# American Institute of Mathematical Sciences

June 2019, 13(3): 449-460. doi: 10.3934/ipi.2019022

## CT image reconstruction on a low dimensional manifold

 1 Biomedical Imaging Center, Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA 2 School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China 3 GE Healthcare Technologies, Waukesha, WI 53188, USA 4 Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, USA

* Corresponding authors: Rongjie Lai

Received  January 2018 Revised  January 2019 Published  March 2019

Fund Project: W. Cong, G. Wang and Q. Yang's work is partially supported by the National Institutes of Health Grant NIH/NIBIB R01 EB016977 and U01 EB017140. R. Lai's work is partially supported by the National Science Foundation NSF DMS-1522645 and an NSF CAREER Award DMS-1752934

The patch manifold of a natural image has a low dimensional structure and accommodates rich structural information. Inspired by the recent work of the low-dimensional manifold model (LDMM), we apply the LDMM for regularizing X-ray computed tomography (CT) image reconstruction. This proposed method recovers detailed structural information of images, significantly enhancing spatial and contrast resolution of CT images. Both numerically simulated data and clinically experimental data are used to evaluate the proposed method. The comparative studies are also performed over the simultaneous algebraic reconstruction technique (SART) incorporated the total variation (TV) regularization to demonstrate the merits of the proposed method. Results indicate that the LDMM-based method enables a more accurate image reconstruction with high fidelity and contrast resolution.

Citation: 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
##### References:
 [1] T. Brox, O. Kleinschmidt and D. Cremers, Efficient nonlocal means for denoising of textural patterns, IEEE Transactions on Image Processing, 17 (2008), 1083-1092. doi: 10.1109/TIP.2008.924281. [2] A. Buades, B. Coll and J.-M. Morel, A non-local algorithm for image denoising, In Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society Conference on, volume 2, pages 60–65. IEEE, 2005. [3] E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Transactions on Information Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083. [4] E. J. Candes, J. K. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 59 (2006), 1207-1223. doi: 10.1002/cpa.20124. [5] 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. [6] I. A. Elbakri and J. A. Fessler, Statistical image reconstruction for polyenergetic x-ray computed tomography, IEEE Transactions on Medical Imaging, 21 (2002), 89-99. doi: 10.1109/42.993128. [7] H. Gao, H. Yu, S. Osher and G. Wang, Multi-energy ct based on a prior rank, intensity and sparsity model (prism), Inverse Problems, 27 (2011), 115012, 22pp. doi: 10.1088/0266-5611/27/11/115012. [8] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation, 7 (2008), 1005-1028. doi: 10.1137/070698592. [9] S. Ha and K. Mueller, Low dose ct image restoration using a database of image patches, Physics in Medicine & Biology, 60 (2015), 869-882. doi: 10.1088/0031-9155/60/2/869. [10] A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, IEEE press New York, 1988. [11] Z. Li, Z. Shi and J. Sun, Point integral method for solving poisson-type equations on manifolds from point clouds with convergence guarantees, Communications in Computational Physics, 22 (2017), 228-258. doi: 10.4208/cicp.111015.250716a. [12] B. De Man, J. Nuyts, P. Dupont, G. Marchal and P. Suetens, An iterative maximum-likelihood polychromatic algorithm for ct, IEEE Transactions on Medical Imaging, 20 (2001), 999-1008. [13] B. De Man, S. Basu, N. Chandra, B. Dunham, P. Edic, M. Iatrou, S. McOlash, P. Sainath, C. Shaughnessy, B. Tower, et al., Catsim: a new computer assisted tomography simulation environment, In Medical Imaging 2007: Physics of Medical Imaging, volume 6510, page 65102G. International Society for Optics and Photonics, 2007. [14] S. Osher, Z. Shi and W. Zhu, Low dimensional manifold model for image processing, SIAM Journal on Imaging Sciences, 10 (2017), 1669-1690. doi: 10.1137/16M1058686. [15] S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Modeling & Simulation, 4 (2005), 460-489. doi: 10.1137/040605412. [16] G. Peyré, Manifold models for signals and images, Computer Vision and Image Understanding, 113 (2009), 249-260. [17] Y. Quan, H. Ji and Z. Shen, Data-driven multi-scale non-local wavelet frame construction and image recovery, Journal of Scientific Computing, 63 (2015), 307-329. doi: 10.1007/s10915-014-9893-2. [18] L. Ritschl, F. Bergner, C. Fleischmann and M. Kachelrieß, Improved total variation-based ct image reconstruction applied to clinical data, Physics in Medicine & Biology, 56 (2011), 1545-1561. doi: 10.1088/0031-9155/56/6/003. [19] E. Y. Sidky, Y. Duchin, X. Pan and C. Ullberg, A constrained, total-variation minimization algorithm for low-intensity x-ray ct, Medical Physics, 38 (2011), S117–S125. doi: 10.1118/1.3560887. [20] J. Tang, B. E. Nett and G.-H. Chen, Performance comparison between total variation (tv)-based compressed sensing and statistical iterative reconstruction algorithms, Physics in Medicine & Biology, 54 (2009), 5781–5804. doi: 10.1088/0031-9155/54/19/008. [21] Q. Xu, H. Yu, X. Mou, L. Zhang, J. Hsieh and G. Wang, Low-dose x-ray ct reconstruction via dictionary learning, IEEE Transactions on Medical Imaging, 31 (2012), 1682-1697. [22] X. Zhang and T. F. Chan, Wavelet inpainting by nonlocal total variation, Inverse problems and Imaging, 4 (2010), 191-210. doi: 10.3934/ipi.2010.4.191.

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##### References:
 [1] T. Brox, O. Kleinschmidt and D. Cremers, Efficient nonlocal means for denoising of textural patterns, IEEE Transactions on Image Processing, 17 (2008), 1083-1092. doi: 10.1109/TIP.2008.924281. [2] A. Buades, B. Coll and J.-M. Morel, A non-local algorithm for image denoising, In Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society Conference on, volume 2, pages 60–65. IEEE, 2005. [3] E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Transactions on Information Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083. [4] E. J. Candes, J. K. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 59 (2006), 1207-1223. doi: 10.1002/cpa.20124. [5] 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. [6] I. A. Elbakri and J. A. Fessler, Statistical image reconstruction for polyenergetic x-ray computed tomography, IEEE Transactions on Medical Imaging, 21 (2002), 89-99. doi: 10.1109/42.993128. [7] H. Gao, H. Yu, S. Osher and G. Wang, Multi-energy ct based on a prior rank, intensity and sparsity model (prism), Inverse Problems, 27 (2011), 115012, 22pp. doi: 10.1088/0266-5611/27/11/115012. [8] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation, 7 (2008), 1005-1028. doi: 10.1137/070698592. [9] S. Ha and K. Mueller, Low dose ct image restoration using a database of image patches, Physics in Medicine & Biology, 60 (2015), 869-882. doi: 10.1088/0031-9155/60/2/869. [10] A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, IEEE press New York, 1988. [11] Z. Li, Z. Shi and J. Sun, Point integral method for solving poisson-type equations on manifolds from point clouds with convergence guarantees, Communications in Computational Physics, 22 (2017), 228-258. doi: 10.4208/cicp.111015.250716a. [12] B. De Man, J. Nuyts, P. Dupont, G. Marchal and P. Suetens, An iterative maximum-likelihood polychromatic algorithm for ct, IEEE Transactions on Medical Imaging, 20 (2001), 999-1008. [13] B. De Man, S. Basu, N. Chandra, B. Dunham, P. Edic, M. Iatrou, S. McOlash, P. Sainath, C. Shaughnessy, B. Tower, et al., Catsim: a new computer assisted tomography simulation environment, In Medical Imaging 2007: Physics of Medical Imaging, volume 6510, page 65102G. International Society for Optics and Photonics, 2007. [14] S. Osher, Z. Shi and W. Zhu, Low dimensional manifold model for image processing, SIAM Journal on Imaging Sciences, 10 (2017), 1669-1690. doi: 10.1137/16M1058686. [15] S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Modeling & Simulation, 4 (2005), 460-489. doi: 10.1137/040605412. [16] G. Peyré, Manifold models for signals and images, Computer Vision and Image Understanding, 113 (2009), 249-260. [17] Y. Quan, H. Ji and Z. Shen, Data-driven multi-scale non-local wavelet frame construction and image recovery, Journal of Scientific Computing, 63 (2015), 307-329. doi: 10.1007/s10915-014-9893-2. [18] L. Ritschl, F. Bergner, C. Fleischmann and M. Kachelrieß, Improved total variation-based ct image reconstruction applied to clinical data, Physics in Medicine & Biology, 56 (2011), 1545-1561. doi: 10.1088/0031-9155/56/6/003. [19] E. Y. Sidky, Y. Duchin, X. Pan and C. Ullberg, A constrained, total-variation minimization algorithm for low-intensity x-ray ct, Medical Physics, 38 (2011), S117–S125. doi: 10.1118/1.3560887. [20] J. Tang, B. E. Nett and G.-H. Chen, Performance comparison between total variation (tv)-based compressed sensing and statistical iterative reconstruction algorithms, Physics in Medicine & Biology, 54 (2009), 5781–5804. doi: 10.1088/0031-9155/54/19/008. [21] Q. Xu, H. Yu, X. Mou, L. Zhang, J. Hsieh and G. Wang, Low-dose x-ray ct reconstruction via dictionary learning, IEEE Transactions on Medical Imaging, 31 (2012), 1682-1697. [22] X. Zhang and T. F. Chan, Wavelet inpainting by nonlocal total variation, Inverse problems and Imaging, 4 (2010), 191-210. doi: 10.3934/ipi.2010.4.191.
The patch manifold of a CT image (left) and the corresponding dimension function of the patch manifold with patch size $16\times 16$ (right)
Comparison of image reconstruction. (a) Ground truth CT images, (b) the reconstructed image using the LDMM-based method, and (c) the reconstructed image using SART with TV
Profiles of reconstructed image. (a) The profiles along the vertical midlines in the phantom and image reconstructed by LDMM-based reconstruction method, (b) the profiles along the horizontal midlines in the phantom and image reconstructed by LDMM-based reconstruction method. (c) The profiles along the vertical midlines in the phantom and image reconstructed by SART+TV reconstruction method, and (d) the profiles along the horizontal vertical midlines in the phantom and image reconstructed by SART+TV reconstruction method
The sinogram simulated from CatSim
The sinogram measured from a clinical x-ray CT scanner
Comparison of CT reconstruction. (a) Ground truth CT images, (b) the reconstructed image using the LDMM-based image reconstruction method, and (c) the reconstructed image using SART with TV
Comparison of CT image reconstructions from clinical CT raw data. (a) The reconstructed image using the LDMM-based method, (b) the reconstructed image using SART with TV, and (c) the reconstructed image using FPB
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