• Previous Article
    Nonlocal regularized CNN for image segmentation
  • IPI Home
  • This Issue
  • Next Article
    Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements
October  2020, 14(5): 867-890. doi: 10.3934/ipi.2020040

A nonconvex truncated regularization and box-constrained model for CT reconstruction

1. 

Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, China

2. 

School of Mathematical Sciences, Nankai University, Tianjin 300071, China

* Corresponding author: Yiming Gao

Received  January 2020 Revised  May 2020 Published  July 2020

Fund Project: This work was supported by Zhejiang Provincial Natural Science Foundation of China (No. LQ20A010007) the National Natural Science Foundation of China (NSFC) (Nos. 11871035, 11531013, 11971138), Recruitment Program of Global Young Expert, and Postdoctoral Science Foundation of China (No. 2019M651002)

X-ray computed tomography has been a useful technology in cancer detection and radiation therapy. However, high radiation dose during CT scans may increase the underlying risk of healthy organs. Usually, sparse-view X-ray projection is an effective method to reduce radiation. In this paper, we propose a constrained nonconvex truncated regularization model for this low-dose CT reconstruction. It preserves sharp edges very well. Although this model is quite complicated to analyze, we establish two useful theoretical results for its minimizers. Motivated by them, an iterative support shrinking algorithm is introduced. To handle more nondifferentiable points of the regularization function except zero point, we use a general proximally linearization technique at them, which is helpful to implement our algorithm. For this algorithm, we prove the convergence of the objective function, and give a lower bound theory of the iterative sequence. Numerical experiments and comparisons demonstrate that our model with the proposed algorithm performs good for low-dose CT reconstruction.

Citation: Weina Wang, Chunlin Wu, Yiming Gao. A nonconvex truncated regularization and box-constrained model for CT reconstruction. Inverse Problems & Imaging, 2020, 14 (5) : 867-890. doi: 10.3934/ipi.2020040
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.   Google Scholar

[2]

C. BaoB. DongL. HouZ. ShenX. Zhang and X. Zhang, Image restoration by minimizing zero norm of wavelet frame coefficients, Inverse Problems, 32 (2016), 115-142.  doi: 10.1088/0266-5611/32/11/115004.  Google Scholar

[3]

M. BhatiaW. C. Karl and A. S. Willsky, A wavelet-based method for multiscale tomographic reconstruction, IEEE Transactions on Medical Imaging, 15 (1996), 92-101.  doi: 10.21236/ADA459987.  Google Scholar

[4]

E. G. BirginJ. M. Martínez and M. Raydan, Nonmonotone spectral projected gradient methods on convex sets, SIAM Journal on Optimization, 10 (2000), 1196-1211.  doi: 10.1137/S1052623497330963.  Google Scholar

[5]

D. J. Brenner and E. J. Hall, Computed Tomography–an increasing source of radiation exposure, New England Journal of Medicine, 357 (2007), 2277-2284.  doi: 10.1056/NEJMra072149.  Google Scholar

[6]

A. CaiL. WangB. YanL. LiH. Zhang and G. Hu, Efficient TpV minimization for circular, cone-beam computed tomography reconstruction via non-convex optimization, Computerized Medical Imaging and Graphics, 45 (2015), 1-10.  doi: 10.1016/j.compmedimag.2015.06.004.  Google Scholar

[7]

R. H. ChanM. Tao and X. Yuan, Constrained total variation deblurring models and fast algorithms based on alternating direction method of multipliers, SIAM Journal on Imaging Sciences, 6 (2013), 680-697.  doi: 10.1137/110860185.  Google Scholar

[8]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Mathematical Programming, 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

[9]

X. ChenM. K. Ng and C. Zhang, Non-Lipschitz $\ell_{p}$-Regularization and Box Constrained Model for Image Restoration, IEEE Transactions on Image Processing, 21 (2012), 4709-4721.  doi: 10.1109/TIP.2012.2214051.  Google Scholar

[10]

X. ChenF. Xu and Y. Ye, Lower bound theory of nonzero entries in solutions of $\ell_{2}$-$\ell_{p}$ minimization, SIAM Journal on Scientific Computing, 32 (2010), 2832-2852.  doi: 10.1137/090761471.  Google Scholar

[11]

Z. ChenX. JinL. Li and G. Wang, A limited-angle CT reconstruction method based on anisotropic TV minimization, Physics in Medicine and Biology, 58 (2013), 2119-2141.  doi: 10.1088/0031-9155/58/7/2119.  Google Scholar

[12]

J. K. ChoiB. Dong and X. Zhang, Limited tomography reconstruction via tight frame and simultaneous sinogram extrapolation, J. Comput. Math., 34 (2016), 575-589.  doi: 10.4208/jcm.1605-m2016-0535.  Google Scholar

[13]

B. DongJ. Li and Z. Shen, X-Ray CT image reconstruction via wavelet frame based regularization and radon domain inpainting, J. Sci. Comput., 54 (2013), 333-349.  doi: 10.1007/s10915-012-9579-6.  Google Scholar

[14]

G. Edgar and T. H. Herman, X-Ray CT image reconstruction via wavelet frame based regularization and radon domain inpainting, Inverse Problems, 33 (2017), 185-208.   Google Scholar

[15]

A. J. EinsteinM. J. Henzlova and R. Sanjay, Estimating risk of cancer associated with radiation exposure from 64-slice computed tomography coronary angiography, Journal of American Medical Association, 298 (2007), 317-323.  doi: 10.1001/jama.298.3.317.  Google Scholar

[16]

E. Y. SidkyR. ChartrandJ. M. Boone and X. Pan, Constrained TpV minimization for enhanced exploitation of gradient sparsity: Application to CT image reconstruction, IEEE Journal of Translational Engineering in Health and Medicine, 2 (2014), 1-18.   Google Scholar

[17]

H. GaoR. LiY. Lin and L. Xing, 4D cone beam CT via spatiotemporal tensor framelet, Medical Physics, 39 (2012), 6943-6946.  doi: 10.1118/1.4762288.  Google Scholar

[18]

Y. Gao and C. Wu, On a general smoothly truncated regularization for variational piecewise constant image restoration: Construction and convergent algorithms, Inverse Problems, 36 (2020), 1-31.  doi: 10.1088/1361-6420/ab6619.  Google Scholar

[19]

R. GordonR. Bender and G. T. Herman, Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography, Journal of Theoretical Biology, 29 (1970), 471-476.  doi: 10.1016/0022-5193(70)90109-8.  Google Scholar

[20]

M. Hintermüller and T. Wu, Nonconvex $TV^{q}$-models in image restoration: Analysis and a trust-region regularization based superlinearly convergent solver, SIAM Journal on Imaging Sciences, 6 (2013), 1385-1415.  doi: 10.1137/110854746.  Google Scholar

[21]

X. JiaB. DongY. Lou and S. B. Jiang, GPU-based iterative cone-beam CT reconstruction using tight frame regularization, Physics in Medicine and Biology, 56 (2011), 3787-3801.  doi: 10.1088/0031-9155/56/13/004.  Google Scholar

[22]

X. JiaY. LouR. LiW. Y. William and S. B. Jiang, GPU-based fast cone beam CT reconstruction from undersampled and noisy projection data via total variation, Medical Physics, 37 (2010), 1757-1760.  doi: 10.1118/1.3371691.  Google Scholar

[23]

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

[24]

H. KimJ. ChenA. WangC. ChuangM. Held and J. Pouliot, Non-local total-variation (NLTV) minimization combined with reweighted L1-norm for compressed sensing CT reconstruction, Physics in Medicine and Biology, 61 (2016), 6878-6891.  doi: 10.1088/0031-9155/61/18/6878.  Google Scholar

[25]

J. K. KimJ. A. Fessler and Z. Zhang, Forward-projection architecture for fast iterative image reconstruction in X-ray CT, IEEE Transactions on Signal Processing, 60 (2012), 5508-5518.  doi: 10.1109/TSP.2012.2208636.  Google Scholar

[26]

J. W. KressL. A. Feldkamp and L. C. Davis, Practical cone-beam algorithm, Journal of the Optical Society of American, 1 (1984), 612-619.   Google Scholar

[27]

M. LaiY. Xu and W. Yin, Improved iteratively reweighted least squares for unconstrained smoothed $\ell_{q}$ minimization, SIAM Journal on Numerical Analysis, 51 (2013), 927-957.  doi: 10.1137/110840364.  Google Scholar

[28]

Y. LiuZ. LiangJ. MaH. LuK. WangH. Zhang and W. Moore, Total variation-stokes strategy for sparse-view X-ray CT image reconstruction, IEEE Transactions on Medical Imaging, 33 (2014), 749-763.   Google Scholar

[29]

Y. LiuJ. MaY. Fan and Z. Liang, Adaptive-weighted total variation minimization for sparse data toward low-dose x-ray computed tomography image reconstruction, Physics in Medicine and Biology, 57 (2012), 7923-7956.  doi: 10.1088/0031-9155/57/23/7923.  Google Scholar

[30]

M. Nikolova, Analysis of the Recovery of Edges in Images and Signals by Minimizing Nonconvex Regularized Least-Squares, SIAM Journal on Multiscale Modeling and Simulation, 4 (2005), 960-991.  doi: 10.1137/040619582.  Google Scholar

[31]

M. NikolovaM. K. Ng and C. P. Tam, Fast Nonconvex Nonsmooth Minimization Methods for Image Restoration and Reconstruction, IEEE Transactions on image processing, 19 (2010), 3073-3088.  doi: 10.1109/TIP.2010.2052275.  Google Scholar

[32]

M. NikolovaM. K. NgS. Zhang and W. K. Ching, Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization, SIAM Journal on Imaging Sciences, 1 (2008), 2-25.  doi: 10.1137/070692285.  Google Scholar

[33]

S. NiuJ. HuangZ. BianD. ZengW. ChenG. YuZ. Liang and J. Ma, Iterative reconstruction for sparse-view x-ray CT using alpha-divergence constrained total generalized variation minimization, Journal of X-ray Science and Technology, 25 (2017), 673-688.  doi: 10.3233/XST-16239.  Google Scholar

[34]

S. Ramani and J. A. Fessler, A splitting-based iterative algorithm for accelerated statistical X-ray CT reconstruction, IEEE Transactions on Medical Imaging, 31 (2012), 677-688.  doi: 10.1109/TMI.2011.2175233.  Google Scholar

[35]

M. RantalaS. VanskaS. JarvenpaaM. KalkeM. LassasJ. Moberg and S. Siltanen, Wavelet-based reconstruction for limited-angle X-ray tomography, IEEE Transactions on Medical Imaging, 25 (2006), 210-217.  doi: 10.1109/TMI.2005.862206.  Google Scholar

[36]

L. I. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[37]

W. P. SegarsM. MaheshT. J. BeckE. C. Frey and B. M. Tsui, Realistic CT simulation using the 4D XCAT phantom, Medical Physics, 35 (2008), 3800-3808.  doi: 10.1118/1.2955743.  Google Scholar

[38]

E. Y. Sidky and X. Pan, Image reconstruction in circular cone-beam CT by constrained, total-variation minimization, Physics in Medicine and Biology, 153 (2008), 4777-4807.   Google Scholar

[39]

J. WangT. Li and L. Xing, Iterative image reconstruction for CBCT using edge-preserving prior, Medical Physics, 36 (2009), 252-260.  doi: 10.1118/1.3036112.  Google Scholar

[40]

W. Wang, C. Wu and X. Tai, A globally convergent algorithm for a constrained non-Lipschitz image restoration model, Journal of Scientific Computing, (2020). doi: 10.1007/s10915-020-01190-4.  Google Scholar

[41]

Z. WangA. C. BovikH. 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

[42]

C. WuZ. Liu and S. Wen, A general truncated regularization framework for contrast-preserving variational signal and image restoration: Motivation and implementation, Science China Mathematics, 61 (2018), 1711-1732.  doi: 10.1007/s11425-017-9260-8.  Google Scholar

[43]

C. Wu and X. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM J. Imaging Sci., 3 (2010), 300-339.  doi: 10.1137/090767558.  Google Scholar

[44]

Q. XuH. YuX. MouL. ZhangH. Jiang and G. Wang, Low-dose X-ray CT reconstruction via dictionary learning, IEEE Transactions on Medical Imaging, 31 (2012), 1682-1697.   Google Scholar

[45]

Y. Xu and W. Yin, A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion, SIAM Journal on Imaging Sciences, 6 (2013), 1758-1789.  doi: 10.1137/120887795.  Google Scholar

[46]

C. ZengR. Jia and C. Wu, An iterative support shrinking algorithm for non-Lipschitz optimization in image restoration, Journal of Mathematical Imaging and Vision, 61 (2018), 122-139.  doi: 10.1007/s10851-018-0830-0.  Google Scholar

[47]

C. Zeng and C. Wu, On the discontinuity of images recovered by noncovex nonsmooth regularized isotropic models with box constraints, Advances in Computational Mathematics, 45 (2018), 589-610.  doi: 10.1007/s10444-018-9629-1.  Google Scholar

[48]

C. Zeng and C. Wu, On the edge recovery property of noncovex nonsmooth regularization in image restoration, SIAM Journal Numerical Analysis, 56 (2018), 1168-1182.  doi: 10.1137/17M1123687.  Google Scholar

[49]

R. Zhan and B. Dong, CT image reconstruction by spatial-radon domain data-driven tight frame regularization, SIAM Journal on Imaging Sciences, 9 (2016), 1063-1083.  doi: 10.1137/16M105928X.  Google Scholar

[50]

X. ZhangM. Bai and M. K. Ng, Nonconvex-TV based image restoration with impulse noise removal, SIAM Journal on Imaging Sciences, 10 (2017), 1627-1667.  doi: 10.1137/16M1076034.  Google Scholar

[51]

Y. ZhangB. Dong and Z. Lu, An efficient algorithm for $\ell_{0}$ minimization in wavelet frame based image restoration, Mathematics of Computation, 82 (2013), 995-1015.  doi: 10.1090/S0025-5718-2012-02631-7.  Google Scholar

[52]

W. ZhouJ. Cai and H. Gao, Adaptive tight frame based medical image reconstruction: A proof-of-concept study for computed tomography, Inverse Problems, 29 (2013), 125-146.  doi: 10.1088/0266-5611/29/12/125006.  Google Scholar

[53]

L. ZhuT. Niu and M. Petrongolo, Iterative CT reconstruction via minimizing adaptively reweighted total variation, Journal of X-ray Science and Technology, 22 (2014), 227-240.  doi: 10.3233/XST-140421.  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.   Google Scholar

[2]

C. BaoB. DongL. HouZ. ShenX. Zhang and X. Zhang, Image restoration by minimizing zero norm of wavelet frame coefficients, Inverse Problems, 32 (2016), 115-142.  doi: 10.1088/0266-5611/32/11/115004.  Google Scholar

[3]

M. BhatiaW. C. Karl and A. S. Willsky, A wavelet-based method for multiscale tomographic reconstruction, IEEE Transactions on Medical Imaging, 15 (1996), 92-101.  doi: 10.21236/ADA459987.  Google Scholar

[4]

E. G. BirginJ. M. Martínez and M. Raydan, Nonmonotone spectral projected gradient methods on convex sets, SIAM Journal on Optimization, 10 (2000), 1196-1211.  doi: 10.1137/S1052623497330963.  Google Scholar

[5]

D. J. Brenner and E. J. Hall, Computed Tomography–an increasing source of radiation exposure, New England Journal of Medicine, 357 (2007), 2277-2284.  doi: 10.1056/NEJMra072149.  Google Scholar

[6]

A. CaiL. WangB. YanL. LiH. Zhang and G. Hu, Efficient TpV minimization for circular, cone-beam computed tomography reconstruction via non-convex optimization, Computerized Medical Imaging and Graphics, 45 (2015), 1-10.  doi: 10.1016/j.compmedimag.2015.06.004.  Google Scholar

[7]

R. H. ChanM. Tao and X. Yuan, Constrained total variation deblurring models and fast algorithms based on alternating direction method of multipliers, SIAM Journal on Imaging Sciences, 6 (2013), 680-697.  doi: 10.1137/110860185.  Google Scholar

[8]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Mathematical Programming, 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

[9]

X. ChenM. K. Ng and C. Zhang, Non-Lipschitz $\ell_{p}$-Regularization and Box Constrained Model for Image Restoration, IEEE Transactions on Image Processing, 21 (2012), 4709-4721.  doi: 10.1109/TIP.2012.2214051.  Google Scholar

[10]

X. ChenF. Xu and Y. Ye, Lower bound theory of nonzero entries in solutions of $\ell_{2}$-$\ell_{p}$ minimization, SIAM Journal on Scientific Computing, 32 (2010), 2832-2852.  doi: 10.1137/090761471.  Google Scholar

[11]

Z. ChenX. JinL. Li and G. Wang, A limited-angle CT reconstruction method based on anisotropic TV minimization, Physics in Medicine and Biology, 58 (2013), 2119-2141.  doi: 10.1088/0031-9155/58/7/2119.  Google Scholar

[12]

J. K. ChoiB. Dong and X. Zhang, Limited tomography reconstruction via tight frame and simultaneous sinogram extrapolation, J. Comput. Math., 34 (2016), 575-589.  doi: 10.4208/jcm.1605-m2016-0535.  Google Scholar

[13]

B. DongJ. Li and Z. Shen, X-Ray CT image reconstruction via wavelet frame based regularization and radon domain inpainting, J. Sci. Comput., 54 (2013), 333-349.  doi: 10.1007/s10915-012-9579-6.  Google Scholar

[14]

G. Edgar and T. H. Herman, X-Ray CT image reconstruction via wavelet frame based regularization and radon domain inpainting, Inverse Problems, 33 (2017), 185-208.   Google Scholar

[15]

A. J. EinsteinM. J. Henzlova and R. Sanjay, Estimating risk of cancer associated with radiation exposure from 64-slice computed tomography coronary angiography, Journal of American Medical Association, 298 (2007), 317-323.  doi: 10.1001/jama.298.3.317.  Google Scholar

[16]

E. Y. SidkyR. ChartrandJ. M. Boone and X. Pan, Constrained TpV minimization for enhanced exploitation of gradient sparsity: Application to CT image reconstruction, IEEE Journal of Translational Engineering in Health and Medicine, 2 (2014), 1-18.   Google Scholar

[17]

H. GaoR. LiY. Lin and L. Xing, 4D cone beam CT via spatiotemporal tensor framelet, Medical Physics, 39 (2012), 6943-6946.  doi: 10.1118/1.4762288.  Google Scholar

[18]

Y. Gao and C. Wu, On a general smoothly truncated regularization for variational piecewise constant image restoration: Construction and convergent algorithms, Inverse Problems, 36 (2020), 1-31.  doi: 10.1088/1361-6420/ab6619.  Google Scholar

[19]

R. GordonR. Bender and G. T. Herman, Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography, Journal of Theoretical Biology, 29 (1970), 471-476.  doi: 10.1016/0022-5193(70)90109-8.  Google Scholar

[20]

M. Hintermüller and T. Wu, Nonconvex $TV^{q}$-models in image restoration: Analysis and a trust-region regularization based superlinearly convergent solver, SIAM Journal on Imaging Sciences, 6 (2013), 1385-1415.  doi: 10.1137/110854746.  Google Scholar

[21]

X. JiaB. DongY. Lou and S. B. Jiang, GPU-based iterative cone-beam CT reconstruction using tight frame regularization, Physics in Medicine and Biology, 56 (2011), 3787-3801.  doi: 10.1088/0031-9155/56/13/004.  Google Scholar

[22]

X. JiaY. LouR. LiW. Y. William and S. B. Jiang, GPU-based fast cone beam CT reconstruction from undersampled and noisy projection data via total variation, Medical Physics, 37 (2010), 1757-1760.  doi: 10.1118/1.3371691.  Google Scholar

[23]

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

[24]

H. KimJ. ChenA. WangC. ChuangM. Held and J. Pouliot, Non-local total-variation (NLTV) minimization combined with reweighted L1-norm for compressed sensing CT reconstruction, Physics in Medicine and Biology, 61 (2016), 6878-6891.  doi: 10.1088/0031-9155/61/18/6878.  Google Scholar

[25]

J. K. KimJ. A. Fessler and Z. Zhang, Forward-projection architecture for fast iterative image reconstruction in X-ray CT, IEEE Transactions on Signal Processing, 60 (2012), 5508-5518.  doi: 10.1109/TSP.2012.2208636.  Google Scholar

[26]

J. W. KressL. A. Feldkamp and L. C. Davis, Practical cone-beam algorithm, Journal of the Optical Society of American, 1 (1984), 612-619.   Google Scholar

[27]

M. LaiY. Xu and W. Yin, Improved iteratively reweighted least squares for unconstrained smoothed $\ell_{q}$ minimization, SIAM Journal on Numerical Analysis, 51 (2013), 927-957.  doi: 10.1137/110840364.  Google Scholar

[28]

Y. LiuZ. LiangJ. MaH. LuK. WangH. Zhang and W. Moore, Total variation-stokes strategy for sparse-view X-ray CT image reconstruction, IEEE Transactions on Medical Imaging, 33 (2014), 749-763.   Google Scholar

[29]

Y. LiuJ. MaY. Fan and Z. Liang, Adaptive-weighted total variation minimization for sparse data toward low-dose x-ray computed tomography image reconstruction, Physics in Medicine and Biology, 57 (2012), 7923-7956.  doi: 10.1088/0031-9155/57/23/7923.  Google Scholar

[30]

M. Nikolova, Analysis of the Recovery of Edges in Images and Signals by Minimizing Nonconvex Regularized Least-Squares, SIAM Journal on Multiscale Modeling and Simulation, 4 (2005), 960-991.  doi: 10.1137/040619582.  Google Scholar

[31]

M. NikolovaM. K. Ng and C. P. Tam, Fast Nonconvex Nonsmooth Minimization Methods for Image Restoration and Reconstruction, IEEE Transactions on image processing, 19 (2010), 3073-3088.  doi: 10.1109/TIP.2010.2052275.  Google Scholar

[32]

M. NikolovaM. K. NgS. Zhang and W. K. Ching, Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization, SIAM Journal on Imaging Sciences, 1 (2008), 2-25.  doi: 10.1137/070692285.  Google Scholar

[33]

S. NiuJ. HuangZ. BianD. ZengW. ChenG. YuZ. Liang and J. Ma, Iterative reconstruction for sparse-view x-ray CT using alpha-divergence constrained total generalized variation minimization, Journal of X-ray Science and Technology, 25 (2017), 673-688.  doi: 10.3233/XST-16239.  Google Scholar

[34]

S. Ramani and J. A. Fessler, A splitting-based iterative algorithm for accelerated statistical X-ray CT reconstruction, IEEE Transactions on Medical Imaging, 31 (2012), 677-688.  doi: 10.1109/TMI.2011.2175233.  Google Scholar

[35]

M. RantalaS. VanskaS. JarvenpaaM. KalkeM. LassasJ. Moberg and S. Siltanen, Wavelet-based reconstruction for limited-angle X-ray tomography, IEEE Transactions on Medical Imaging, 25 (2006), 210-217.  doi: 10.1109/TMI.2005.862206.  Google Scholar

[36]

L. I. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[37]

W. P. SegarsM. MaheshT. J. BeckE. C. Frey and B. M. Tsui, Realistic CT simulation using the 4D XCAT phantom, Medical Physics, 35 (2008), 3800-3808.  doi: 10.1118/1.2955743.  Google Scholar

[38]

E. Y. Sidky and X. Pan, Image reconstruction in circular cone-beam CT by constrained, total-variation minimization, Physics in Medicine and Biology, 153 (2008), 4777-4807.   Google Scholar

[39]

J. WangT. Li and L. Xing, Iterative image reconstruction for CBCT using edge-preserving prior, Medical Physics, 36 (2009), 252-260.  doi: 10.1118/1.3036112.  Google Scholar

[40]

W. Wang, C. Wu and X. Tai, A globally convergent algorithm for a constrained non-Lipschitz image restoration model, Journal of Scientific Computing, (2020). doi: 10.1007/s10915-020-01190-4.  Google Scholar

[41]

Z. WangA. C. BovikH. 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

[42]

C. WuZ. Liu and S. Wen, A general truncated regularization framework for contrast-preserving variational signal and image restoration: Motivation and implementation, Science China Mathematics, 61 (2018), 1711-1732.  doi: 10.1007/s11425-017-9260-8.  Google Scholar

[43]

C. Wu and X. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM J. Imaging Sci., 3 (2010), 300-339.  doi: 10.1137/090767558.  Google Scholar

[44]

Q. XuH. YuX. MouL. ZhangH. Jiang and G. Wang, Low-dose X-ray CT reconstruction via dictionary learning, IEEE Transactions on Medical Imaging, 31 (2012), 1682-1697.   Google Scholar

[45]

Y. Xu and W. Yin, A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion, SIAM Journal on Imaging Sciences, 6 (2013), 1758-1789.  doi: 10.1137/120887795.  Google Scholar

[46]

C. ZengR. Jia and C. Wu, An iterative support shrinking algorithm for non-Lipschitz optimization in image restoration, Journal of Mathematical Imaging and Vision, 61 (2018), 122-139.  doi: 10.1007/s10851-018-0830-0.  Google Scholar

[47]

C. Zeng and C. Wu, On the discontinuity of images recovered by noncovex nonsmooth regularized isotropic models with box constraints, Advances in Computational Mathematics, 45 (2018), 589-610.  doi: 10.1007/s10444-018-9629-1.  Google Scholar

[48]

C. Zeng and C. Wu, On the edge recovery property of noncovex nonsmooth regularization in image restoration, SIAM Journal Numerical Analysis, 56 (2018), 1168-1182.  doi: 10.1137/17M1123687.  Google Scholar

[49]

R. Zhan and B. Dong, CT image reconstruction by spatial-radon domain data-driven tight frame regularization, SIAM Journal on Imaging Sciences, 9 (2016), 1063-1083.  doi: 10.1137/16M105928X.  Google Scholar

[50]

X. ZhangM. Bai and M. K. Ng, Nonconvex-TV based image restoration with impulse noise removal, SIAM Journal on Imaging Sciences, 10 (2017), 1627-1667.  doi: 10.1137/16M1076034.  Google Scholar

[51]

Y. ZhangB. Dong and Z. Lu, An efficient algorithm for $\ell_{0}$ minimization in wavelet frame based image restoration, Mathematics of Computation, 82 (2013), 995-1015.  doi: 10.1090/S0025-5718-2012-02631-7.  Google Scholar

[52]

W. ZhouJ. Cai and H. Gao, Adaptive tight frame based medical image reconstruction: A proof-of-concept study for computed tomography, Inverse Problems, 29 (2013), 125-146.  doi: 10.1088/0266-5611/29/12/125006.  Google Scholar

[53]

L. ZhuT. Niu and M. Petrongolo, Iterative CT reconstruction via minimizing adaptively reweighted total variation, Journal of X-ray Science and Technology, 22 (2014), 227-240.  doi: 10.3233/XST-140421.  Google Scholar

Figure 1.  Three different $ \varphi(t) $. (a) $ \varphi(t) = t,\ 0 \leq t \leq 0.5; \ \varphi(t) = 0.5,\ t > 0.5 $. (b) $ \varphi(t) = t^{0.5},\ 0 \leq t \leq 0.5; \ \varphi(t) = 0.5^{0.5},\ t > 0.5 $. (c) $ \varphi(t) = \mathrm{log}(t+1),\ 0 \leq t \leq 1; \ \varphi(t) = \mathrm{log}(t^{0.5}+1),\ 1 < t \leq 2; \ \varphi(t) = \mathrm{log}(2^{0.5}+1),\ t >2 $
Figure 2.  Test images. (a) $ 128\times 128 $; (b) $ 256\times 256 $
Figure 3.  The visualization of PSNR values versus parameters $ a $ and $ \tau $ of trunc-LN, $ p $ and $ \tau $ of trunc-$ \ell_p $ for "Shepp-Logan'' image with 36 projections
Figure 4.  CT reconstruction comparisons for "Shepp-Logan'' image with $ \sigma = 0.01\|g\|_\infty $. The first, second and third rows are the reconstructed results when $ N = 18 $, 36 and 72, respectively. The PSNR and SSIM values are attached in the brackets
Figure 5.  CT reconstruction comparisons for "NCAT'' image with $ \sigma = 0.01\|g\|_\infty $. The first, second and third rows are the reconstructed results when $ N = 18 $, 36 and 72, respectively. The PSNR and SSIM values are attached in the brackets
Figure 6.  The first row shows the ground truth. The second and third rows show the zoom-in views of the "Shepp-Logan'' and "NCAT'' image reconstructions with 36 projections
Figure 7.  The residual errors of the reconstructions by three methods with 36 projections for "Shepp-Logan'' and "NCAT''
Figure 8.  The reconstruction comparisons of 60th and 80th rows of "Shepp-Logan'' image for the three methods with 36 projections
Figure 9.  The values $ F(u_k) $ versus the iteration number for "Shepp-Logan'' image with 36 projections
Figure 10.  CT reconstruction comparisons for real chest CT image with projection number $ N = 60 $. The PSNR and SSIM values are attached in the brackets
Figure 11.  A simple example. (a) The illustration of an image and its gradient. (b) Pixel values in black and norms of gradient in red
Table 1.  More quantitative comparisons of reconstruction on "Shepp-Logan'' in terms of PSNR and SSIM
$ N_p $ noise level TV TW-$ \ell_0 $ trunc-LN trunc-$ \ell_p $
PSNR/SSIM PSNR/SSIM PSNR/SSIM PSNR/SSIM
$ 18 $ $ 0.005\|g\|_\infty $ 36.28/0.9688 39.31/0.9871 43.36/0.9902 43.66/0.9924
$ 0.01\|g\|_\infty $ 31.88/0.9159 34.67/0.9729 39.19/0.9800 38.24/0.9794
$ 0.02\|g\|_\infty $ 27.52/0.8389 29.54/0.9460 32.08/0.9507 31.53/0.9561
$ 36 $ $ 0.005\|g\|_\infty $ 39.49/0.9860 44.52/0.9941 46.99/0.9959 49.52/0.9991
$ 0.01\|g\|_\infty $ 34.45/0.9564 39.84/0.9860 42.80/0.9897 42.87/0.9910
$ 0.02\|g\|_\infty $ 29.71/0.8972 33.56/0.9695 38.09/0.9776 37.43/0.9769
$ 72 $ $ 0.005\|g\|_\infty $ 43.27/0.9916 49.93/0.9984 50.20/0.9980 52.92/0.9989
$ 0.01\|g\|_\infty $ 37.80/0.9779 43.95/0.9942 45.61/0.9954 46.41/0.9954
$ 0.02\|g\|_\infty $ 32.54/0.9393 37.36/0.9758 41.07/0.9862 40.58/0.9868
$ N_p $ noise level TV TW-$ \ell_0 $ trunc-LN trunc-$ \ell_p $
PSNR/SSIM PSNR/SSIM PSNR/SSIM PSNR/SSIM
$ 18 $ $ 0.005\|g\|_\infty $ 36.28/0.9688 39.31/0.9871 43.36/0.9902 43.66/0.9924
$ 0.01\|g\|_\infty $ 31.88/0.9159 34.67/0.9729 39.19/0.9800 38.24/0.9794
$ 0.02\|g\|_\infty $ 27.52/0.8389 29.54/0.9460 32.08/0.9507 31.53/0.9561
$ 36 $ $ 0.005\|g\|_\infty $ 39.49/0.9860 44.52/0.9941 46.99/0.9959 49.52/0.9991
$ 0.01\|g\|_\infty $ 34.45/0.9564 39.84/0.9860 42.80/0.9897 42.87/0.9910
$ 0.02\|g\|_\infty $ 29.71/0.8972 33.56/0.9695 38.09/0.9776 37.43/0.9769
$ 72 $ $ 0.005\|g\|_\infty $ 43.27/0.9916 49.93/0.9984 50.20/0.9980 52.92/0.9989
$ 0.01\|g\|_\infty $ 37.80/0.9779 43.95/0.9942 45.61/0.9954 46.41/0.9954
$ 0.02\|g\|_\infty $ 32.54/0.9393 37.36/0.9758 41.07/0.9862 40.58/0.9868
Table 2.  More quantitative comparisons of reconstruction on "NCAT'' in terms of PSNR and SSIM
$ N_p $ noise level TV TW-$ \ell_0 $ trunc-LN trunc-$ \ell_p $
PSNR/SSIM PSNR/SSIM PSNR/SSIM PSNR/SSIM
$ 18 $ $ 0.005\|g\|_\infty $ 30.89/0.9580 29.97/0.9704 33.27/0.9875 33.90/0.9887
$ 0.01\|g\|_\infty $ 28.15/0.9141 26.74/0.9399 28.52/0.9630 28.44/0.9658
$ 0.02\|g\|_\infty $ 25.39/0.8220 24.12/0.8956 24.69/0.9189 24.72/0.9216
$ 36 $ $ 0.005\|g\|_\infty $ 33.49/0.9703 34.01/0.9877 38.36/0.9954 38.99/0.9964
$ 0.01\|g\|_\infty $ 30.20/0.9339 29.48/0.9668 31.32/0.9816 31.79/0.9834
$ 0.02\|g\|_\infty $ 27.08/0.8448 25.88/0.9210 27.02/0.9520 27.19/0.9548
$ 72 $ $ 0.005\|g\|_\infty $ 35.78/0.9721 37.81/0.9943 42.29/0.9980 42.76/0.9985
$ 0.01\|g\|_\infty $ 31.70/0.9340 31.86/0.9795 34.25/0.9902 34.40/0.9908
$ 0.02\|g\|_\infty $ 27.89/0.8276 27.44/0.9410 28.94/0.9694 29.18/0.9714
$ N_p $ noise level TV TW-$ \ell_0 $ trunc-LN trunc-$ \ell_p $
PSNR/SSIM PSNR/SSIM PSNR/SSIM PSNR/SSIM
$ 18 $ $ 0.005\|g\|_\infty $ 30.89/0.9580 29.97/0.9704 33.27/0.9875 33.90/0.9887
$ 0.01\|g\|_\infty $ 28.15/0.9141 26.74/0.9399 28.52/0.9630 28.44/0.9658
$ 0.02\|g\|_\infty $ 25.39/0.8220 24.12/0.8956 24.69/0.9189 24.72/0.9216
$ 36 $ $ 0.005\|g\|_\infty $ 33.49/0.9703 34.01/0.9877 38.36/0.9954 38.99/0.9964
$ 0.01\|g\|_\infty $ 30.20/0.9339 29.48/0.9668 31.32/0.9816 31.79/0.9834
$ 0.02\|g\|_\infty $ 27.08/0.8448 25.88/0.9210 27.02/0.9520 27.19/0.9548
$ 72 $ $ 0.005\|g\|_\infty $ 35.78/0.9721 37.81/0.9943 42.29/0.9980 42.76/0.9985
$ 0.01\|g\|_\infty $ 31.70/0.9340 31.86/0.9795 34.25/0.9902 34.40/0.9908
$ 0.02\|g\|_\infty $ 27.89/0.8276 27.44/0.9410 28.94/0.9694 29.18/0.9714
[1]

Jussi Toivanen, Alexander Meaney, Samuli Siltanen, Ville Kolehmainen. Joint reconstruction in low dose multi-energy CT. Inverse Problems & Imaging, 2020, 14 (4) : 607-629. doi: 10.3934/ipi.2020028

[2]

Neil K. Chada, Claudia Schillings, Simon Weissmann. On the incorporation of box-constraints for ensemble Kalman inversion. Foundations of Data Science, 2019, 1 (4) : 433-456. doi: 10.3934/fods.2019018

[3]

Wenxiang Cong, Ge Wang, Qingsong Yang, Jia Li, Jiang Hsieh, Rongjie Lai. CT image reconstruction on a low dimensional manifold. Inverse Problems & Imaging, 2019, 13 (3) : 449-460. doi: 10.3934/ipi.2019022

[4]

Daniil Kazantsev, William M. Thompson, William R. B. Lionheart, Geert Van Eyndhoven, Anders P. Kaestner, Katherine J. Dobson, Philip J. Withers, Peter D. Lee. 4D-CT reconstruction with unified spatial-temporal patch-based regularization. Inverse Problems & Imaging, 2015, 9 (2) : 447-467. doi: 10.3934/ipi.2015.9.447

[5]

Jing Zhou, Zhibin Deng. A low-dimensional SDP relaxation based spatial branch and bound method for nonconvex quadratic programs. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2087-2102. doi: 10.3934/jimo.2019044

[6]

Philipp Hungerländer, Barbara Kaltenbacher, Franz Rendl. Regularization of inverse problems via box constrained minimization. Inverse Problems & Imaging, 2020, 14 (3) : 437-461. doi: 10.3934/ipi.2020021

[7]

David Yang Gao. Solutions and optimality criteria to box constrained nonconvex minimization problems. Journal of Industrial & Management Optimization, 2007, 3 (2) : 293-304. doi: 10.3934/jimo.2007.3.293

[8]

Robert Azencott, Bernhard G. Bodmann, Tasadduk Chowdhury, Demetrio Labate, Anando Sen, Daniel Vera. ROI reconstruction from truncated cone-beam projections. Inverse Problems & Imaging, 2018, 12 (1) : 29-57. doi: 10.3934/ipi.2018002

[9]

Srimanta Bhattacharya, Sushmita Ruj, Bimal Roy. Combinatorial batch codes: A lower bound and optimal constructions. Advances in Mathematics of Communications, 2012, 6 (2) : 165-174. doi: 10.3934/amc.2012.6.165

[10]

Chui-Jie Wu. Large optimal truncated low-dimensional dynamical systems. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 559-583. doi: 10.3934/dcds.1996.2.559

[11]

M. Delgado Pineda, E. A. Galperin, P. Jiménez Guerra. MAPLE code of the cubic algorithm for multiobjective optimization with box constraints. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 407-424. doi: 10.3934/naco.2013.3.407

[12]

Hanwool Na, Myeongmin Kang, Miyoun Jung, Myungjoo Kang. Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parameters. Inverse Problems & Imaging, 2019, 13 (1) : 117-147. doi: 10.3934/ipi.2019007

[13]

Bartomeu Coll, Joan Duran, Catalina Sbert. Half-linear regularization for nonconvex image restoration models. Inverse Problems & Imaging, 2015, 9 (2) : 337-370. doi: 10.3934/ipi.2015.9.337

[14]

Yuan Wang, Zhi-Feng Pang, Yuping Duan, Ke Chen. Image retinex based on the nonconvex TV-type regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020050

[15]

Florent Foucaud, Tero Laihonen, Aline Parreau. An improved lower bound for $(1,\leq 2)$-identifying codes in the king grid. Advances in Mathematics of Communications, 2014, 8 (1) : 35-52. doi: 10.3934/amc.2014.8.35

[16]

Geng Chen, Ronghua Pan, Shengguo Zhu. A polygonal scheme and the lower bound on density for the isentropic gas dynamics. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4259-4277. doi: 10.3934/dcds.2019172

[17]

Aixian Zhang, Zhengchun Zhou, Keqin Feng. A lower bound on the average Hamming correlation of frequency-hopping sequence sets. Advances in Mathematics of Communications, 2015, 9 (1) : 55-62. doi: 10.3934/amc.2015.9.55

[18]

Pablo Blanc. A lower bound for the principal eigenvalue of fully nonlinear elliptic operators. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3613-3623. doi: 10.3934/cpaa.2020158

[19]

Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 585-606. doi: 10.3934/dcds.2019024

[20]

Chengxiang Wang, Li Zeng. Error bounds and stability in the $l_{0}$ regularized for CT reconstruction from small projections. Inverse Problems & Imaging, 2016, 10 (3) : 829-853. doi: 10.3934/ipi.2016023

2019 Impact Factor: 1.373

Article outline

Figures and Tables

[Back to Top]