# American Institute of Mathematical Sciences

2017, 11(6): 917-948. doi: 10.3934/ipi.2017043

## Wavelet tight frame and prior image-based image reconstruction from limited-angle projection data

 1 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China 2 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 3 Engineering Research Center of Industrial Computed Tomography, Nondestructive Testing of the Education Ministry of China, Chongqing University, Chongqing 400044, China

1The corresponding author: drlizeng@cqu.edu.cn

Received  December 2015 Revised  August 2017 Published  September 2017

The limited-angle projection data of an object, in some practical applications of computed tomography (CT), are obtained due to the restriction of scanning condition. In these situations, since the projection data are incomplete, some limited-angle artifacts will be presented near the edges of reconstructed image using some classical reconstruction algorithms, such as filtered backprojection (FBP). The reconstructed image can be fine approximated by sparse coefficients under a proper wavelet tight frame, and the quality of reconstructed image can be improved by an available prior image. To deal with limited-angle CT reconstruction problem, we propose a minimization model that is based on wavelet tight frame and a prior image, and perform this minimization problem efficiently by iteratively minimizing separately. Moreover, we show that each bounded sequence, which is generated by our method, converges to a critical or a stationary point. The experimental results indicate that our algorithm can efficiently suppress artifacts and noise and preserve the edges of reconstructed image, what's more, the introduced prior image will not miss the important information that is not included in the prior image.

Citation: Chengxiang Wang, Li Zeng, Yumeng Guo, Lingli Zhang. Wavelet tight frame and prior image-based image reconstruction from limited-angle projection data. Inverse Problems & Imaging, 2017, 11 (6) : 917-948. doi: 10.3934/ipi.2017043
##### References:
 [1] G. Bachar, J. H. Siewerdsen, M. J. Daly, D. A. Jaffray and J. C. Irish, Image quality and localization accuracy in C-arm tomosynthesis-guided head and neck surgery, Med. Phys., 34 (2007), 4664-4677. doi: 10.1118/1.2799492. [2] C. Bao, H. Ji and Z. Shen, Convergence analysis for iterative data-driven tight frame construction scheme, Appl. Comput. Harmon. Anal., 38 (2015), 510-523. doi: 10.1016/j.acha.2014.06.007. [3] J. G. Bian, J. H. Siewerdsen, X. Han, E. Y. Sidky, J. L. Prince, C. A. Pelizzari and X. C. Pan, Evaluation of sparse-view reconstruction from flat-panel-detector cone-beam CT, Physics in Medicine and Biology, 55 (2010), 6575-6599. doi: 10.1088/0031-9155/55/22/001. [4] J. Bolte, S. Sabach and M. Teboulle, Proximal alternating linearized minimization for nonconvex and nonsmooth problems, Mathematical Programming, 146 (2014), 459-494. doi: 10.1007/s10107-013-0701-9. [5] T. M. Buzug, Computed Tomography: From Photon Statistics to Modern Cone-beam CT, 1nd edition, Springer-Verlag, Berlin Heidelberg, 2008. [6] J. F. Cai, H. Ji, Z. W. Shen and G. B. Ye, Data-driven tight frame construction and image denoising, Appl. Comput. Harmon. Anal., 37 (2014), 89-105. doi: 10.1016/j.acha.2013.10.001. [7] J. F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Modeling and Simulation, 8 (2009), 337-369. doi: 10.1137/090753504. [8] 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. [9] G. H. Chen, Time-resolved interventional cardiac C-arm cone-beam CT: An application of the PICCS algorithm, IEEE Transactions on Medical Imaging, 31 (2012), 907-923. doi: 10.1109/TMI.2011.2172951. [10] Z. Q. Chen, X. Jin, L. Li and G. Wang, A limited-angle CT reconstruction method based on anisotropic TV minimization, Phys. Med. Biol., 58 (2013), 2119-2141. doi: 10.1088/0031-9155/58/7/2119. [11] B. Dong and Z. Shen, MRA-based wavelet frames and applications: Image segmentation and surface reconstruction In SPIE Defense, Security, and Sensing, International Society for Optics and Photonics, (2012), 840102. [12] M. M. Eger and P. E. Danielsson, Scanning of logs with linear cone-beam tomography, Computers and Electronics in Agriculture, 41 (2003), 45-62. doi: 10.1016/S0168-1699(03)00041-3. [13] J. M. Fadili and G. Peyré, Total variation projection with first order schemes, IEEE Transactions on Image Processing, 20 (2011), 657-669. doi: 10.1109/TIP.2010.2072512. [14] J. Frikel, Sparse regularization in limited angle tomography, Appl. Comput. Harmon. Anal., 34 (2013), 117-141. doi: 10.1016/j.acha.2012.03.005. [15] H. Gao, J. F. Cai, Z. W. Shen and H. Zhao, Robust principal component analysis-based four-dimensional computed tomography, Phys. Med. Biol., 56 (2011), 3781-3798. doi: 10.1088/0031-9155/56/11/002. [16] H. Gao, R. Li, Y. Lin and L. Xing, 4D cone beam CT via spatiotemporal tensor framelet, Medical Physics, 39 (2012), 6943-6946. doi: 10.1118/1.4762288. [17] H. Gao, L. Zhang, Z. Chen, Y. Xing, J. Cheng and Z. Qi, Direct filtered-backprojection-type reconstruction from a straight-line trajectory, Optical Engineering, 46 (2007), 057003. doi: 10.1117/1.2739624. [18] B. Han, G. Kutyniok and Z. Shen, Adaptive multiresolution analysis structures and shearlet systems, SIAM. J. Numer. Anal., 49 (2011), 1921-1946. doi: 10.1137/090780912. [19] P. C. Hansen, E. Y. Sidky and X. C. Pan, Accelerated gradient methods for total-variation-based CT image reconstruction, arXiv: 1105. 4002. [20] G. T. Herman and R. Davidi, Image reconstruction from a small number of projections, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/4/045011. [21] X. Jia, B. Dong, Y. Lou and S. B. jiang, GPU-based iterative cone-beam CT reconstruction using tight frame regularization, Phys. Med. Biol., 56 (2010), 3787-3806. doi: 10.1088/0031-9155/56/13/004. [22] V. Kolehmainen, S. Siltanen, S. Jarvenpaa, J. P. Kaipio, P. Koistinenand, M. Lassas, J. Pirttila and E. Somersalo, Statistical inversion for medical x-ray tomography with few radiographs: Ⅱ. Application to dental radiology, Phys. Med. Biol., 48 (2003), 1465-1490. doi: 10.1088/0031-9155/48/10/315. [23] S. J. LaRoque, E. Y. Sidky and X. C. Pan, Accurate image reconstruction from few-view and limited-angle data in diffraction tomography, JOSA A, 25 (2008), 1772-1782. doi: 10.1364/JOSAA.25.001772. [24] X. Lu, Y. Sun and Y. Yuan, Image reconstruction by an alternating minimisation, Neurocomputing, 74 (2011), 661-670. doi: 10.1016/j.neucom.2010.08.003. [25] X. Lu, Y. Sun and Y. Yuan, Optimization for limited angle tomography in medical image processing, Phys. Med. Biol., 44 (2011), 2427-2435. doi: 10.1016/j.patcog.2010.12.016. [26] M. G. Lubner, Prospective evaluation of prior image constrained compressed sensing (PICCS) algorithm in abdominal CT: A comparison of reduced dose with standard dose imaging, Abdominal Imaging, 40 (2015), 207-221. doi: 10.1007/s00261-014-0178-x. [27] F. Natterer, The Mathmetics of Computed Tomography 1nd edition, B. G. Teubner, Stuttgart. , 1986. [28] B. Nett, J. Tang, S. Leng and G. H. Chen, Tomosynthesis via total variation minimization reconstruction and prior image constrained compressed sensing (PICCS) on a C-arm system, Medical Imaging. International Society for Optics and Photonics, 6913 (2008), 1-10. doi: 10.1117/12.771294. [29] E. T. Quinto, Exterior and limited-angle tomography in non-destructive evaluation, Inverse Problems, 14 (1998), 1339-1353. doi: 10.1088/0266-5611/14/2/009. [30] A. Ron and Z. Shen, Affine systems in $L_{2}(R^{d})$: The analysis of the analysis operator, Journal of Functional Analysis, 148 (1997), 408-447. doi: 10.1006/jfan.1996.3079. [31] L. Shen, Y. Xu and X. Zeng, Wavelet inpainting with the $\ell_{0}$ sparse regularization, Appl. Comput. Harmon. Anal., 41 (2016), 26-53. doi: 10.1016/j.acha.2015.03.001. [32] Z. Shen, Wavelet frames and image restorations, Proceedings of the International congress of Mathematicians, 4 (2010), 2834-2863. [33] E. Y. Sidky, C. M. Kao and X. C. Pan, Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT, arXiv: 0904. 4495. [34] E. Y. Sidky and X. C. Pan, Accurate image reconstruction in circular cone-beam computed tomography by total variation minimization: A preliminary investigation, IEEE Nuclear Science Symposium Conference Record, 5 (2006), 2904-2907. doi: 10.1109/NSSMIC.2006.356484. [35] E. Y. Sidky and X. C. Pan, Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Physics in Medicine and Biology, 53 (2008), 3572277-3572284. doi: 10.1088/0031-9155/53/17/021. [36] W. P. Segars, D. S. Lalush and B. M. Tsui, A realistic splinebased dynamic heart phantom, IEEE Trans. Nucl. Sci., 46 (1999), 503-506. doi: 10.1109/23.775570. [37] M. Storath, A. Weinmann, J. Frikeland and M. Unser, Joint image reconstruction and segmentation using the Potts model, Inverse Problems, 31 (2015), 025003. doi: 10.1088/0266-5611/31/2/025003. [38] J. Tang, J. Hsieh and G. H. Chen, Temporal resolution improvement in cardiac CT using PICCS (TRI-PICCS): Performance studies, Medical Physics, 37 (2010), 4377-4388. doi: 10.1118/1.3460318. [39] A. Tingberg, X-ray tomosynthesis: A review of its use for breast and chest imaging, Radiation Protection Dosimetry, 139 (2010), 100-107. doi: 10.1093/rpd/ncq099. [40] H. K. Tuy, An inversion formula for cone-beam reconstruction, SIAM J. Appl. Math., 43 (1983), 546-552. doi: 10.1137/0143035. [41] Z. Wang, A. Bovik, H. Sheikhand and E. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans Image Process, 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861. [42] Z. Wang, Z. Huang, Z. Chen, L. Zhang, X. Jiang, K. Kang, H. Yin, Z. Wang and M. Stampanoni, Low-dose multiple-information retrieval algorithm for x-ray grating-based imaging, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 635 (2011), 103-107. [43] F. Yang, Y. Shen and Z. S. Liu, The proximal alternating iterative hard thresholding method for $\ell_{0}$ minimization, with complexity $O(\frac{1}{\sqrt{k}})$, Journal of Computational and Applied Mathematics, 311 (2017), 115-129. doi: 10.1016/j.cam.2016.07.013. [44] W. Yu and L. Zeng, $\ell_{0}$ gradient minimization based image reconstruction for limited-angle computed tomography, PLoS ONE, 10 (2015), e0130793. doi: 10.1371/journal.pone.0130793. [45] L. Zeng, J. Q. Guo and B. D. Liu, Limited-angle cone-beam computed tomography image reconstruction by total variation minimization and piecewise-constant modification, Journal of Inverse and Ill-Posed Problems, 21 (2013), 735-754. doi: 10.1515/jip-2011-0010. [46] Y. Zhang, B. Dong and Z. S. Lu, $\ell_{0}$ Minimization for wavelet frame based image restoration, Mathematics of Computation, 82 (2013), 995-1015. doi: 10.1090/S0025-5718-2012-02631-7. [47] B. Zhao, H. Gao, H. Ding and S. Molloi, Tight-frame based iterative image reconstruction for spectral breast CT, Medical Physics, 40 (2013), 031905. doi: 10.1118/1.4790468. [48] W. Zhou, J. F. Cai and H. Gao, Adaptive tight frame based medical image reconstruction: a proof-of-concept study for computed tomography, Inverse problems, 29 (2013), 1-18. doi: 10.1088/0266-5611/29/12/125006. [49] chest phantom website, http://lgdv.cs.fau.de/External/vollib/.

show all references

##### References:
 [1] G. Bachar, J. H. Siewerdsen, M. J. Daly, D. A. Jaffray and J. C. Irish, Image quality and localization accuracy in C-arm tomosynthesis-guided head and neck surgery, Med. Phys., 34 (2007), 4664-4677. doi: 10.1118/1.2799492. [2] C. Bao, H. Ji and Z. Shen, Convergence analysis for iterative data-driven tight frame construction scheme, Appl. Comput. Harmon. Anal., 38 (2015), 510-523. doi: 10.1016/j.acha.2014.06.007. [3] J. G. Bian, J. H. Siewerdsen, X. Han, E. Y. Sidky, J. L. Prince, C. A. Pelizzari and X. C. Pan, Evaluation of sparse-view reconstruction from flat-panel-detector cone-beam CT, Physics in Medicine and Biology, 55 (2010), 6575-6599. doi: 10.1088/0031-9155/55/22/001. [4] J. Bolte, S. Sabach and M. Teboulle, Proximal alternating linearized minimization for nonconvex and nonsmooth problems, Mathematical Programming, 146 (2014), 459-494. doi: 10.1007/s10107-013-0701-9. [5] T. M. Buzug, Computed Tomography: From Photon Statistics to Modern Cone-beam CT, 1nd edition, Springer-Verlag, Berlin Heidelberg, 2008. [6] J. F. Cai, H. Ji, Z. W. Shen and G. B. Ye, Data-driven tight frame construction and image denoising, Appl. Comput. Harmon. Anal., 37 (2014), 89-105. doi: 10.1016/j.acha.2013.10.001. [7] J. F. Cai, S. Osher and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Modeling and Simulation, 8 (2009), 337-369. doi: 10.1137/090753504. [8] 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. [9] G. H. Chen, Time-resolved interventional cardiac C-arm cone-beam CT: An application of the PICCS algorithm, IEEE Transactions on Medical Imaging, 31 (2012), 907-923. doi: 10.1109/TMI.2011.2172951. [10] Z. Q. Chen, X. Jin, L. Li and G. Wang, A limited-angle CT reconstruction method based on anisotropic TV minimization, Phys. Med. Biol., 58 (2013), 2119-2141. doi: 10.1088/0031-9155/58/7/2119. [11] B. Dong and Z. Shen, MRA-based wavelet frames and applications: Image segmentation and surface reconstruction In SPIE Defense, Security, and Sensing, International Society for Optics and Photonics, (2012), 840102. [12] M. M. Eger and P. E. Danielsson, Scanning of logs with linear cone-beam tomography, Computers and Electronics in Agriculture, 41 (2003), 45-62. doi: 10.1016/S0168-1699(03)00041-3. [13] J. M. Fadili and G. Peyré, Total variation projection with first order schemes, IEEE Transactions on Image Processing, 20 (2011), 657-669. doi: 10.1109/TIP.2010.2072512. [14] J. Frikel, Sparse regularization in limited angle tomography, Appl. Comput. Harmon. Anal., 34 (2013), 117-141. doi: 10.1016/j.acha.2012.03.005. [15] H. Gao, J. F. Cai, Z. W. Shen and H. Zhao, Robust principal component analysis-based four-dimensional computed tomography, Phys. Med. Biol., 56 (2011), 3781-3798. doi: 10.1088/0031-9155/56/11/002. [16] H. Gao, R. Li, Y. Lin and L. Xing, 4D cone beam CT via spatiotemporal tensor framelet, Medical Physics, 39 (2012), 6943-6946. doi: 10.1118/1.4762288. [17] H. Gao, L. Zhang, Z. Chen, Y. Xing, J. Cheng and Z. Qi, Direct filtered-backprojection-type reconstruction from a straight-line trajectory, Optical Engineering, 46 (2007), 057003. doi: 10.1117/1.2739624. [18] B. Han, G. Kutyniok and Z. Shen, Adaptive multiresolution analysis structures and shearlet systems, SIAM. J. Numer. Anal., 49 (2011), 1921-1946. doi: 10.1137/090780912. [19] P. C. Hansen, E. Y. Sidky and X. C. Pan, Accelerated gradient methods for total-variation-based CT image reconstruction, arXiv: 1105. 4002. [20] G. T. Herman and R. Davidi, Image reconstruction from a small number of projections, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/4/045011. [21] X. Jia, B. Dong, Y. Lou and S. B. jiang, GPU-based iterative cone-beam CT reconstruction using tight frame regularization, Phys. Med. Biol., 56 (2010), 3787-3806. doi: 10.1088/0031-9155/56/13/004. [22] V. Kolehmainen, S. Siltanen, S. Jarvenpaa, J. P. Kaipio, P. Koistinenand, M. Lassas, J. Pirttila and E. Somersalo, Statistical inversion for medical x-ray tomography with few radiographs: Ⅱ. Application to dental radiology, Phys. Med. Biol., 48 (2003), 1465-1490. doi: 10.1088/0031-9155/48/10/315. [23] S. J. LaRoque, E. Y. Sidky and X. C. Pan, Accurate image reconstruction from few-view and limited-angle data in diffraction tomography, JOSA A, 25 (2008), 1772-1782. doi: 10.1364/JOSAA.25.001772. [24] X. Lu, Y. Sun and Y. Yuan, Image reconstruction by an alternating minimisation, Neurocomputing, 74 (2011), 661-670. doi: 10.1016/j.neucom.2010.08.003. [25] X. Lu, Y. Sun and Y. Yuan, Optimization for limited angle tomography in medical image processing, Phys. Med. Biol., 44 (2011), 2427-2435. doi: 10.1016/j.patcog.2010.12.016. [26] M. G. Lubner, Prospective evaluation of prior image constrained compressed sensing (PICCS) algorithm in abdominal CT: A comparison of reduced dose with standard dose imaging, Abdominal Imaging, 40 (2015), 207-221. doi: 10.1007/s00261-014-0178-x. [27] F. Natterer, The Mathmetics of Computed Tomography 1nd edition, B. G. Teubner, Stuttgart. , 1986. [28] B. Nett, J. Tang, S. Leng and G. H. Chen, Tomosynthesis via total variation minimization reconstruction and prior image constrained compressed sensing (PICCS) on a C-arm system, Medical Imaging. International Society for Optics and Photonics, 6913 (2008), 1-10. doi: 10.1117/12.771294. [29] E. T. Quinto, Exterior and limited-angle tomography in non-destructive evaluation, Inverse Problems, 14 (1998), 1339-1353. doi: 10.1088/0266-5611/14/2/009. [30] A. Ron and Z. Shen, Affine systems in $L_{2}(R^{d})$: The analysis of the analysis operator, Journal of Functional Analysis, 148 (1997), 408-447. doi: 10.1006/jfan.1996.3079. [31] L. Shen, Y. Xu and X. Zeng, Wavelet inpainting with the $\ell_{0}$ sparse regularization, Appl. Comput. Harmon. Anal., 41 (2016), 26-53. doi: 10.1016/j.acha.2015.03.001. [32] Z. Shen, Wavelet frames and image restorations, Proceedings of the International congress of Mathematicians, 4 (2010), 2834-2863. [33] E. Y. Sidky, C. M. Kao and X. C. Pan, Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT, arXiv: 0904. 4495. [34] E. Y. Sidky and X. C. Pan, Accurate image reconstruction in circular cone-beam computed tomography by total variation minimization: A preliminary investigation, IEEE Nuclear Science Symposium Conference Record, 5 (2006), 2904-2907. doi: 10.1109/NSSMIC.2006.356484. [35] E. Y. Sidky and X. C. Pan, Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Physics in Medicine and Biology, 53 (2008), 3572277-3572284. doi: 10.1088/0031-9155/53/17/021. [36] W. P. Segars, D. S. Lalush and B. M. Tsui, A realistic splinebased dynamic heart phantom, IEEE Trans. Nucl. Sci., 46 (1999), 503-506. doi: 10.1109/23.775570. [37] M. Storath, A. Weinmann, J. Frikeland and M. Unser, Joint image reconstruction and segmentation using the Potts model, Inverse Problems, 31 (2015), 025003. doi: 10.1088/0266-5611/31/2/025003. [38] J. Tang, J. Hsieh and G. H. Chen, Temporal resolution improvement in cardiac CT using PICCS (TRI-PICCS): Performance studies, Medical Physics, 37 (2010), 4377-4388. doi: 10.1118/1.3460318. [39] A. Tingberg, X-ray tomosynthesis: A review of its use for breast and chest imaging, Radiation Protection Dosimetry, 139 (2010), 100-107. doi: 10.1093/rpd/ncq099. [40] H. K. Tuy, An inversion formula for cone-beam reconstruction, SIAM J. Appl. Math., 43 (1983), 546-552. doi: 10.1137/0143035. [41] Z. Wang, A. Bovik, H. Sheikhand and E. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans Image Process, 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861. [42] Z. Wang, Z. Huang, Z. Chen, L. Zhang, X. Jiang, K. Kang, H. Yin, Z. Wang and M. Stampanoni, Low-dose multiple-information retrieval algorithm for x-ray grating-based imaging, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 635 (2011), 103-107. [43] F. Yang, Y. Shen and Z. S. Liu, The proximal alternating iterative hard thresholding method for $\ell_{0}$ minimization, with complexity $O(\frac{1}{\sqrt{k}})$, Journal of Computational and Applied Mathematics, 311 (2017), 115-129. doi: 10.1016/j.cam.2016.07.013. [44] W. Yu and L. Zeng, $\ell_{0}$ gradient minimization based image reconstruction for limited-angle computed tomography, PLoS ONE, 10 (2015), e0130793. doi: 10.1371/journal.pone.0130793. [45] L. Zeng, J. Q. Guo and B. D. Liu, Limited-angle cone-beam computed tomography image reconstruction by total variation minimization and piecewise-constant modification, Journal of Inverse and Ill-Posed Problems, 21 (2013), 735-754. doi: 10.1515/jip-2011-0010. [46] Y. Zhang, B. Dong and Z. S. Lu, $\ell_{0}$ Minimization for wavelet frame based image restoration, Mathematics of Computation, 82 (2013), 995-1015. doi: 10.1090/S0025-5718-2012-02631-7. [47] B. Zhao, H. Gao, H. Ding and S. Molloi, Tight-frame based iterative image reconstruction for spectral breast CT, Medical Physics, 40 (2013), 031905. doi: 10.1118/1.4790468. [48] W. Zhou, J. F. Cai and H. Gao, Adaptive tight frame based medical image reconstruction: a proof-of-concept study for computed tomography, Inverse problems, 29 (2013), 1-18. doi: 10.1088/0266-5611/29/12/125006. [49] chest phantom website, http://lgdv.cs.fau.de/External/vollib/.
Scanning geometry of limited-angle CT. $\textrm{S}$ denotes the X-ray source, $\textrm{o}$ denotes the rotation center of object, $\textrm{D}$ denotes the detector, and $\theta$ denotes the rotation angle which is less than $180^{0}$ plus a fan-angle
Reconstructed result of NCAT phantom using FBP algorithm for the scanning angle $[0,120^{0}]$. The limited-angle artifacts are labelled by the red rectangles
(a) and (b) are the reconstructed results using FBP algorithm for scanning range $[0,360^{0}]$ and $[0,120^{0}]$, respectively
The wavelet transform of (a) of Figure 3 under B-spline frame
The wavelet transform of (b) of Figure 3 under B-spline frame
NCAT phantom and prior image. The first column is NCAT phantom, the second column is the prior image and the third column is the absolute value of the difference between phantom and prior image. The prior image is the same with NCAT phantom
The reconstructed results for different scanning ranges using FBP algorithm, PICCS algorithm and our algorithm. The display window is $[0,255]$. The noise levels are $0.5\%\|g\|_{\infty}$
The reconstructed results for different scanning ranges using FBP algorithm, PICCS algorithm and our algorithm. The display window is $[0,255]$. The noise levels are $1\%\|g\|_{\infty}$
Phantom, prior image, and the absolute value of the difference between phantom and prior image. Three circles are labelled by red rectangles
The reconstructed results for the different scanning ranges using FBP algorithm, PICCS algorithm and our algorithm. Three circles are labelled by red rectangles. The display window is $[0,255]$
Phantom and prior image. The first column is the phantom, the subsequent columns are the prior image and the absolute value of the difference between phantom and prior image
The reconstructed results using PICCS algorithm and our algorithm. The display window for the first row is $[0,1]$ and for the second row is $[0.8, 0.9]$
The reconstructed results for the different scanning ranges using $\ell_{1}-\ell_{0}$ algorithm and $\ell_{2}-\ell_{0}$ algorithm. Three circles are labelled by red rectangles. The display window is $[0,255]$
The reconstructed results of simulation phantom for the different scanning ranges using $\ell_{1}-\ell_{0}$ algorithm and $\ell_{2}-\ell_{0}$ algorithm. The display window for the first row is $[0,1]$ and for the second row is $[0.8, 0.9]$
The reconstructed results of gear using FBP algorithm and our algorithm. ROI is labelled by red rectangle. The display window is $[0, 0.0096]mm^{-1}$
The zoom-in view of ROI of Figure 15
Geometrical scanning parameters of simulated CT system
 The distance between source and object center $981mm$ The angle interval of two adjacent projection views $1^{0}$ The angle interval of two adjacent rays $0.00329^{0}$ The diameter of field of view $143.6222mm$ Detector numbers $256$ Pixel size $0.5632\times0.5632mm^{2}$ Image size $256\times256$
 The distance between source and object center $981mm$ The angle interval of two adjacent projection views $1^{0}$ The angle interval of two adjacent rays $0.00329^{0}$ The diameter of field of view $143.6222mm$ Detector numbers $256$ Pixel size $0.5632\times0.5632mm^{2}$ Image size $256\times256$
Quantitatively characterize the reconstruction quality
 Scanning ranges Variances Algorithm RMSE PSNR MSSIM $0\sim 360^{0}$ $0.5\% \|g\|_{\infty}$ FBP 8.403 29.64 0.9921 $1\% \|g\|_{\infty}$ FBP 12.88 25.93 0.9800 $0\sim 100^{0}$ $0.5\% \|g\|_{\infty}$ our algorithm 2.116 41.62 0.9995 PICCS 3.743 36.67 0.9985 $1\% \|g\|_{\infty}$ our algorithm 4.747 34.60 0.9973 PICCS 5.953 32.64 0.9953 $0\sim 80^{0}$ $0.5\% \|g\|_{\infty}$ our algorithm 2.240 41.12 0.9994 PICCS 4.087 35.90 0.9984 $1\% \|g\|_{\infty}$ our algorithm 4.258 35.55 0.9978 PICCS 4.915 32.69 0.9953 $0\sim 60^{0}$ $0.5\% \|g\|_{\infty}$ our algorithm 1.843 42.82 0.9996 PICCS 5.905 32.71 0.9956 $1\% \|g\|_{\infty}$ our algorithm 3.566 37.09 0.9985 PICCS 6.250 32.21 0.9952
 Scanning ranges Variances Algorithm RMSE PSNR MSSIM $0\sim 360^{0}$ $0.5\% \|g\|_{\infty}$ FBP 8.403 29.64 0.9921 $1\% \|g\|_{\infty}$ FBP 12.88 25.93 0.9800 $0\sim 100^{0}$ $0.5\% \|g\|_{\infty}$ our algorithm 2.116 41.62 0.9995 PICCS 3.743 36.67 0.9985 $1\% \|g\|_{\infty}$ our algorithm 4.747 34.60 0.9973 PICCS 5.953 32.64 0.9953 $0\sim 80^{0}$ $0.5\% \|g\|_{\infty}$ our algorithm 2.240 41.12 0.9994 PICCS 4.087 35.90 0.9984 $1\% \|g\|_{\infty}$ our algorithm 4.258 35.55 0.9978 PICCS 4.915 32.69 0.9953 $0\sim 60^{0}$ $0.5\% \|g\|_{\infty}$ our algorithm 1.843 42.82 0.9996 PICCS 5.905 32.71 0.9956 $1\% \|g\|_{\infty}$ our algorithm 3.566 37.09 0.9985 PICCS 6.250 32.21 0.9952
Geometrical scanning parameters of simulated CT system
 The distance between source and object center $981mm$ The angle interval of two adjacent projection views $0.703^{0}$ The angle interval of two adjacent rays $0.0005^{0}$ The diameter of field of view $279.5mm$ Detector numbers $560$ Pixel size $0.5\times0.5mm^{2}$ Image size $512\times512$
 The distance between source and object center $981mm$ The angle interval of two adjacent projection views $0.703^{0}$ The angle interval of two adjacent rays $0.0005^{0}$ The diameter of field of view $279.5mm$ Detector numbers $560$ Pixel size $0.5\times0.5mm^{2}$ Image size $512\times512$
Quantitatively characterize the reconstruction quality
 Scanning ranges Algorithm RMSE PSNR MSSIM $0\sim 360^{0}$ FBP 4.978 34.19 0.9961 $0\sim 120^{0}$ our algorithm 3.607 36.99 0.9979 PICCS 4.208 35.65 0.9972 $0\sim 100^{0}$ our algorithm 3.901 36.31 0.9976 PICCS 4.190 35.69 0.9972 $0\sim 80^{0}$ our algorithm 3.693 36.78 0.9979 PICCS 4.177 35.71 0.9972
 Scanning ranges Algorithm RMSE PSNR MSSIM $0\sim 360^{0}$ FBP 4.978 34.19 0.9961 $0\sim 120^{0}$ our algorithm 3.607 36.99 0.9979 PICCS 4.208 35.65 0.9972 $0\sim 100^{0}$ our algorithm 3.901 36.31 0.9976 PICCS 4.190 35.69 0.9972 $0\sim 80^{0}$ our algorithm 3.693 36.78 0.9979 PICCS 4.177 35.71 0.9972
The parameters of simulated phantom
 $I$ $h$ $v$ $x_{0}$ $y_{0}$ $r$ 1 0.74 0.74 0 0 0 -1 0.5 0.5 0 0 0 -1 0.1 0.1 0.43 0.43 0 -1 0.1 0.1 -0.43 -0.43 0 -1 0.1 0.1 -0.43 0.43 0 -1 0.1 0.1 0.43 -0.43 0 -1 0.12 0.006 0.25 0.55 -18 -1 0.08 0.006 0.25 -0.55 -240 -1 0.08 0.006 -0.55 0.3 20
 $I$ $h$ $v$ $x_{0}$ $y_{0}$ $r$ 1 0.74 0.74 0 0 0 -1 0.5 0.5 0 0 0 -1 0.1 0.1 0.43 0.43 0 -1 0.1 0.1 -0.43 -0.43 0 -1 0.1 0.1 -0.43 0.43 0 -1 0.1 0.1 0.43 -0.43 0 -1 0.12 0.006 0.25 0.55 -18 -1 0.08 0.006 0.25 -0.55 -240 -1 0.08 0.006 -0.55 0.3 20
Quantitatively characterize the reconstruction quality
 Scanning ranges Algorithm RMSE PSNR MSSIM $0\sim 120^{0}$ our algorithm 0.040 27.97 0.9949 PICCS 0.057 28.26 0.9897 $0\sim 100^{0}$ our algorithm 0.0378 28.45 0.9954 PICCS 0.0546 28.65 0.9906
 Scanning ranges Algorithm RMSE PSNR MSSIM $0\sim 120^{0}$ our algorithm 0.040 27.97 0.9949 PICCS 0.057 28.26 0.9897 $0\sim 100^{0}$ our algorithm 0.0378 28.45 0.9954 PICCS 0.0546 28.65 0.9906
Quantitatively characterize the reconstruction quality
 Scanning ranges Algorithm RMSE PSNR MSSIM $0\sim 120^{0}$ $\ell_{1}-\ell_{0}$ 3.725 36.71 0.9978 $0\sim 120^{0}$ $\ell_{2}-\ell_{0}$ 3.607 36.99 0.9979 $0\sim 100^{0}$ $\ell_{1}-\ell_{0}$ 3.856 36.41 0.9977 $0\sim 100^{0}$ $\ell_{2}-\ell_{0}$ 3.901 36.31 0.9976
 Scanning ranges Algorithm RMSE PSNR MSSIM $0\sim 120^{0}$ $\ell_{1}-\ell_{0}$ 3.725 36.71 0.9978 $0\sim 120^{0}$ $\ell_{2}-\ell_{0}$ 3.607 36.99 0.9979 $0\sim 100^{0}$ $\ell_{1}-\ell_{0}$ 3.856 36.41 0.9977 $0\sim 100^{0}$ $\ell_{2}-\ell_{0}$ 3.901 36.31 0.9976
Quantitatively characterize the reconstruction quality
 Scanning ranges Algorithm RMSE PSNR MSSIM $0\sim 120^{0}$ $\ell_{1}-\ell_{0}$ 0.0404 27.86 0.9948 $0\sim 120^{0}$ $\ell_{2}-\ell_{0}$ 0.0400 27.97 0.9949 $0\sim 100^{0}$ $\ell_{1}-\ell_{0}$ 0.0383 28.34 0.9954 $0\sim 100^{0}$ $\ell_{2}-\ell_{0}$ 0.0378 28.45 0.9954
 Scanning ranges Algorithm RMSE PSNR MSSIM $0\sim 120^{0}$ $\ell_{1}-\ell_{0}$ 0.0404 27.86 0.9948 $0\sim 120^{0}$ $\ell_{2}-\ell_{0}$ 0.0400 27.97 0.9949 $0\sim 100^{0}$ $\ell_{1}-\ell_{0}$ 0.0383 28.34 0.9954 $0\sim 100^{0}$ $\ell_{2}-\ell_{0}$ 0.0378 28.45 0.9954
Quantitatively characterize the reconstruction quality
 Scanning ranges Algorithm RMSE PSNR MSSIM $0\sim 360^{0}$ FBP 6.559 31.79 0.9965 $0\sim 80^{0}$ our algorithm 4.543 34.98 0.9983
 Scanning ranges Algorithm RMSE PSNR MSSIM $0\sim 360^{0}$ FBP 6.559 31.79 0.9965 $0\sim 80^{0}$ our algorithm 4.543 34.98 0.9983
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