American Institute of Mathematical Sciences

Advanced
doi: 10.3934/ipi.2021019

Limited-angle CT reconstruction with generalized shrinkage operators as regularizers

Xiaojuan Deng 1,2, , Xing Zhao 1,2, , Mengfei Li 3, and  Hongwei Li 1,2,,

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

Received  January 2020 Revised  September 2020 Published  February 2021

Fund Project: 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

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.

Keywords: Limited-angle, CT reconstruction, visible boundary, generalized shrinkage operator, regularizer, alternating minimization.
Mathematics Subject Classification: Primary: 65K10, 47A52; Secondary: 94A08.
Citation: Xiaojuan Deng, Xing Zhao, Mengfei Li, Hongwei Li. Limited-angle CT reconstruction with generalized shrinkage operators as regularizers. Inverse Problems & Imaging, doi: 10.3934/ipi.2021019
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.  Google Scholar

[2]

D. P. Bertsekas, Incremental gradient, subgradient, and proximal methods for convex optimization: A survey, Optimization for Machine Learning, 2010 (2011), 3. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

R. Chartrand, Shrinkage mappings and their induced penalty functions, in International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, 2014, 1026–1029. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[24]

X. Li, Z. Zhu, A. M.-C. So and J. D. Lee, Incremental methods for weakly convex optimization, arXiv: 1907.11687. Google Scholar

[25]

A. K. Louis, Incomplete data problems in X-ray computerized tomography, Numerische Mathematik, 48 (1986), 251-262.  doi: 10.1007/BF01389474.  Google Scholar

[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, Digital tomosynthesis in breast imaging., Radiology, 205 (1997), 399-406.  doi: 10.1148/radiology.205.2.9356620.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[31]

L. H. Thomas, Elliptic problems in linear difference equations over a network, Watson Sci. Comput. Lab. Rept., Columbia University, New York, 1. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

Z. Wang and A. C. Bovik, A universal image quality index, IEEE Signal Processing Letters, 9 (2002), 81-84.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

L. Xu, C. Lu, Y. Xu and J. Jia, Image smoothing via L0 gradient minimization, ACM Transactions on Graphics (TOG), 30 (2011), 174. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[43]

J. Zhao, Z. Chen, L. Zhang and X. Jin, Unsupervised learnable sinogram inpainting network (SIN) for limited angle CT reconstruction, arXiv: 1811.03911. Google Scholar

show all references

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.  Google Scholar

[2]

D. P. Bertsekas, Incremental gradient, subgradient, and proximal methods for convex optimization: A survey, Optimization for Machine Learning, 2010 (2011), 3. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

R. Chartrand, Shrinkage mappings and their induced penalty functions, in International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, 2014, 1026–1029. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[24]

X. Li, Z. Zhu, A. M.-C. So and J. D. Lee, Incremental methods for weakly convex optimization, arXiv: 1907.11687. Google Scholar

[25]

A. K. Louis, Incomplete data problems in X-ray computerized tomography, Numerische Mathematik, 48 (1986), 251-262.  doi: 10.1007/BF01389474.  Google Scholar

[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, Digital tomosynthesis in breast imaging., Radiology, 205 (1997), 399-406.  doi: 10.1148/radiology.205.2.9356620.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[31]

L. H. Thomas, Elliptic problems in linear difference equations over a network, Watson Sci. Comput. Lab. Rept., Columbia University, New York, 1. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

Z. Wang and A. C. Bovik, A universal image quality index, IEEE Signal Processing Letters, 9 (2002), 81-84.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

L. Xu, C. Lu, Y. Xu and J. Jia, Image smoothing via L0 gradient minimization, ACM Transactions on Graphics (TOG), 30 (2011), 174. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[43]

J. Zhao, Z. Chen, L. Zhang and X. Jin, Unsupervised learnable sinogram inpainting network (SIN) for limited angle CT reconstruction, arXiv: 1811.03911. Google Scholar

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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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
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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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 $
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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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 $
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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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 $
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 $)
PSNR 16.886 51.7035 45.4061 53.4988
SSIM 0.6822 0.9755 0.9710 0.9808
UQI 0.8155 0.9999 0.9998 0.9999
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SART AEDS($ \ell_0, \ell_0 $) AEDS($ \ell_0, \ell_1 $) GAEDS($ p, q $)
PSNR 16.886 51.7035 45.4061 53.4988
SSIM 0.6822 0.9755 0.9710 0.9808
UQI 0.8155 0.9999 0.9998 0.9999
Table 6.  Reconstruction parameters used by the competing algorithms for the rhombus phantom
AEDS($ \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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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 $
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 $)
150 degree PSNR 31.7800 34.4700 35.7580 35.5110
SSIM 0.99366 0.99496 0.99616 0.99615
UQI 0.99030 0.99477 0.99609 0.99590
140 degree PSNR 29.4930 33.7360 33.5110 34.1700
SSIM 0.98854 0.99426 0.99330 0.99361
UQI 0.98333 0.99382 0.99339 0.99433
130 degree PSNR 27.2680 32.7270 32.2330 33.7820
SSIM 0.98347 0.99272 0.98837 0.99425
UQI 0.97162 0.99218 0.99105 0.99388
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SART AEDS($ \ell_0, \ell_0 $) AEDS($ \ell_0, \ell_1 $) GAEDS($ p, q $)
150 degree PSNR 31.7800 34.4700 35.7580 35.5110
SSIM 0.99366 0.99496 0.99616 0.99615
UQI 0.99030 0.99477 0.99609 0.99590
140 degree PSNR 29.4930 33.7360 33.5110 34.1700
SSIM 0.98854 0.99426 0.99330 0.99361
UQI 0.98333 0.99382 0.99339 0.99433
130 degree PSNR 27.2680 32.7270 32.2330 33.7820
SSIM 0.98347 0.99272 0.98837 0.99425
UQI 0.97162 0.99218 0.99105 0.99388
Table 8.  Geometrical parameters of the real scanning device
Parameter Value
Voltage 140 kV
Current 160 mA
Distance of X-ray source to rotation center 311.49 mm
Width of detector unit 0.127 mm
Number of detector units 1920
Distance of rotation center to detector 697.82 mm
Scanning angular interval 0.2 degree
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Parameter Value
Voltage 140 kV
Current 160 mA
Distance of X-ray source to rotation center 311.49 mm
Width of detector unit 0.127 mm
Number of detector units 1920
Distance of rotation center to detector 697.82 mm
Scanning angular interval 0.2 degree
Table 9.  Reconstruction parameters for the real data test
AEDS($ \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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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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