Limited-angle CT reconstruction with generalized shrinkage operators as regularizers

Xiaojuan Deng; Xing Zhao; Mengfei Li; Hongwei Li

doi:10.3934/ipi.2021019

Article Contents

2021, Volume 15, Issue 6: 1287-1306. Doi: 10.3934/ipi.2021019 open access

This issue Previous Article Next Article RWRM: Residual Wasserstein regularization model for image restoration

Limited-angle CT reconstruction with generalized shrinkage operators as regularizers

1.
School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China
2.
Beijing Advanced Innovation Center for Imaging Technology, Capital Normal University, Beijing, 100048, China
3.
Division of Ionizing Radiation, National Institute of Metrology, Beijing 100029, China

^* Corresponding author: Hongwei Li
^* Corresponding author: Hongwei Li

Received: December 31, 2019

Revised: August 31, 2020

Early access: February 2021

Published: December 2021

This research was supported by National Natural Science Foundation of China (NSFC) (61971292, 61827809, 61901127 and 61871275) and key research project of the Academy for Multidisciplinary Studies, Capital Normal University. The authors are also grateful to Beijing Advanced Innovation Center for Imaging Technology for funding this this research work

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Abstract

Limited-angle reconstruction is a very important but challenging problem in the field of computed tomography (CT) which has been extensively studied for many years. However, some difficulties still remain. Based on the theory of visible and invisible boundary developed by Quinto et.al, we propose a reconstruction model for limited-angle CT, which encodes the visible edges as priors to recover the invisible ones. The new model utilizes generalized shrinkage operators as regularizers to perform edge-preserving smoothing such that the visible edges are employed as anchors to recover piecewise-constant or piecewise-smooth reconstructions, while noises and artifacts are suppressed or removed. This work extends our previous research on limited-angle reconstruction which employs gradient $ \ell_0 $ and $ \ell_1 $ norm regularizers. The effectiveness of the proposed model and its corresponding solving algorithm shall be verified by numerical experiments with simulated data as well as real data.

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Mathematics Subject Classification: Primary: 65K10, 47A52; Secondary: 94A08.

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Figure 1. Illustration of the fan-beam scanning configuration used in this paper for limited-angle reconstruction

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Figure 2. (a) Shepp-Logan phantom; (b) reconstructed image from 120-degree data ($ [\pi/6, 5\pi/6] $) with SART (10 iterations). The display window is set to [0, 0.6]

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Figure 3. (a)-(b) show the reconstructions from full- and limited-angle ($ [\pi/6, 5\pi/6] $) data, respectively; (c)-(f) show reconstructions of the four competing algorithms with limited-angle data, respectively, and 2000 iterations are performed to guarantee convergence. The display window is set to [0, 0.6]

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Figure 4. (a)-(b) show the reconstructions from full- and limited-angle ($ [\pi/6, 5\pi/6] $) data, respectively; (c)-(f) reconstructed images from limited-angle data by using GAEDS algorithm (with different $ (p, q) $), and 2000 iterations are performed to guarantee convergence. The display window is set to [0, 0.6]

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Figure 5. (a) Rectangular phantom; (b)-(c) noisy reconstruction with SART (10 iterations) from full-and limited-angle ($ [\pi/4, 3\pi/4] $) data; (d), (f) and (h) are reconstructed images from 90-degree data, by applying AEDS($ \ell_0, \ell_0 $), AEDS($ \ell_0, \ell_1 $) and GAEDS($ p, q $), respectively; (e), (g) and (i) are the corresponding residual images. The display window is set to [0, 1]

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Figure 6. (a) Rasterized image of the designed rhombus phantom by using CTSim; (b) full-angle SART reconstruction (10 iterations). The display window is set to [0, 0.22]

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Figure 7. The first row shows the reconstructions with different angular ranges by applying SART (10 iterations), while the second, third and fourth rows show the results of AEDS($ \ell_0, \ell_0 $), AEDS($ \ell_0, \ell_1 $) and GAEDS($ p, q $), respectively. Note that 1000 iterations are performed for both AEDS and GAEDS algorithms, which are enough for convergence. The display window is set to [0, 0.22]

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Figure 8. (a) Photograph of the flat object; (b) the reference image reconstructed from the full-angle data; (c) reconstructed image from 120-degree data ($ [\pi/6, 5\pi/6] $) with SART (10 iterations); (d)-(f) show reconstructions with AEDS($ \ell_0, \ell_0 $), AEDS($ \ell_0, \ell_1 $) and GAEDS($ p, q $), respectively. Note that 1000 iterations are performed for each algorithm. The display window is set to [0, 0.06]

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Figure 9. Energy curves and increments curves of GAEDS($ p, q $) for the experiment with the Shepp-Logan phantom. (a) shows the energy curves; (b) shows the $ D(n) $ curves

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Table 1. Geometrical parameters of discrete simulations

Parameter	Value
Distance of X-ray source to rotation center	600 mm
Width of detector unit	0.25 mm
Number of detector units	1118
Distance of rotation center to detector	1739.63 mm
Scanning Angular Interval	0.5 degree

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Table 2. Reconstruction parameters used by the competing algorithms on the Shepp-Logan phantom ($ I_0 = 1\times10^6 $)

SART+ATV$ _{\ell_1} $	SART+ATV$ _{\ell_p} $	GAEDS($ p, q $)	GAEDS($ p, q $)
$ p=1 $	$ p=-0.5 $	$ p=-0.5 $	$ p=-\infty $
$ q=1 $	$ q=-0.5 $	$ q=-0.5 $	$ q=-\infty $
$ \lambda_1=0.02 $	$ \lambda_1=0.04 $	$ \lambda_1=0.06 $	$ \lambda_1=0.0004 $
$ \lambda_2=0.02 $	$ \lambda_2=0.04 $	$ \lambda_2=0.02 $	$ \lambda_2=0.0001 $

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Table 3. Reconstruction parameters used by the GAEDS algorithm with different $ (p, q) $ on the Shepp-Logan phantom ($ I_0 = 1\times10^5 $)

GAEDS($ p, q $)	GAEDS($ p, q $)	GAEDS($ p, q $)	GAEDS($ p, q $)
$ p=-\infty $	$ p=-\infty $	$ p=-\infty $	$ p=-0.5 $
$ q=-\infty $	$ q=-0.5 $	$ q=1 $	$ q=-0.5 $
$ \lambda_1=0.0004 $	$ \lambda_1=0.0003 $	$ \lambda_1=0.0006 $	$ \lambda_1=0.035 $
$ \lambda_2=0.0002 $	$ \lambda_2=0.03 $	$ \lambda_2=0.03 $	$ \lambda_2=0.013 $

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Table 4. Reconstruction parameters used by the competing algorithms on the rectangular phantom

AEDS($ \ell_0, \ell_0 $)	AEDS($ \ell_0, \ell_1 $)	GAEDS($ p, q $)
		$ p=-\infty, q=-0.5 $
$ \lambda_1=0.003 $	$ \lambda_1=0.004 $	$ \lambda_1=0.003 $
$ \lambda_2=0.0003 $	$ \lambda_2=0.006 $	$ \lambda_2=0.02 $

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Table 5. PSNR, SSIM and UQI of the 90-degree reconstructions for the rectangular phantom

	SART	AEDS($ \ell_0, \ell_0 $)	AEDS($ \ell_0, \ell_1 $)	GAEDS($ p, q $)
PSNR	16.886	51.7035	45.4061	53.4988
SSIM	0.6822	0.9755	0.9710	0.9808
UQI	0.8155	0.9999	0.9998	0.9999

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Table 6. Reconstruction parameters used by the competing algorithms for the rhombus phantom

	AEDS($ \ell_0, \ell_0 $)	AEDS($ \ell_0, \ell_1 $)	GAEDS($ p, q $)
Rhombus(150°)			$ p=-\infty, q=0.8 $
	$ \lambda_1=0.00003 $	$ \lambda_1=0.00003 $	$ \lambda_1=0.000033 $
	$ \lambda_2=0.000005 $	$ \lambda_2=0.01 $	$ \lambda_2=0.018 $
Rhombus(140°)			$ p=-\infty, q=0.9 $
	$ \lambda_1=0.00003 $	$ \lambda_1=0.00003 $	$ \lambda_1=0.000033 $
	$ \lambda_2=0.000008 $	$ \lambda_2=0.01 $	$ \lambda_2=0.03 $
Rhombus(130°)			$ p=-\infty, q=0.8 $
	$ \lambda_1=0.00001 $	$ \lambda_1=0.00005 $	$ \lambda_1=0.00001 $
	$ \lambda_2=0.000005 $	$ \lambda_2=0.0125 $	$ \lambda_2=0.01 $

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Table 7. PSNR, SSIM and UQI of the 90-degree reconstructions for the rhombus phantom

		SART	AEDS($ \ell_0, \ell_0 $)	AEDS($ \ell_0, \ell_1 $)	GAEDS($ p, q $)
150 degree	PSNR	31.7800	34.4700	35.7580	35.5110
	SSIM	0.99366	0.99496	0.99616	0.99615
	UQI	0.99030	0.99477	0.99609	0.99590
140 degree	PSNR	29.4930	33.7360	33.5110	34.1700
	SSIM	0.98854	0.99426	0.99330	0.99361
	UQI	0.98333	0.99382	0.99339	0.99433
130 degree	PSNR	27.2680	32.7270	32.2330	33.7820
	SSIM	0.98347	0.99272	0.98837	0.99425
	UQI	0.97162	0.99218	0.99105	0.99388

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Table 8. Geometrical parameters of the real scanning device

Parameter	Value
Voltage	140 kV
Current	160 mA
Distance of X-ray source to rotation center	311.49 mm
Width of detector unit	0.127 mm
Number of detector units	1920
Distance of rotation center to detector	697.82 mm
Scanning angular interval	0.2 degree

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Table 9. Reconstruction parameters for the real data test

AEDS($ \ell_0, \ell_0 $)	AEDS($ \ell_0, \ell_1 $)	GAEDS($ p, q $)
		$ p=-\infty, q=-0.5 $
$ \lambda_1=0.00003 $	$ \lambda_1=0.00007 $	$ \lambda_1=0.00003 $
$ \lambda_2=0.000005 $	$ \lambda_2=0.005 $	$ \lambda_2=0.005 $

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References

[1]	A. H. Andersen and A. C. Kak, Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm, Ultrasonic Imaging, 6 (1984), 81-94. doi: 10.1177/016173468400600107.
[2]	D. P. Bertsekas, Incremental gradient, subgradient, and proximal methods for convex optimization: A survey, Optimization for Machine Learning, 2010 (2011), 3.
[3]	D. P. Bertsekas, Incremental proximal methods for large scale convex optimization, Mathematical Programming, 129 (2011), 163-195. doi: 10.1007/s10107-011-0472-0.
[4]	S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers, Now Foundations and Trends® in Machine learning, 2011. doi: 10.1561/9781601984616.
[5]	A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145. doi: 10.1007/s10851-010-0251-1.
[6]	R. Chartrand, Fast algorithms for nonconvex compressive sensing: MRI reconstruction from very few data, in International Symposium on Biomedical Imaging: From Nano to Macro, IEEE, 2009, 262–265.
[7]	R. Chartrand, Shrinkage mappings and their induced penalty functions, in International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, 2014, 1026–1029.
[8]	F. Chen, L. Shen and B. W. Suter, Computing the proximity operator of the $\ell_p$ norm with 0 < p < 1, IET Signal Processing, 10 (2016), 557-565.
[9]	G.-H. Chen, J. Tang and S. Leng, Prior image constrained compressed sensing (PICCS): A method to accurately reconstruct dynamic CT images from highly undersampled projection data sets, Medical Physics, 35 (2008), 660-663. doi: 10.1118/1.2836423.
[10]	Z. Chen, X. Jin, L. Li and G. Wang, A limited-angle CT reconstruction method based on anisotropic TV minimization, Physics in Medicine & Biology, 58 (2013), 2119. doi: 10.1088/0031-9155/58/7/2119.
[11]	R. Fahrig, R. Dixon, T. Payne, R. L. Morin, A. Ganguly and N. Strobel, Dose and image quality for a cone-beam C-arm CT system, Medical Physics, 33 (2006), 4541-4550. doi: 10.1118/1.2370508.
[12]	J. Frikel and E. T. Quinto, Characterization and reduction of artifacts in limited angle tomography, Inverse Problems, 29 (2013), 125007, 21 pp. doi: 10.1088/0266-5611/29/12/125007.
[13]	P. Gilbert, Iterative methods for the three-dimensional reconstruction of an object from projections, Journal of Theoretical Biology, 36 (1972), 105-117. doi: 10.1016/0022-5193(72)90180-4.
[14]	T. Goldstein and S. Osher, The split bregman method for L1 regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891.
[15]	C. Gong, L. Zeng and C. Wang, Image reconstruction model for limited-angle ct based on prior image induced relative total variation, Applied Mathematical Modelling, 74 (2019), 586-605. doi: 10.1016/j.apm.2019.05.020.
[16]	R. Gordon, R. Bender and G. T. Herman, Algebraic reconstruction techniques (art) for three-dimensional electron microscopy and x-ray photography, Journal of Theoretical Biology, 29 (1970), 471-481. doi: 10.1016/0022-5193(70)90109-8.
[17]	G. T. Gullberg, The reconstruction of fan-beam data by filtering the back-projection, Computer Graphics and Image Processing, 10 (1979), 30-47. doi: 10.1016/0146-664X(79)90033-9.
[18]	Q. Huynh-Thu and M. Ghanbari, Scope of validity of psnr in image/video quality assessment, Electronics Letters, 44 (2008), 800-801. doi: 10.1049/el:20080522.
[19]	F. Jacobs, E. Sundermann, B. De Sutter, M. Christiaens and I. Lemahieu, A fast algorithm to calculate the exact radiological path through a pixel or voxel space, Journal of Computing and Information Technology, 6 (1998), 89-94.
[20]	M. Jiang and G. Wang, Convergence of the simultaneous algebraic reconstruction technique (SART), IEEE Transactions on Image Processing, 12 (2003), 957-961. doi: 10.1109/TIP.2003.815295.
[21]	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.
[22]	J. Kaipio and E. Somersalo, Statistical inverse problems: discretization, model reduction and inverse crimes, Journal of Computational and Applied Mathematics, 198 (2007), 493-504. doi: 10.1016/j.cam.2005.09.027.
[23]	H. Kudo and T. Saito, Sinogram recovery with the method of convex projections for limited-data reconstruction in computed tomography, Journal of the Optical Society of America A, 8 (1991), 1148-1160.
[24]	X. Li, Z. Zhu, A. M.-C. So and J. D. Lee, Incremental methods for weakly convex optimization, arXiv: 1907.11687.
[25]	A. K. Louis, Incomplete data problems in X-ray computerized tomography, Numerische Mathematik, 48 (1986), 251-262. doi: 10.1007/BF01389474.
[26]	L. T. Niklason, B. T. Christian, L. E. Niklason, D. B. Kopans, D. E. Castleberry, B. H. Opsahl-Ong, C. E. Landberg, P. J. Slanetz, A. A. Giardino and R. Moore, et al., Digital tomosynthesis in breast imaging., Radiology, 205 (1997), 399-406. doi: 10.1148/radiology.205.2.9356620.
[27]	E. T. Quinto, Artifacts and visible singularities in limited data X-ray tomography, Sensing and Imaging, 18 (2017), 9. doi: 10.1007/s11220-017-0158-7.
[28]	N. A. B. Riis, J. Frøsig, Y. Dong and P. C. Hansen, Limited-data x-ray CT for underwater pipeline inspection, Inverse Problems, 34 (2018), 034002, 16 pp. doi: 10.1088/1361-6420/aaa49c.
[29]	E. Y. Sidky, C.-M. Kao and X. Pan, Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT, Journal of X-ray Science and Technology, 14 (2006), 119-139.
[30]	S. Tan, Y. Zhang, G. Wang, X. Mou, G. Cao, Z. Wu and H. Yu, Tensor-based dictionary learning for dynamic tomographic reconstruction, Physics in Medicine & Biology, 60 (2015), 2803. doi: 10.1088/0031-9155/60/7/2803.
[31]	L. H. Thomas, Elliptic problems in linear difference equations over a network, Watson Sci. Comput. Lab. Rept., Columbia University, New York, 1.
[32]	C. Wang, L. Zeng, Y. Guo and L. Zhang, Wavelet tight frame and prior image-based image reconstruction from limited-angle projection data, Inverse Problems & Imaging, 11 (2017), 917-948. doi: 10.3934/ipi.2017043.
[33]	T. Wang, K. Nakamoto, H. Zhang and H. Liu, Reweighted anisotropic total variation minimization for limited-angle CT reconstruction, IEEE Transactions on Nuclear Science, 64 (2017), 2742-2760. doi: 10.1109/TNS.2017.2750199.
[34]	Z. Wang and A. C. Bovik, A universal image quality index, IEEE Signal Processing Letters, 9 (2002), 81-84.
[35]	Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861.
[36]	J. Woodworth and R. Chartrand, Compressed sensing recovery via nonconvex shrinkage penalties, Inverse Problems, 32 (2016), 075004, 25 pp. doi: 10.1088/0266-5611/32/7/075004.
[37]	C. Wu and X.-C. Tai, Augmented lagrangian method, dual methods, and split bregman iteration for ROF, vectorial TV, and high order models, SIAM Journal on Imaging Sciences, 3 (2010), 300-339. doi: 10.1137/090767558.
[38]	D. Wu and L. Zeng, Limited-angle reverse helical cone-beam CT for pipeline with low rank decomposition, Optics Communications, 328 (2014), 109-115. doi: 10.1016/j.optcom.2014.04.077.
[39]	J. Xu, Y. Zhao, H. Li and P. Zhang, An image reconstruction model regularized by edge-preserving diffusion and smoothing for limited-angle computed tomography, Inverse Problems, 35 (2019), 085004, 34 pp. doi: 10.1088/1361-6420/ab08f9.
[40]	L. Xu, C. Lu, Y. Xu and J. Jia, Image smoothing via L0 gradient minimization, ACM Transactions on Graphics (TOG), 30 (2011), 174.
[41]	X. Xue, S. Zhao, Y. Zhao and P. Zhang, Image reconstruction for limited-angle computed tomography with curvature constraint, Measurement Science and Technology, 30 (2019), 125401.
[42]	L. Zhang, L. Zeng and Y. Guo, $l_0$ regularization based on a prior image incorporated non-local means for limited-angle x-ray ct reconstruction, Journal of X-ray science and technology, 26 (2018), 481-498. doi: 10.3233/XST-17334.
[43]	J. Zhao, Z. Chen, L. Zhang and X. Jin, Unsupervised learnable sinogram inpainting network (SIN) for limited angle CT reconstruction, arXiv: 1811.03911.