# American Institute of Mathematical Sciences

February  2012, 6(1): 133-146. doi: 10.3934/ipi.2012.6.133

## The order of convergence for Landweber Scheme with $\alpha,\beta$-rule

 1 Department of Mathematics, Shanghai Maritime University, Shanghai 200135, China 2 LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China

Received  February 2011 Revised  September 2011 Published  February 2012

The Landweber scheme is widely used in various image reconstruction problems. In previous works, $\alpha,\beta$-rule is suggested to stop the Landweber iteration so as to get proper iteration results. The order of convergence of discrepancy principal (DP rule), which is a special case of $\alpha,\beta$-rule, with constant relaxation coefficient $\lambda$ satisfying $0<\lambda\sigma_1^2<1,~(\|A\|_{V,W}=\sigma_1>0)$ has been studied. A sufficient condition for convergence of Landweber scheme is that the value $\lambda_m\sigma_1^2$ should be lied in a closed interval, i.e. $0<\varepsilon\leq\lambda_m\sigma_1^2\leq2-\varepsilon$, $(0<\varepsilon<1)$. In this paper, we mainly investigate the order of convergence of the $\alpha,\beta$-rule with variable relaxation coefficient $\lambda_m$ satisfying $0 < \varepsilon\leq\lambda_m \sigma_1^2 \leq 2-\varepsilon$. According to the order of convergence, we can conclude that $\alpha,\beta$-rule is the optimal rule for the Landweber scheme.
Citation: Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems and Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133
##### References:
 [1] Y. Censor and T. Elfving, Block-iterative algorithms with diagonally scaled oblique projections for the linear feasibility problem, SIAM Journal on Matrix Analysis and Applications, 24 (2002), 40-58. doi: 10.1137/S089547980138705X. [2] M. Jiang and G. Wang, Convergence studies on iterative algorithms for image reconstruction, IEEE Transactions on Medical Imaging, 22 (2003), 569-579. doi: 10.1109/TMI.2003.812253. [3] A. Andersen and A. Kak, Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm, Ultrasonic Imaging, 6 (1984), 81-94. doi: 10.1016/0161-7346(84)90008-7. [4] G. Cimmino, Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari, La Ricerca Scientifica, series II, Anno IX, XVI (1938), 326-333. [5] Y. Censor, D. Gordon and R. Gordon, Component avering: An efficient iterative parallel algorithm for large and sparse unstructured problems, Parallel Computing, 27 (2001), 777-808. doi: 10.1016/S0167-8191(00)00100-9. [6] L. Landweber, An iteration formula for Fredholm integral equations of the first kind, American Journal of Mathematics, 73 (1951), 615-624. doi: 10.2307/2372313. [7] R. J. Santos, Equivalence of regularization and truncated iteration for general ill-posed problems, Linear Algebra and its Applications, 236 (1996), 25-33. doi: 10.1016/0024-3795(94)00114-6. [8] F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction," SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001. [9] Y. Xiao, D. Michalski, Y. Censor and J. M. Galvin, Inherent smoothness of intensity patterns for intensity modulated radiation therapy generated by simultaneous projection algorithms, Physics in Medicine and Biology, 49 (2004), 3227-3245. doi: 10.1088/0031-9155/49/14/015. [10] P .D. Acton, S. R. Choi, K. Plossl and H. F. Kung, Quantification of dopamine transporters in the mouse brain using ultra-high resolution single-photon emission tomography, European Journal of Nuclear Medicine and Molecular Imaging, 29 (2002), 691-698. doi: 10.1007/s00259-002-0903-5. [11] W. Q. Yang and L. H. Peng, Image reconstruction algorithms for electrical capacitance tomography, Measurement Science & Technology, 14 (2003), R1-R13. doi: 10.1088/0957-0233/14/1/201. [12] J. Wang and Y. B. Zheng, On the convergence of generalized simultaneous iterative reconstruction algorithms, IEEE Transactions on Image Processing, 16 (2007), 1-6. doi: 10.1109/TIP.2006.887725. [13] G. Qu, C. Wang and M. Jiang, Necessary and sufficient convergence conditions for algebraic image reconstruction algorithms, IEEE Transactions on Image Processing, 18 (2009), 435-440. doi: 10.1109/TIP.2008.2008076. [14] G. Qu and M. Jiang, Landweber scheme for compact operator equation in Hilbert space and its applications, Communications in Numerical Methods in Engineering, 25 (2009), 771-786. doi: 10.1002/cnm.1196. [15] T. Zhang and B. Yu, Boosting with early stopping: Convergence and consistency, Annals of Statistics, 33 (2005), 1538-1579. doi: 10.1214/009053605000000255. [16] T. Elfving and T. Nikazad, Stopping rules for Landweber-type interation, Inverse Problems, 23 (2007), 1417-1432. doi: 10.1088/0266-5611/23/4/004. [17] A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems," Applied Mathematical Sciences, 120, Springer-Verlag, New York, 1996. [18] T. Hein and K. Kazimierski, Modifed Landweber iteration in Banach spaces-Convergence and convergence rates, Numerical Functional Analysis and Optimization, 31 (2010), 1158-1184. [19] M. Jiang, Iterative Algebraic Algorithms for Image Reconstruction, in " Medical Imaging Systems Technology," Vol. I, World Scientific, Singapore, (2005), 351-382. [20] C. W. Groetsch, "Inverse Problems in the Mathematical Sciences (Theory & Practice of Applied Geophysics)," Informatica International, Inc., 1993. [21] S. Aja-Fernández, R. Estépar, C. Alberola-López and C. Westin, Image quality assessment based on local variance, Proceedings IEEE Engineering in Medicine and Biology Society, 1 (2006), 4815-4818.

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##### References:
 [1] Y. Censor and T. Elfving, Block-iterative algorithms with diagonally scaled oblique projections for the linear feasibility problem, SIAM Journal on Matrix Analysis and Applications, 24 (2002), 40-58. doi: 10.1137/S089547980138705X. [2] M. Jiang and G. Wang, Convergence studies on iterative algorithms for image reconstruction, IEEE Transactions on Medical Imaging, 22 (2003), 569-579. doi: 10.1109/TMI.2003.812253. [3] A. Andersen and A. Kak, Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm, Ultrasonic Imaging, 6 (1984), 81-94. doi: 10.1016/0161-7346(84)90008-7. [4] G. Cimmino, Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari, La Ricerca Scientifica, series II, Anno IX, XVI (1938), 326-333. [5] Y. Censor, D. Gordon and R. Gordon, Component avering: An efficient iterative parallel algorithm for large and sparse unstructured problems, Parallel Computing, 27 (2001), 777-808. doi: 10.1016/S0167-8191(00)00100-9. [6] L. Landweber, An iteration formula for Fredholm integral equations of the first kind, American Journal of Mathematics, 73 (1951), 615-624. doi: 10.2307/2372313. [7] R. J. Santos, Equivalence of regularization and truncated iteration for general ill-posed problems, Linear Algebra and its Applications, 236 (1996), 25-33. doi: 10.1016/0024-3795(94)00114-6. [8] F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction," SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001. [9] Y. Xiao, D. Michalski, Y. Censor and J. M. Galvin, Inherent smoothness of intensity patterns for intensity modulated radiation therapy generated by simultaneous projection algorithms, Physics in Medicine and Biology, 49 (2004), 3227-3245. doi: 10.1088/0031-9155/49/14/015. [10] P .D. Acton, S. R. Choi, K. Plossl and H. F. Kung, Quantification of dopamine transporters in the mouse brain using ultra-high resolution single-photon emission tomography, European Journal of Nuclear Medicine and Molecular Imaging, 29 (2002), 691-698. doi: 10.1007/s00259-002-0903-5. [11] W. Q. Yang and L. H. Peng, Image reconstruction algorithms for electrical capacitance tomography, Measurement Science & Technology, 14 (2003), R1-R13. doi: 10.1088/0957-0233/14/1/201. [12] J. Wang and Y. B. Zheng, On the convergence of generalized simultaneous iterative reconstruction algorithms, IEEE Transactions on Image Processing, 16 (2007), 1-6. doi: 10.1109/TIP.2006.887725. [13] G. Qu, C. Wang and M. Jiang, Necessary and sufficient convergence conditions for algebraic image reconstruction algorithms, IEEE Transactions on Image Processing, 18 (2009), 435-440. doi: 10.1109/TIP.2008.2008076. [14] G. Qu and M. Jiang, Landweber scheme for compact operator equation in Hilbert space and its applications, Communications in Numerical Methods in Engineering, 25 (2009), 771-786. doi: 10.1002/cnm.1196. [15] T. Zhang and B. Yu, Boosting with early stopping: Convergence and consistency, Annals of Statistics, 33 (2005), 1538-1579. doi: 10.1214/009053605000000255. [16] T. Elfving and T. Nikazad, Stopping rules for Landweber-type interation, Inverse Problems, 23 (2007), 1417-1432. doi: 10.1088/0266-5611/23/4/004. [17] A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems," Applied Mathematical Sciences, 120, Springer-Verlag, New York, 1996. [18] T. Hein and K. Kazimierski, Modifed Landweber iteration in Banach spaces-Convergence and convergence rates, Numerical Functional Analysis and Optimization, 31 (2010), 1158-1184. [19] M. Jiang, Iterative Algebraic Algorithms for Image Reconstruction, in " Medical Imaging Systems Technology," Vol. I, World Scientific, Singapore, (2005), 351-382. [20] C. W. Groetsch, "Inverse Problems in the Mathematical Sciences (Theory & Practice of Applied Geophysics)," Informatica International, Inc., 1993. [21] S. Aja-Fernández, R. Estépar, C. Alberola-López and C. Westin, Image quality assessment based on local variance, Proceedings IEEE Engineering in Medicine and Biology Society, 1 (2006), 4815-4818.
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