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Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries
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 |
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. |
show all references
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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