Block-iterative algorithms
The recently presented sequential unconstrained minimization algorithm, SUMMA, is extended to provide a framework for the derivation of block‐iterative, or partial‐gradient, optimization methods. This block‐iterative SUMMA (BI‐SUMMA) includes, and is motivated by, block‐iterative versions of the alg...
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| Vydané v: | International transactions in operational research Ročník 16; číslo 4; s. 427 - 463 |
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| Médium: | Journal Article |
| Jazyk: | English |
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Oxford, UK
Blackwell Publishing Ltd
01.07.2009
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| ISSN: | 0969-6016, 1475-3995 |
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| Abstract | The recently presented sequential unconstrained minimization algorithm, SUMMA, is extended to provide a framework for the derivation of block‐iterative, or partial‐gradient, optimization methods. This block‐iterative SUMMA (BI‐SUMMA) includes, and is motivated by, block‐iterative versions of the algebraic reconstruction technique (ART) and its multiplicative variant, the MART. The BI‐SUMMA approach is designed to provide computationally tractable and quickly convergent algorithms. The rescaled block‐iterative expectation maximization maximum likelihood method (RBI‐EMML) is closely related to the RBI‐MART, but is not a particular case of BI‐SUMMA. |
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| AbstractList | The recently presented sequential unconstrained minimization algorithm, SUMMA, is extended to provide a framework for the derivation of block‐iterative, or partial‐gradient, optimization methods. This block‐iterative SUMMA (BI‐SUMMA) includes, and is motivated by, block‐iterative versions of the algebraic reconstruction technique (ART) and its multiplicative variant, the MART. The BI‐SUMMA approach is designed to provide computationally tractable and quickly convergent algorithms. The rescaled block‐iterative expectation maximization maximum likelihood method (RBI‐EMML) is closely related to the RBI‐MART, but is not a particular case of BI‐SUMMA. The recently presented sequential unconstrained minimization algorithm, SUMMA, is extended to provide a framework for the derivation of block-iterative, or partial-gradient, optimization methods. This block-iterative SUMMA (BI-SUMMA) includes, and is motivated by, block-iterative versions of the algebraic reconstruction technique (ART) and its multiplicative variant, the MART. The BI-SUMMA approach is designed to provide computationally tractable and quickly convergent algorithms. The rescaled block-iterative expectation maximization maximum likelihood method (RBI-EMML) is closely related to the RBI-MART, but is not a particular case of BI-SUMMA. [PUBLICATION ABSTRACT] |
| Author | Byrne, Charles |
| Author_xml | – sequence: 1 givenname: Charles surname: Byrne fullname: Byrne, Charles email: Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell, MA, USA Charles_Byrne@uml.edu organization: Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell, MA, USAE-mail: Charles_Byrne@uml.edu |
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| Cites_doi | 10.1088/0266-5611/18/2/310 10.1007/BF00940051 10.1016/0024-3795(90)90203-O 10.1016/S1570-579X(01)80010-0 10.1137/S1052623495287022 10.4153/CJM-1954-037-2 10.1080/02522667.1987.10698894 10.1214/aoms/1177729694 10.1109/83.623192 10.1364/JOSAA.23.001292 10.1109/83.499919 10.1109/TMI.1982.4307558 10.1016/S1570-579X(01)80008-2 10.1109/42.363108 10.1016/0041-5553(67)90040-7 10.1088/0266-5611/14/6/006 10.1109/42.241889 10.1137/1032122 10.1109/83.210869 10.4153/CJM-1954-038-x 10.1088/0266-5611/20/1/006 10.1090/conm/204/02620 10.1016/0022-5193(70)90109-8 10.1007/BF01436376 10.1137/1.9781611971316 10.1088/0266-5611/24/1/015013 10.1109/TIP.2004.841193 10.1214/aoms/1177692379 10.1137/050626090 10.1109/42.538946 10.1016/0024-3795(81)90139-7 10.1109/TMI.1987.4307810 10.1201/b10651 10.1109/23.106689 10.1080/01621459.1985.10477119 10.2307/2032162 10.1007/BF03007664 10.1287/moor.17.3.670 10.1016/0041-5553(67)90113-9 10.1023/A:1013349430987 10.1137/S0036144593251710 10.1109/42.921483 |
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| References | 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. Tanabe, K., 1971. Projection method for solving a singular system of linear equations and its applications. Numerical Mathematics 17, 203-214. Wernick, M., Aarsvold, J. (eds) 2004. Emission Tomography: The Fundamentals of PET and SPECT. Elsevier Academic Press, San Diego. 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. Byrne, C., 1993. Iterative image reconstruction algorithms based on cross-entropy minimization. IEEE Transactions on Image Processing IP-2, 96-103. Dax, A., 1990. The convergence of linear stationary iterative processes for solving singular unstructured systems of linear equations. SIAM Review 32, 611-635. 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. Golshtein, E., Tretyakov, N., 1996. Modified Lagrangians and Monotone Maps in Optimization. John Wiley and Sons Inc., New York. Byrne, C., 1997. Convergent block-iterative algorithms for image reconstruction from inconsistent data. IEEE Transactions on Image Processing IP-6, 1296-1304. Byrne, C., 2002. Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Problems 18, 441-453. 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. Motzkin, T., Schoenberg, I., 1954. The relaxation method for linear inequalities. Canadian Journal of Mathematics 6, 393-404. Herman, G.T., Meyer, L., 1993. Algebraic reconstruction techniques can be made computationally efficient. IEEE Transactions on Medical Imaging 12, 600-609. Hudson, H.M., Larkin, R.S., 1994. Accelerated image reconstruction using ordered subsets of projection data. IEEE Transactions on Medical Imaging 13, 601-609. 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. Darroch, J., Ratcliff, D., 1972. Generalized iterative scaling for log-linear models. Annals of Mathematical Statistics 43, 1470-1480. 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. Censor, Y., Zenios, S.A., 1997. Parallel Optimization: Theory, Algorithms and Applications. 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IEEE Transactions on Medical Imaging MI-6, 2, 106-114. Teboulle, M., 1992. Entropic proximal mappings with applications to nonlinear programming. Mathematics of Operations Research 17, 3, 670-690. Agmon, S., 1954. The relaxation method for linear inequalities. Canadian Journal of Mathematics 6, 382-392. Bertsekas, D.P., 1997. A new class of incremental gradient methods for least squares problems. SIAM Journal on Optimization 7, 913-926. Hudson, M., Hutton, B., Larkin, R., 1992. Accelerated EM reconstruction using ordered subsets. Journal of Nuclear Medicine 33. p. 960. Mann, W., 1953. Mean value methods in iteration. Proceedings of the American Mathmatical Society 4, 506-510. 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. Schmidlin, P., 1972. Iterative separation of sections in tomographic scintigrams. Nuclear Medicine 11, 1, 1-6. 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. Byrne, C., 2007. Applied Iterative Methods. AK Peters Publishers, Wellesley, MA. Lange, K., Carson, R., 1984. EM reconstruction algorithms for emission and transmission tomography. Journal of Computer Assisted Tomography 8, 306-316. Byrne, C., 2005b. Signal Processing: A Mathematical Approach. AK Peters Publishers, Wellesley, MA. Censor, Y., Zenios, S.A., 1992. Proximal minimization algorithm with D-functions. Journal of Optimization Theory and Applications 73, 3, 451-464. 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. Vardi, Y., Shepp, L.A., Kaufman, L., 1985. A statistical model for positron emission tomography. Journal of the American Statistical Association 80, 8-20. Byrne, C., 2005a. Choosing parameters in block-iterative or ordered-subset reconstruction algorithms. IEEE Transactions on Image Processing 14, 3, 321-327. Censor, Y., Segman, J., 1987. On block-iterative maximization. Journal of Information and Optimization Sciences 8, 275-291. 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. Cimmino, G., 1938. Calcolo approssimato per soluzioni dei sistemi di equazioni lineari. La Ricerca Scientifica XVI, Series II, Anno IX 1, 326-333. Baillon, J.-B., Haddad, G., 1977. Quelques propriétés des operateurs angle-bornes et n-cycliquement monotones. Israel Journal of Mathematics 26, 137-150. Byrne, C., 1998. Iterative algorithms for deblurring and deconvolution with constraints. Inverse Problems 14, 1455-1467. Byrne, C., 2004. A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 20, 103-120. Byrne, C., 1996. Block-iterative methods for image reconstruction from projections. IEEE Transactions on Image Processing IP-5, 792-794. Kullback, S., Leibler, R., 1951. On information and sufficiency. Annals of Mathematical Statistics 22, 79-86. Bauschke, H., Borwein, J., 1996. On projection algorithms for solving convex feasibility problems. SIAM Review 38, 3, 367-426. 2005a; 14 1996; IP‐5 2004; 20 1990; 32 2002; 18 1977; 26 1987; 8 1990; 37 1992; 17 1985; 80 1997 1953; 4 2007 1996 1951; 22 1972; 43 2004 1996; 38 1992; 33 1996; 15 1981; 40 1997; 7 1992; 73 2001; 105 1967; 7 1938; 1 1997; 204 1993; 12 1997; IP‐6 2005b 2001 1990 2006; 23 1971; 17 1984; 8 2001; TMI‐20 2001; 8 2005; 4 1954; 6 1987; MI‐6 1993; IP‐2 2008; 24 1994; 13 1982; MI‐1 1970; 29 1972; 11 1990; 130 1998; 14 e_1_2_11_30_1 e_1_2_11_36_1 Wernick M. (e_1_2_11_51_1) 2004 e_1_2_11_13_1 e_1_2_11_34_1 e_1_2_11_11_1 Censor Y. (e_1_2_11_23_1) 1997 e_1_2_11_29_1 e_1_2_11_6_1 e_1_2_11_27_1 e_1_2_11_4_1 e_1_2_11_48_1 e_1_2_11_2_1 Vardi Y. (e_1_2_11_50_1) 1985; 80 e_1_2_11_20_1 Golshtein E. (e_1_2_11_32_1) 1996 e_1_2_11_47_1 e_1_2_11_8_1 e_1_2_11_22_1 e_1_2_11_43_1 e_1_2_11_15_1 Lange K. (e_1_2_11_41_1) 1984; 8 e_1_2_11_19_1 e_1_2_11_10_1 e_1_2_11_31_1 Schmidlin P. (e_1_2_11_45_1) 1972; 11 e_1_2_11_14_1 e_1_2_11_35_1 e_1_2_11_12_1 e_1_2_11_33_1 e_1_2_11_7_1 e_1_2_11_28_1 e_1_2_11_5_1 e_1_2_11_26_1 Hudson M. (e_1_2_11_38_1) 1992; 33 e_1_2_11_3_1 e_1_2_11_49_1 Cimmino G. (e_1_2_11_24_1) 1938; 1 Byrne C. (e_1_2_11_17_1) 2005 e_1_2_11_21_1 e_1_2_11_44_1 e_1_2_11_46_1 e_1_2_11_25_1 e_1_2_11_40_1 e_1_2_11_9_1 e_1_2_11_42_1 e_1_2_11_18_1 e_1_2_11_16_1 e_1_2_11_37_1 e_1_2_11_39_1 |
| References_xml | – reference: 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. – reference: Baillon, J.-B., Haddad, G., 1977. Quelques propriétés des operateurs angle-bornes et n-cycliquement monotones. Israel Journal of Mathematics 26, 137-150. – reference: 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. – reference: 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. – reference: Cimmino, G., 1938. Calcolo approssimato per soluzioni dei sistemi di equazioni lineari. La Ricerca Scientifica XVI, Series II, Anno IX 1, 326-333. – reference: Motzkin, T., Schoenberg, I., 1954. The relaxation method for linear inequalities. Canadian Journal of Mathematics 6, 393-404. – reference: Censor, Y., Zenios, S.A., 1992. Proximal minimization algorithm with D-functions. Journal of Optimization Theory and Applications 73, 3, 451-464. – reference: Byrne, C., 2002. Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Problems 18, 441-453. – reference: Byrne, C., 2005b. Signal Processing: A Mathematical Approach. AK Peters Publishers, Wellesley, MA. – reference: Darroch, J., Ratcliff, D., 1972. Generalized iterative scaling for log-linear models. Annals of Mathematical Statistics 43, 1470-1480. – reference: Lange, K., Carson, R., 1984. EM reconstruction algorithms for emission and transmission tomography. Journal of Computer Assisted Tomography 8, 306-316. – reference: 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. – reference: Censor, Y., Zenios, S.A., 1997. Parallel Optimization: Theory, Algorithms and Applications. Oxford University Press, New York. – reference: 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. – reference: Censor, Y., Segman, J., 1987. On block-iterative maximization. Journal of Information and Optimization Sciences 8, 275-291. – reference: Tanabe, K., 1971. Projection method for solving a singular system of linear equations and its applications. Numerical Mathematics 17, 203-214. – reference: Byrne, C., 1993. Iterative image reconstruction algorithms based on cross-entropy minimization. IEEE Transactions on Image Processing IP-2, 96-103. – reference: Bauschke, H., Borwein, J., 1996. On projection algorithms for solving convex feasibility problems. SIAM Review 38, 3, 367-426. – reference: Hudson, H.M., Larkin, R.S., 1994. Accelerated image reconstruction using ordered subsets of projection data. IEEE Transactions on Medical Imaging 13, 601-609. – reference: Teboulle, M., 1992. Entropic proximal mappings with applications to nonlinear programming. Mathematics of Operations Research 17, 3, 670-690. – reference: Byrne, C., 2007. Applied Iterative Methods. AK Peters Publishers, Wellesley, MA. – reference: Hudson, M., Hutton, B., Larkin, R., 1992. Accelerated EM reconstruction using ordered subsets. Journal of Nuclear Medicine 33. p. 960. – reference: Bertsekas, D.P., 1997. A new class of incremental gradient methods for least squares problems. SIAM Journal on Optimization 7, 913-926. – reference: Byrne, C., 1998. Iterative algorithms for deblurring and deconvolution with constraints. Inverse Problems 14, 1455-1467. – reference: 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. – reference: Byrne, C., 2005a. Choosing parameters in block-iterative or ordered-subset reconstruction algorithms. IEEE Transactions on Image Processing 14, 3, 321-327. – reference: Agmon, S., 1954. The relaxation method for linear inequalities. Canadian Journal of Mathematics 6, 382-392. – reference: 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. 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