Sparse-view X-ray CT based on a box-constrained nonlinear weighted anisotropic TV regularization

Huiying Li; Yizhuang Song; Huiying Li; Yizhuang Song

doi:10.3934/mbe.2024223

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 4: 5047-5067. doi: 10.3934/mbe.2024223

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Sparse-view X-ray CT based on a box-constrained nonlinear weighted anisotropic TV regularization

Huiying Li ,
Yizhuang Song ^,

School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China

Received: 11 December 2023 Revised: 09 February 2024 Accepted: 26 February 2024 Published: 04 March 2024

Sparse-view computed tomography (CT) is an important way to reduce the negative effect of radiation exposure in medical imaging by skipping some X-ray projections. However, due to violating the Nyquist/Shannon sampling criterion, there are severe streaking artifacts in the reconstructed CT images that could mislead diagnosis. Noting the ill-posedness nature of the corresponding inverse problem in a sparse-view CT, minimizing an energy functional composed by an image fidelity term together with properly chosen regularization terms is widely used to reconstruct a medical meaningful attenuation image. In this paper, we propose a regularization, called the box-constrained nonlinear weighted anisotropic total variation (box-constrained NWATV), and minimize the regularization term accompanying the least square fitting using an alternative direction method of multipliers (ADMM) type method. The proposed method is validated through the Shepp-Logan phantom model, alongisde the actual walnut X-ray projections provided by Finnish Inverse Problems Society and the human lung images. The experimental results show that the reconstruction speed of the proposed method is significantly accelerated compared to the existing $ L_1/L_2 $ regularization method. Precisely, the central processing unit (CPU) time is reduced more than 8 times.
- sparse-view CT,
- computed tomography,
- box-constrained anisotropic TV,
- regularization,
- inverse problems
Citation: Huiying Li, Yizhuang Song. Sparse-view X-ray CT based on a box-constrained nonlinear weighted anisotropic TV regularization[J]. Mathematical Biosciences and Engineering, 2024, 21(4): 5047-5067. doi: 10.3934/mbe.2024223

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Abstract

Sparse-view computed tomography (CT) is an important way to reduce the negative effect of radiation exposure in medical imaging by skipping some X-ray projections. However, due to violating the Nyquist/Shannon sampling criterion, there are severe streaking artifacts in the reconstructed CT images that could mislead diagnosis. Noting the ill-posedness nature of the corresponding inverse problem in a sparse-view CT, minimizing an energy functional composed by an image fidelity term together with properly chosen regularization terms is widely used to reconstruct a medical meaningful attenuation image. In this paper, we propose a regularization, called the box-constrained nonlinear weighted anisotropic total variation (box-constrained NWATV), and minimize the regularization term accompanying the least square fitting using an alternative direction method of multipliers (ADMM) type method. The proposed method is validated through the Shepp-Logan phantom model, alongisde the actual walnut X-ray projections provided by Finnish Inverse Problems Society and the human lung images. The experimental results show that the reconstruction speed of the proposed method is significantly accelerated compared to the existing $ L_1/L_2 $ regularization method. Precisely, the central processing unit (CPU) time is reduced more than 8 times.

References

[1]	P. C. Hansen, J. S. Jørgensen, W. R. B. Lionheart, Computed Tomography: Algorithms, Insight, and Just Enough Theory, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2021. https://doi.org/10.1137/1.9781611976670
[2]	F. Natterer, The Mathematics of Computerized Tomography, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001. https://doi.org/10.1137/1.9780898719284
[3]	C. M. Hyun, T. Bayaraa, S. M. Lee, H. Jung, J. K. Seo, Deep learning for dental cone-beam computed tomography, in Deep Learning and Medical Applications (eds. J.K. Seo), Springer Nature Singapore, (2023), 101–175. https://doi.org/10.1007/978-981-99-1839-3_3
[4]	A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001. https://doi.org/10.1137/1.9780898719277
[5]	E. J. Hall, D. J. Brenner, Cancer risks from diagnostic radiology, Brit. J. Radiol., 81 (2008), 362–378. https://doi.org/10.1259/bjr/01948454 doi: 10.1259/bjr/01948454
[6]	M. Chen, Y. F. Pu, Y. C. Bai, Low-dose CT image denoising using residual convolutional network with fractional TV loss, Neurocomputing, 452 (2021), 510–520. https://doi.org/10.1016/j.neucom.2020.10.004 doi: 10.1016/j.neucom.2020.10.004
[7]	Z. Hu, D. Liang, D. Xia, H. Zheng, Compressive sampling in computed tomography: method and application, Nucl. Instrum. Meth. A., 748 (2014), 26–32. https://doi.org/10.1016/j.nima.2014.02.026 doi: 10.1016/j.nima.2014.02.026
[8]	K. H. Jin, M. T. McCann, E. Froustey, M. Unser, Deep convolutional neural network for inverse problems in imaging, IEEE. T. Image. Process., 26 (2017), 4509–4522. https://doi.org/10.1109/TIP.2017.2713099 doi: 10.1109/TIP.2017.2713099
[9]	Y. Han, J. C. Ye, Framing U-Net via deep convolutional framelets: Application to sparse-view CT, IEEE Trans. Med. Imag., 37 (2018), 1418–1429. https://doi.org/10.1109/TMI.2018.2823768 doi: 10.1109/TMI.2018.2823768
[10]	S. Xie, X. Zheng, Y. Chen, L. Xie, J. Liu, Y. Zhang, et al., Artifact removal using improved GoogLeNet for sparse-view CT reconstruction, Sci. Rep., 8 (2018), 1–9. https://doi.org/10.1038/s41598-018-25153-w doi: 10.1038/s41598-018-25153-w
[11]	E. Y. Sidky, I. Lorente, J. G. Brankov, X. Pan, Do CNNs solve the CT inverse problem?, IEEE Trans. Biomed. Eng., 68 (2021), 1799–1810. https://doi.org/10.1109/TBME.2020.3020741 doi: 10.1109/TBME.2020.3020741
[12]	A. Faridani, Introduction to the mathematics of computed tomography, in Inside Out: Inverse Problems and Applications, Cambridge Univ. Press, (2003), 1–46. https://doi.org/10.4171/PRIMS/47.1.1
[13]	S. J. LaRoque, E. Y. Sidky, X. Pan, Accurate image reconstruction from few-view and limited-angle data in diffraction tomography, J. Opt. Soc. Am. A, 25 (2008), 1772–1782. https://doi.org/10.1364/JOSAA.25.001772 doi: 10.1364/JOSAA.25.001772
[14]	M. Burger, S. Osher, A guide to the TV zoo, in Level Set and PDE Based Reconstruction Methods in Imaging (eds. M.Burger and S.Osher), Springer International Publishing, (2013), 1–70. https://doi.org/10.1007/978-3-319-01712-9_1
[15]	X. Jin, L. Li, Z. Chen, L. Zhang, Y. Xing, Anisotropic total variation for limited-angle CT reconstruction, in IEEE Nuclear Science Symposuim & Medical Imaging Conference, (2010), 2232–2238. https://doi.org/10.1109/NSSMIC.2010.5874180
[16]	Z. Tian, X. Jia, K. Yuan, T. Pan, S. B. Jiang, Low-dose CT reconstruction via edge-preserving total variation regularization, Phys. Med. Biol., 56 (2011), 5949–5967. https://doi.org/10.1088/0031-9155/56/18/011 doi: 10.1088/0031-9155/56/18/011
[17]	Y. Liu, J. Ma, Y. Fan, Z. Liang, Adaptive-weighted total variation minimization for sparse data toward low-dose x-ray computed tomography image reconstruction, Phys. Med. Biol., 57 (2012), 7923–7956. https://doi.org/10.1088/0031-9155/57/23/7923 doi: 10.1088/0031-9155/57/23/7923
[18]	Y. Xi, P. Zhou, H. Yu, T. Zhang, L. Zhang, Z. Qiao, et al., Adaptive-weighted high order TV algorithm for sparse-view CT reconstruction, Med. Phys., 50 (2023), 5568–5584. https://doi.org/10.1002/mp.16371 doi: 10.1002/mp.16371
[19]	Y. Wang, Z. Qi, A new adaptive-weighted total variation sparse-view computed tomography image reconstruction with local improved gradient information, J. X-Ray Sci. Technol., 26 (2018), 957–975. https://doi.org/10.3233/XST-180412 doi: 10.3233/XST-180412
[20]	S. Niu, Y. Gao, Z. Bian, J. Huang, W. Chen, G. Yu, et al., Sparse-view X-ray CT reconstruction via total generalized variation regularization, Phys. Med. Biol., 59 (2014), 2997–3017. https://doi.org/10.1088/0031-9155/59/12/2997 doi: 10.1088/0031-9155/59/12/2997
[21]	Z. Qu, X. Zhao, J. Pan, P. Chen, Sparse-view CT reconstruction based on gradient directional total variation, Meas. Sci. Technol., 30 (2019), 1–11. https://doi.org/10.1088/1361-6501/ab09c6 doi: 10.1088/1361-6501/ab09c6
[22]	L. Zhang, H. Zhao, W. Ma, J. Jiang, L. Zhang, J. Li, et al., Resolution and noise performance of sparse view X-ray CT reconstruction via Lp-norm regularization, Phys. Medica., 52 (2018), 72–80. https://doi.org/10.1016/j.ejmp.2018.04.396 doi: 10.1016/j.ejmp.2018.04.396
[23]	C. Wang, M. Tao, J. Nagy, Y. Lou, Limited-angle CT reconstruction via the $L_1/L_2$ minimization, SIAM J. Imaging. Sci., 14 (2021), 749–777. https://doi.org/10.1137/20M1341490 doi: 10.1137/20M1341490
[24]	Y. Song, Y. Wang, D. Liu, A nonlinear weighted anisotropic total variation regularization for electrical impedance tomography, IEEE Trans. Instrum. Meas., 71 (2022), 1–13. https://doi.org/10.1109/TIM.2022.3220288 doi: 10.1109/TIM.2022.3220288
[25]	J. Radon, On the determination of functions from their integral values along certain manifolds, IEEE Trans. Med. Imag., 5 (1986), 170–176. https://doi.org/10.1109/TMI.1986.4307775 doi: 10.1109/TMI.1986.4307775
[26]	S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends. Mach. Learn., 3 (2011), 1–122. https://doi.org/10.1561/2200000016 doi: 10.1561/2200000016
[27]	I. Daubechies, M. Defrise, C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pur. Appl. Math., 57 (2004), 1413–1457. https://doi.org/10.1002/cpa.20042 doi: 10.1002/cpa.20042
[28]	Z. Wang, A. C. Bovik, H. R. Sheikh, E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE. T. Image. Process., 13 (2004), 600–612. https://doi.org/10.1109/TIP.2003.819861 doi: 10.1109/TIP.2003.819861
[29]	Y. Saad, M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), 856–869. https://doi.org/10.1137/0907058 doi: 10.1137/0907058
[30]	P. C. Hansen, J. S. Jørgensen, AIR Tools Ⅱ: Algebraic iterative reconstruction methods, improved implementation, Numer. Algor., 79 (2018), 107–137. https://doi.org/10.1007/s11075-017-0430-x doi: 10.1007/s11075-017-0430-x
[31]	K. Hämäläinen, L. Harhanen, A. Kallonen, A. Kujanpää, E. Niemi, S. Siltanen, Tomographic X-ray data of a walnut, preprint, arXiv: 1502.04064. https://doi.org/10.48550/arXiv.1502.04064
[32]	O. Grove, A. E. Berglund, M. B. Schabath, H. J. W. L. Aerts, A. Dekker, H. Wang, et al., Quantitative computed tomographic descriptors associate tumor shape complexity and intratumor heterogeneity with prognosis in lung adenocarcinoma, PLoS ONE, 10 (2015), 1–14. https://doi.org/10.1371/journal.pone.0118261 doi: 10.1371/journal.pone.0118261
[33]	G. Wang, J. C. Ye, B. D. Man, Deep learning for tomographic image reconstruction, Nat. Mach. Intell., 2 (2020), 737–748. https://doi.org/10.1038/s42256-020-00273-z doi: 10.1038/s42256-020-00273-z
[34]	V. Antun, F. Renna, C. Poon, B. Adcock, A. C. Hansen, On instabilities of deep learning in image reconstruction and the potential costs of AI, P. Natl. A. Sci., 117 (2020), 30088–30095. https://doi.org/10.1073/pnas.1907377117 doi: 10.1073/pnas.1907377117
[35]	S. Li, Q. Cao, Y. Chen, Y. Hu, L. Luo, C. Toumoulin, Dictionary learning based sinogram inpainting for CT sparse reconstruction, Optik, 125 (2014), 2862–2867. https://doi.org/10.1016/j.ijleo.2014.01.003 doi: 10.1016/j.ijleo.2014.01.003
[36]	E. Kobler, A. Effland, K. Kunisch, T. Pock, Total deep variation: A stable regularization method for inverse problems, IEEE. T. Pattern. Anal., 44 (2022), 9163–9180. https://doi.org/10.1109/TPAMI.2021.3124086 doi: 10.1109/TPAMI.2021.3124086
[37]	J. Xu, F. Noo, Convex optimization algorithms in medical image reconstruction—in the age of AI, Phys. Med. Biol., 67 (2022), 1–77. https://doi.org/10.1088/1361-6560/ac3842 doi: 10.1088/1361-6560/ac3842
[38]	J. Kaipio, E. Somersalo, Statistical and Computational Inverse Problems, Springer, New York, 2005. https://doi.org/10.1007/b138659
[39]	L. Xu, L. Li, W. Wang, Y. Gao, CT image reconstruction algorithms based on the Hanke Raus parameter choice rule, Inverse Probl. Sci. Eng., 28 (2020), 87–103. https://doi.org/10.1080/17415977.2019.1628739 doi: 10.1080/17415977.2019.1628739
[40]	H. Park, J. K. Choi, J. K. Seo, Characterization of metal artifacts in X-ray computed tomography, Commun. Pure Appl. Math., 70 (2017), 2191–2217. https://doi.org/10.1002/cpa.21680 doi: 10.1002/cpa.21680
[41]	S. M. Lee, J. K. Seo, Y. E. Chung, J. Baek, H. S. Park, Technical Note: A model-based sinogram correction for beam hardening artifact reduction in CT, Med. Phys., 44 (2017), e147–e152. https://doi.org/10.1002/mp.12218 doi: 10.1002/mp.12218

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