., Borwein, J., 1996. On projection algorithms for solving convex feasibility problems. SIAM Review 38, 3, 367–426.
4. Bauschke, H., Borwein, J., Lewis, A., 1997. The method of cyclic projections for closed convex sets in Hilbert space. Contemporary Mathematics: Recent Developments in Optimization Theory and Nonlinear Analysis, American Mathematical Society 204, 1–38.
5. Bertsekas, D.P., 1997. A new class of incremental gradient methods for least squares problems. SIAM Journal on Optimization 7, 913–926.
6. Bregman, L.M., 1967. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Mathematical Physics 7, 200–217.
7. Browne, J., De Pierro, A., 1996. A row-action alternative to the EM algorithm for maximizing likelihoods in emission tomography. IEEE Transactions on Medical Imaging 15, 687–699.
8. Byrne, C., 1993. Iterative image reconstruction algorithms based on cross-entropy minimization. IEEE Transactions on Image Processing IP-2, 96–103.
9. Byrne, C., 1996. Block-iterative methods for image reconstruction from projections. IEEE Transactions on Image Processing IP-5, 792–794.
10. Byrne, C., 1997. Convergent block-iterative algorithms for image reconstruction from inconsistent data. IEEE Transactions on Image Processing IP-6, 1296–1304.
11. Byrne, C., 1998. Iterative algorithms for deblurring and deconvolution with constraints. Inverse Problems 14, 1455–1467.
12. Byrne, C., 2001. Bregman-Legendre multidistance projection algorithms for convex feasibility and optimization. In Butnariu, D., Censor, Y., Reich, S. (eds) Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, Studies in Computational Mathematics, Vol. 8. Elsevier Publishers, Amsterdam, 87–100.
13. Byrne, C., 2002. Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Problems 18, 441–453.
14. Byrne, C., 2004. A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 20, 103–120.
15. Byrne, C., 2005a. Choosing parameters in block-iterative or ordered-subset reconstruction algorithms. IEEE Transactions on Image Processing 14, 3, 321–327.
16. Byrne, C., 2005b. Signal Processing: A Mathematical Approach. AK Peters Publishers, Wellesley, MA.
17. Byrne, C., 2007. Applied Iterative Methods. AK Peters Publishers, Wellesley, MA.
18. Byrne, C., 2008. Sequential unconstrained minimization algorithms for constrained optimization. Inverse Problems 24, 1–27
19. Byrne, C., Censor, Y., 2001. Proximity function minimization using multiple Bregman projections, with applications to split feasibility and Kullback–Leibler distance minimization. Annals of Operations Research 105, 77–98.
20. Censor, Y., Segman, J., 1987. On block-iterative maximization. Journal of Information and Optimization Sciences 8, 275–291.
21. Censor, Y., Zenios, S.A., 1992. Proximal minimization algorithm with D-functions. Journal of Optimization Theory and Applications 73, 3, 451–464.
22. Censor, Y., Zenios, S.A., 1997. Parallel Optimization: Theory, Algorithms and Applications. Oxford University Press, New York.
23. Cimmino, G., 1938. Calcolo approssimato per soluzioni dei sistemi di equazioni lineari. La Ricerca Scientifica XVI, Series II, Anno IX 1, 326–333.
24. Combettes, P., 2001. Quasi-Fejérian Analysis of some optimization algorithms. In Butnariu, D., Censor, Y., Reich, S. (eds) Inherently Parallel Algorithms in Feasibility and Optimization and their Applications. Studies in Computational Mathematics. Elsevier Publishers, Amsterdam, 115–152.
25. Combettes, P., Wajs, V., 2005. Signal recovery by proximal forward-backward splitting. Multiscale Modeling and Simulation 4, 4, 1168–1200.
26. Darroch, J., Ratcliff, D., 1972. Generalized iterative scaling for log-linear models. Annals of Mathematical Statistics 43, 1470–1480.
27. Dax, A., 1990. The convergence of linear stationary iterative processes for solving singular unstructured systems of linear equations. SIAM Review 32, 611–635.
28. De Pierro, A., Iusem, A., 1990. On the asymptotic behavior of some alternate smoothing series expansion iterative methods. Linear Algebra and its Applications 130, 3–24.
29. Eggermont, P.P.B., Herman, G.T., Lent, A., 1981. Iterative algorithms for large partitioned linear systems, with applications to image reconstruction. Linear Algebra and its Applications 40, 37–67.
30. Fiacco, A., McCormick, G., 1990. Nonlinear Programming: Sequential Unconstrained Minimization Techniques. SIAM Classics in Mathematics (reissue). Society of Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
31. Golshtein, E., Tretyakov, N., 1996. Modified Lagrangians and Monotone Maps in Optimization. John Wiley and Sons Inc., New York.
32. Gordon, R., Bender, R., Herman, G.T., 1970. Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography. Journal of Theoretical Biology 29, 471–481.
33. Gubin, L.G., Polyak, B.T., Raik, E.V., 1967. The method of projections for finding the common point of convex sets. USSR Computational Mathematics and Mathematical Physics 7, 1–24.
34. Herman, G.T., Meyer, L., 1993. Algebraic reconstruction techniques can be made computationally efficient. IEEE Transactions on Medical Imaging 12, 600–609.
35. Holte, S., Schmidlin, P., Linden, A., Rosenqvist, G., Eriksson, L., 1990. Iterative image reconstruction for positron emission tomography: a study of convergence and quantitation problems. IEEE Transactions on Nuclear Science 37, 629–635.
36. Hudson, H.M., Larkin, R.S., 1994. Accelerated image reconstruction using ordered subsets of projection data. IEEE Transactions on Medical Imaging 13, 601–609.
37. Hudson, M., Hutton, B., Larkin, R., 1992. Accelerated EM reconstruction using ordered subsets. Journal of Nuclear Medicine 33. p. 960.
38. Kullback, S., Leibler, R., 1951. On information and sufficiency. Annals of Mathematical Statistics 22, 79–86.
39. Lange, K., Bahn, M., Little, R., 1987. A theoretical study of some maximum likelihood algorithms for emission and transmission tomography. IEEE Transactions on Medical Imaging MI-6, 2, 106–114.
40. Lange, K., Carson, R., 1984. EM reconstruction algorithms for emission and transmission tomography. Journal of Computer Assisted Tomography 8, 306–316.
41. Mann, W., 1953. Mean value methods in iteration. Proceedings of the American Mathmatical Society 4, 506–510.
42. Motzkin, T., Schoenberg, I., 1954. The relaxation method for linear inequalities. Canadian Journal of Mathematics 6, 393–404.
43. Narayanan, M., Byrne, C., King, M., 2001. An interior point iterative maximum-likelihood reconstruction algorithm incorporating upper and lower bounds with application to SPECT transmission imaging. IEEE Transactions on Medical Imaging TMI-20, 4, 342–353.
44. Schmidlin, P., 1972. Iterative separation of sections in tomographic scintigrams. Nuclear Medicine 11, 1, 1–6.
45. Shepp, L., Vardi, Y., 1982. Maximum likelihood reconstruction for emission tomography. IEEE Transactions on Medical Imaging MI-1, 113–122.
46. Shieh, M., Byrne, C., Testorf, M., Fiddy, M., 2006. Iterative image reconstruction using prior knowledge. Journal of the Optical Society of America A 23, 6, 1292–1300.
47. Tanabe, K., 1971. Projection method for solving a singular system of linear equations and its applications. Numerical Mathematics 17, 203–214.
48. Teboulle, M., 1992. Entropic proximal mappings with applications to nonlinear programming. Mathematics of Operations Research 17, 3, 670–690.
49. Vardi, Y., Shepp, L.A., Kaufman, L., 1985. A statistical model for positron emission tomography. Journal of the American Statistical Association 80, 8–20.
50. Wernick, M., Aarsvold, J. (eds) 2004. Emission Tomography: The Fundamentals of PET and SPECT. Elsevier Academic Press, San Diego.