# American Institute of Mathematical Sciences

January 2018, 12(1): 239-259. doi: 10.3934/ipi.2018010

## A scaled gradient method for digital tomographic image reconstruction

 1 Department of Mathematics, Shanghai University, Shanghai 200444, China 2 Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA

* Corresponding author: James G. Nagy

Received  March 2017 Revised  July 2017 Published  December 2017

Fund Project: The first author is supported by grant no. 15ZR1416300 from the Shanghai Municipal Natural Science Foundation, the third author is supported by grant no. DMS-1522760 from the US National Science Foundation

Digital tomographic image reconstruction uses multiple x-ray projections obtained along a range of different incident angles to reconstruct a 3D representation of an object. For example, computed tomography (CT) generally refers to the situation when a full set of angles are used (e.g., 360 degrees) while tomosynthesis refers to the case when only a limited (e.g., 30 degrees) angular range is used. In either case, most existing reconstruction algorithms assume that the x-ray source is monoenergetic. This results in a simplified linear forward model, which is easy to solve but can result in artifacts in the reconstructed images. It has been shown that these artifacts can be reduced by using a more accurate polyenergetic assumption for the x-ray source, but the polyenergetic model requires solving a large-scale nonlinear inverse problem. In addition to reducing artifacts, a full polyenergetic model can be used to extract additional information about the materials of the object; that is, to provide a mechanism for quantitative imaging. In this paper, we develop an approach to solve the nonlinear image reconstruction problem by incorporating total variation (TV) regularization. The corresponding optimization problem is then solved by using a scaled gradient descent method. The proposed algorithm is based on KKT conditions and Nesterov's acceleration strategy. Experimental results on reconstructed polyenergetic image data illustrate the effectiveness of this proposed approach.

Citation: Jianjun Zhang, Yunyi Hu, James G. Nagy. A scaled gradient method for digital tomographic image reconstruction. Inverse Problems & Imaging, 2018, 12 (1) : 239-259. doi: 10.3934/ipi.2018010
##### References:
 [1] R. E. Alvarez and A. Macovski, Energy-selective reconstructions in x-ray computerized tomography, Phys. Med. Biol., 21 (1976), 733-744. [2] R. E. Alvarez, J. A. Seibert and S. K. Thompson, Comparison of dual energy detector system performance, Med. Phys., 31 (2004), 556-565. doi: 10.1118/1.1645679. [3] R. F. Barber, E. Y. Sidky, T. G. Schmidt and X.-C Pan, An algorithm for constrained one-step inversion of spectral CT data, Phys. Med. Biol., 61 (2016), 3784-3818. [4] M. Bazalova, J. Carrier, L. Beaulieu and F. Verhaegen, Dual-energy CT-based material extraction for tissue segmentation in Monte Carlo dose calculations, Phys. Med. Biol., 53 (2008), 2439-2456. doi: 10.1088/0031-9155/53/9/015. [5] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imag. Sci., 2 (2009), 183-202. doi: 10.1137/080716542. [6] A. Beck and M. Teboulle, Gradient-based algorithms with applications to signal recovery problems, in Convex Optimization in Signal Processing and Communications (eds. Y. Eldar and D. Palomar), Cambridge University Press, (2010), 42-88. [7] M. J. Berger, J. H. Bubbell, S. M. Seltzer, J. Chang, J. S. Coursey, R. Sukumar, D. S. Zucker and K. Olsen, XCOM: Photon Cross Sections Database National Institutes of Standards, 2009. http://www.nist.gov/pml/data/xcom/index.cfm. doi: 10.6028/NBS.IR.87-3597. [8] D. P. Bertsekas, Nonlinear Programming 2nd edition, Athena Scientific, Belmont, Mass., 1999. [9] R. Brooks and G. Di Chiro, Beam hardening in x-ray reconstructive tomography, Phys.Med. Biol., 21 (1976), 390-398. doi: 10.1088/0031-9155/21/3/004. [10] V. M. Bustamante, J. G. Nagy, S. S. J. Feng and I. Sechopoulos, Iterative breast tomosynthesis image reconstruction, SIAM J. Sci. Comput., 35 (2013), S192-S208. doi: 10.1137/120881440. [11] R. Chan and J. Ma, A multiplicative iterative algorithm for box-constrained penalized likelihood image restoration, IEEE trans. Image Process., 21 (2012), 3168-3181. doi: 10.1109/TIP.2012.2188811. [12] T. F. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods SIAM, Philadelphia, 2005. [13] J. Chung, J. G. Nagy and I. Sechopoulos, Numerical algorithms for polyenergetic digital breast tomosynthesis reconstruction, SIAM J. Imaging Sci., 3 (2010), 133-152. doi: 10.1137/090749633. [14] B. De Man, J. Nuyts, P. Dupont, G. Marchal and P. Suetens, An iterative maximumlikelihood polychromatic algorithm for CT, IEEE Trans. Med. Imaging, 20 (2001), 999-1008. [15] I. A. Elbakri and J. A. Fessler, Statistical image reconstruction for polyenergetic x-ray computed tomography, IEEE Trans. Med. Imaging, 21 (2002), 89-99. doi: 10.1109/42.993128. [16] I. A. Elbakri and J. A. Fessler, Segmentation-free statistical image reconstruction for polyenergetic x-ray computed tomography with experimental validation, Phys. Med. Biol., 48 (2003), 2453-2477. doi: 10.1088/0031-9155/48/15/314. [17] C. L. Epstein, Introduction to the Mathematics of Medical Imaging 2nd edition, SIAM, Philadelphia, 2008. [18] G. R. Hammerstein, D. W. Miller, D. R. White, M. E. Masterson, H. Q. Woodard and J. S. Laughlin, Absorbed radiation dose in mammography, Radiology, 130 (1979), 485-491. doi: 10.1148/130.2.485. [19] P. C. Hansen and M. Saxild-Hansen, AIR tools -A MATLAB package of algebraic iterative reconstruction methods, J. Comput. Appl. Math., 236 (2012), 2167-2178. doi: 10.1016/j.cam.2011.09.039. [20] B. Heismann and M. Balda, Quantitative image-based spectral reconstruction for computed tomography, Med. Phys., 36 (2009), 4471-4485. doi: 10.1118/1.3213534. [21] A. Karellas, J. Lo and C. Orton, Point/counterpoint: Cone beam x-ray CT will be superior to digital x-ray tomosynthesis in imaging the breast and delineating cancer, Med. Phys., 35 (2008), 409-411. doi: 10.1118/1.2825612. [22] S. Kullback and R. A. Leibler, On information and sufficiency, Annals of Math. Statis., 22 (1951), 79-86. doi: 10.1214/aoms/1177729694. [23] D. D. Lee and H. S. Seung, Algorithms for non-negative matrix factorization, NIPS, (2000), 556-562. [24] A. Macovski, R. E. Alvarez, J. L.-H. Chan, J. P. Stonestrom and L. M. Zatz, Energy dependent reconstruction in x-ray computerized tomography, Comput. Biol.Med., 6 (1976), 335-336. doi: 10.1016/0010-4825(76)90069-X. [25] Y. E. Nesterov, A method for solving the convex programming problem with convergence rate O(1/k2), Dokl. Akad. Nauk SSSR, 269 (1983), 543-547. [26] J. Noh, J. A. Fessler and P. E. Kinahan, Statistical sinogram restoration in dual-energy CT for PET attenuation correction, IEEE Trans.Med. Imaging, 28 (2009), 1688-1702. [27] J. A. O'Sullivan and J. Benac, Alternating minimization algorithms for transmission tomography, IEEE Trans. Med. Imaging, 26 (2007), 283-297. doi: 10.1109/TMI.2006.886806. [28] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. [29] I. Sechopoulos, A review of breast tomosynthesis. Part Ⅰ. The image acquisition process Med. Phys. 40 (2013), 014301, 12pp. doi: 10.1118/1.4770279. [30] I. Sechopoulos, A review of breast tomosynthesis. Part Ⅱ. Image reconstruction, processing and analysis, and advanced applications Med. Phys. 40 (2013), 014302, 17pp. doi: 10.1118/1.4770281. [31] P. Stenner, T. Berkus and M. Kachelriess, Emperical dual energy calibration (EDEC) for cone-beam computed tomography, Med. Phys., 34 (2007), 3630-3641. [32] P. Sukovic and N. H. Clinthorne, Penalized weighted least-squares image reconstruction for dual energy x-ray transmission tomography, IEEE Trans. Med. Imaging, 19 (2000), 1075-1081. doi: 10.1109/NSSMIC.2000.949389. [33] C. R. Vogel, Computational Methods for Inverse Problems SIAM, Philadelphia, 2002.

show all references

##### References:
 [1] R. E. Alvarez and A. Macovski, Energy-selective reconstructions in x-ray computerized tomography, Phys. Med. Biol., 21 (1976), 733-744. [2] R. E. Alvarez, J. A. Seibert and S. K. Thompson, Comparison of dual energy detector system performance, Med. Phys., 31 (2004), 556-565. doi: 10.1118/1.1645679. [3] R. F. Barber, E. Y. Sidky, T. G. Schmidt and X.-C Pan, An algorithm for constrained one-step inversion of spectral CT data, Phys. Med. Biol., 61 (2016), 3784-3818. [4] M. Bazalova, J. Carrier, L. Beaulieu and F. Verhaegen, Dual-energy CT-based material extraction for tissue segmentation in Monte Carlo dose calculations, Phys. Med. Biol., 53 (2008), 2439-2456. doi: 10.1088/0031-9155/53/9/015. [5] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imag. Sci., 2 (2009), 183-202. doi: 10.1137/080716542. [6] A. Beck and M. Teboulle, Gradient-based algorithms with applications to signal recovery problems, in Convex Optimization in Signal Processing and Communications (eds. Y. Eldar and D. Palomar), Cambridge University Press, (2010), 42-88. [7] M. J. Berger, J. H. Bubbell, S. M. Seltzer, J. Chang, J. S. Coursey, R. Sukumar, D. S. Zucker and K. Olsen, XCOM: Photon Cross Sections Database National Institutes of Standards, 2009. http://www.nist.gov/pml/data/xcom/index.cfm. doi: 10.6028/NBS.IR.87-3597. [8] D. P. Bertsekas, Nonlinear Programming 2nd edition, Athena Scientific, Belmont, Mass., 1999. [9] R. Brooks and G. Di Chiro, Beam hardening in x-ray reconstructive tomography, Phys.Med. Biol., 21 (1976), 390-398. doi: 10.1088/0031-9155/21/3/004. [10] V. M. Bustamante, J. G. Nagy, S. S. J. Feng and I. Sechopoulos, Iterative breast tomosynthesis image reconstruction, SIAM J. Sci. Comput., 35 (2013), S192-S208. doi: 10.1137/120881440. [11] R. Chan and J. Ma, A multiplicative iterative algorithm for box-constrained penalized likelihood image restoration, IEEE trans. Image Process., 21 (2012), 3168-3181. doi: 10.1109/TIP.2012.2188811. [12] T. F. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods SIAM, Philadelphia, 2005. [13] J. Chung, J. G. Nagy and I. Sechopoulos, Numerical algorithms for polyenergetic digital breast tomosynthesis reconstruction, SIAM J. Imaging Sci., 3 (2010), 133-152. doi: 10.1137/090749633. [14] B. De Man, J. Nuyts, P. Dupont, G. Marchal and P. Suetens, An iterative maximumlikelihood polychromatic algorithm for CT, IEEE Trans. Med. Imaging, 20 (2001), 999-1008. [15] I. A. Elbakri and J. A. Fessler, Statistical image reconstruction for polyenergetic x-ray computed tomography, IEEE Trans. Med. Imaging, 21 (2002), 89-99. doi: 10.1109/42.993128. [16] I. A. Elbakri and J. A. Fessler, Segmentation-free statistical image reconstruction for polyenergetic x-ray computed tomography with experimental validation, Phys. Med. Biol., 48 (2003), 2453-2477. doi: 10.1088/0031-9155/48/15/314. [17] C. L. Epstein, Introduction to the Mathematics of Medical Imaging 2nd edition, SIAM, Philadelphia, 2008. [18] G. R. Hammerstein, D. W. Miller, D. R. White, M. E. Masterson, H. Q. Woodard and J. S. Laughlin, Absorbed radiation dose in mammography, Radiology, 130 (1979), 485-491. doi: 10.1148/130.2.485. [19] P. C. Hansen and M. Saxild-Hansen, AIR tools -A MATLAB package of algebraic iterative reconstruction methods, J. Comput. Appl. Math., 236 (2012), 2167-2178. doi: 10.1016/j.cam.2011.09.039. [20] B. Heismann and M. Balda, Quantitative image-based spectral reconstruction for computed tomography, Med. Phys., 36 (2009), 4471-4485. doi: 10.1118/1.3213534. [21] A. Karellas, J. Lo and C. Orton, Point/counterpoint: Cone beam x-ray CT will be superior to digital x-ray tomosynthesis in imaging the breast and delineating cancer, Med. Phys., 35 (2008), 409-411. doi: 10.1118/1.2825612. [22] S. Kullback and R. A. Leibler, On information and sufficiency, Annals of Math. Statis., 22 (1951), 79-86. doi: 10.1214/aoms/1177729694. [23] D. D. Lee and H. S. Seung, Algorithms for non-negative matrix factorization, NIPS, (2000), 556-562. [24] A. Macovski, R. E. Alvarez, J. L.-H. Chan, J. P. Stonestrom and L. M. Zatz, Energy dependent reconstruction in x-ray computerized tomography, Comput. Biol.Med., 6 (1976), 335-336. doi: 10.1016/0010-4825(76)90069-X. [25] Y. E. Nesterov, A method for solving the convex programming problem with convergence rate O(1/k2), Dokl. Akad. Nauk SSSR, 269 (1983), 543-547. [26] J. Noh, J. A. Fessler and P. E. Kinahan, Statistical sinogram restoration in dual-energy CT for PET attenuation correction, IEEE Trans.Med. Imaging, 28 (2009), 1688-1702. [27] J. A. O'Sullivan and J. Benac, Alternating minimization algorithms for transmission tomography, IEEE Trans. Med. Imaging, 26 (2007), 283-297. doi: 10.1109/TMI.2006.886806. [28] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. [29] I. Sechopoulos, A review of breast tomosynthesis. Part Ⅰ. The image acquisition process Med. Phys. 40 (2013), 014301, 12pp. doi: 10.1118/1.4770279. [30] I. Sechopoulos, A review of breast tomosynthesis. Part Ⅱ. Image reconstruction, processing and analysis, and advanced applications Med. Phys. 40 (2013), 014302, 17pp. doi: 10.1118/1.4770281. [31] P. Stenner, T. Berkus and M. Kachelriess, Emperical dual energy calibration (EDEC) for cone-beam computed tomography, Med. Phys., 34 (2007), 3630-3641. [32] P. Sukovic and N. H. Clinthorne, Penalized weighted least-squares image reconstruction for dual energy x-ray transmission tomography, IEEE Trans. Med. Imaging, 19 (2000), 1075-1081. doi: 10.1109/NSSMIC.2000.949389. [33] C. R. Vogel, Computational Methods for Inverse Problems SIAM, Philadelphia, 2002.
From left to right and upper to bottom: original first material, reconstructed first material, original second material, reconstructed second material, sinogram image and RErr with iteration
From left to right and upper to bottom: original first material, reconstructed first material, original second material, reconstructed second material, sinogram image and RErr with iteration
From left to right and upper to bottom: original first material, reconstructed first material, original second material, reconstructed second material, sinogram image and RErr with iteration
From left to right and upper to bottom: original first material, reconstructed first material, original second material, reconstructed second material, sinogram image and RErr with iteration
From left to right and upper to bottom: sum along each row for first material, sum along each column for first material, sum along each row for second material, sum along each column for second material, relative error of reduced resolution image along rows and columns for first material, relative error of reduced resolution image along rows and columns for second material
From left to right and upper to bottom: sum along each row for first material, sum along each column for first material, sum along each row for second material, sum along each column for second material, relative error of reduced resolution image along rows and columns for first material, relative error of reduced resolution image along rows and columns for second material
From left to right and upper to bottom: original first material, reconstructed first material, original second material, reconstructed second material, sinogram image and RErr with iteration
From left to right and upper to bottom: original first material, reconstructed first material, original second material, reconstructed second material, sinogram image and RErr with iteration
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