# American Institute of Mathematical Sciences

December  2018, 12(6): 1429-1442. doi: 10.3934/ipi.2018060

## Tomographic reconstruction methods for decomposing directional components

 1 Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark 2 College of Mathematics and Statistics, Shenzhen University, Shenzhen, Guangdong, China

* Corresponding author: Yiqiu Dong

Received  February 2018 Revised  May 2018 Published  October 2018

Fund Project: The work was supported by Advanced Grant 291405 from the European Research Council and Grant 11701388 from the National Natural Science Foundation of China.

X-ray computed tomography technique has been used in many different practical applications. Often after reconstruction we need segment or decompose objects into different components. In this paper, we propose two new reconstruction methods that can decompose objects at the same time. By incorporating direction information, the proposed methods can decompose objects into various directional components. Furthermore, we propose an algorithm to obtain the direction information of the object directly from its CT measurements. We demonstrate the proposed methods on simulated and real samples to show their practical applicability. The numerical results show the differences between the two methods and effectiveness as dealing with fibre-crack decomposition problems.

Citation: Rasmus Dalgas Kongskov, Yiqiu Dong. Tomographic reconstruction methods for decomposing directional components. Inverse Problems and Imaging, 2018, 12 (6) : 1429-1442. doi: 10.3934/ipi.2018060
##### References:
 [1] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Springer Science + Business Media, 2006. [2] J. F. Aujol, G. Aubert, L. Blanc-Féraud and A. Chambolle, Image decomposition into a bounded variation component and an oscillating component, J. Math. Imaging Vis., 22 (2005), 71-88.  doi: 10.1007/s10851-005-4783-8. [3] J. F. Aujol, G. Gilboa, T. Chan and S. Osher, Structure-texture image decomposition-modeling, algorithms, and parameter selection, Int. J. Comput. Vis., 67 (2006), 111-136. [4] I. Bayram and M. E. Kamasak, A directional total variation, Eur. Signal Process. Conf., 19 (2012), 265-269. [5] A. Bovik, Handbook of Image and Video Processing, Academic Press, 2000. [6] T. M. Buzug, Computed Tomography : From Photon Statistics to Modern Cone-Beam CT, Springer, 2008. [7] A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188.  doi: 10.1007/s002110050258. [8] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis., 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1. [9] A. H. Delaney and Y. Bresler, Globally convergent edge-preserving regularized reconstruction: An application to limited-angle tomography, IEEE Trans. Image Process., 7 (1998), 204-221. [10] G. R. Easley, D. Labate and F. Colonna, Shearlet-based total variation diffusion for denoising, IEEE Trans. Image Process., 18 (2009), 260-268.  doi: 10.1109/TIP.2008.2008070. [11] S. Esedoglu and S. J. Osher, Decomposition of images by the anisotropic Rudin-Osher-Fatemi model, Commun. Pure Appl. Math., 57 (2004), 1609-1626.  doi: 10.1002/cpa.20045. [12] J. Gilles, Noisy image decomposition: A new structure, texture and noise model based on local adaptivity, J. Math. Imaging Vis., 28 (2007), 285-295.  doi: 10.1007/s10851-007-0020-y. [13] M. Holler and K. Kunisch, On infimal convolution of total variation type functionals and applications, SIAM J. Imaging Sci., 7 (2014), 2258-2300.  doi: 10.1137/130948793. [14] K. M. Jespersen, J. Zangenberg, T. Lowe, P. J. Withers and L. P. Mikkelsen, Fatigue damage assessment of uni-directional non-crimp fabric reinforced polyester composite using X-ray computed tomography, Compos. Sci. Technol., 136 (2016), 94-103. [15] R. D. Kongskov and Y. Dong, Directional total generalized variation regularization for impulse noise removal, Proceedings of Scale Sp. Var. Methods Comput. Vis. 2017, LNCS 10302 (2017), 221–231, URL http://link.springer.com/10.1007/978-3-642-24785-9. [16] R. D. Kongskov, Y. Dong and K. Knudsen, Directional total generalized variation regularization, arXiv: 1701.02675 [17] V. P. Krishnan and E. T. Quinto, Handbook of Mathematical Methods in Imaging, Springer Science + Business Media, 2015. [18] J. Li, C. Miao, Z. Shen, G. Wang and H. Yu, Robust frame based X-ray CT reconstruction, J. Comput. Math., 34 (2016), 683-704.  doi: 10.4208/jcm.1608-m2016-0499. [19] Y. Lou, T. Zeng, S. Osher and J. Xin, A weighted difference of anisotropic and isotropic total variation model for image processing, SIAM J. Imaging Sci., 8 (2015), 1798-1823.  doi: 10.1137/14098435X. [20] Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, vol. 22, American Mathematical Society, 2001. doi: 10.1090/ulect/022. [21] W. J. Palenstijn, K. J. Batenburg and J. Sijbers, Performance improvements for iterative electron tomography reconstruction using graphics processing units (GPUs), J. Struct. Biol., 176 (2011), 250-253. [22] E. T. Quinto, Singularities of the X-ray transform and limited data tomography in $\mathbb{R}^2$ and $\mathbb{R}^3$, SIAM J. Math. Anal., 24 (1993), 1215-1225.  doi: 10.1137/0524069. [23] J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Akad. Wiss., 69 (1917), 262-277. [24] J. E. Rouse, Characterisation of Impact Damage in Carbon Fibre Reinforced Plastics by 3D X-Ray Tomography, PhD thesis, University of Manchester, 2012. [25] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D Nonlinear Phenom., 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F. [26] S. R. Sandoghchi, G. T. Jasion, N. V. Wheeler, S. Jain, Z. Lian, J. P. Wooler, R. P. Boardman, N. K. Baddela, Y. Chen, J. R. Hayes, E. N. Fokoua, T. Bradley, D. R. Gray, S. M. Mousavi, M. N. Petrovich, F. Poletti and D. J. Richardson, X-ray tomography for structural analysis of microstructured and multimaterial optical fibers and preforms, Opt. Express, 22 (2014), 26181. [27] O. Scherzer, Handbook of Mathematical Methods in Imaging, Springer Science + Business Media, 2010. [28] E. Y. Sidky and X. Pan, Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Phys. Med. Biol., 53 (2008), 4777-4807. [29] J. L. Starck, M. Elad and D. L. Donoho, Image decomposition via the combination of sparse representation and a variational approach, IEEE Trans. Image Process., 14 (2005), 1570-1582.  doi: 10.1109/TIP.2005.852206. [30] W. van Aarle, W. J. Palenstijn, J. Cant, E. Janssens, F. Bleichrodt, A. Dabravolski, J. De Beenhouwer, K. J. Batenburg and J. Sijbers, Fast and flexible X-ray tomography using the ASTRA toolbox, Opt. Express, 24 (2016), 25129-25147. [31] W. van Aarle, W. J. Palenstijn, J. De Beenhouwer, T. Altantzis, S. Bals, K. J. Batenburg and J. Sijbers, The ASTRA Toolbox: A platform for advanced algorithm development in electron tomography, Ultramicroscopy, 157 (2015), 35-47. [32] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612. [33] J. Weickert, Anisotropic Diffusion in Image Processing, Teubner-Verlag, Stuttgart, Germany, 1998. [34] J. Weickert, Coherence-enhancing diffusion filtering, Int. J. Comput. Vis., 31 (1999), 111-127.

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##### References:
 [1] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Springer Science + Business Media, 2006. [2] J. F. Aujol, G. Aubert, L. Blanc-Féraud and A. Chambolle, Image decomposition into a bounded variation component and an oscillating component, J. Math. Imaging Vis., 22 (2005), 71-88.  doi: 10.1007/s10851-005-4783-8. [3] J. F. Aujol, G. Gilboa, T. Chan and S. Osher, Structure-texture image decomposition-modeling, algorithms, and parameter selection, Int. J. Comput. Vis., 67 (2006), 111-136. [4] I. Bayram and M. E. Kamasak, A directional total variation, Eur. Signal Process. Conf., 19 (2012), 265-269. [5] A. Bovik, Handbook of Image and Video Processing, Academic Press, 2000. [6] T. M. Buzug, Computed Tomography : From Photon Statistics to Modern Cone-Beam CT, Springer, 2008. [7] A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188.  doi: 10.1007/s002110050258. [8] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis., 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1. [9] A. H. Delaney and Y. Bresler, Globally convergent edge-preserving regularized reconstruction: An application to limited-angle tomography, IEEE Trans. Image Process., 7 (1998), 204-221. [10] G. R. Easley, D. Labate and F. Colonna, Shearlet-based total variation diffusion for denoising, IEEE Trans. Image Process., 18 (2009), 260-268.  doi: 10.1109/TIP.2008.2008070. [11] S. Esedoglu and S. J. Osher, Decomposition of images by the anisotropic Rudin-Osher-Fatemi model, Commun. Pure Appl. Math., 57 (2004), 1609-1626.  doi: 10.1002/cpa.20045. [12] J. Gilles, Noisy image decomposition: A new structure, texture and noise model based on local adaptivity, J. Math. Imaging Vis., 28 (2007), 285-295.  doi: 10.1007/s10851-007-0020-y. [13] M. Holler and K. Kunisch, On infimal convolution of total variation type functionals and applications, SIAM J. Imaging Sci., 7 (2014), 2258-2300.  doi: 10.1137/130948793. [14] K. M. Jespersen, J. Zangenberg, T. Lowe, P. J. Withers and L. P. Mikkelsen, Fatigue damage assessment of uni-directional non-crimp fabric reinforced polyester composite using X-ray computed tomography, Compos. Sci. Technol., 136 (2016), 94-103. [15] R. D. Kongskov and Y. Dong, Directional total generalized variation regularization for impulse noise removal, Proceedings of Scale Sp. Var. Methods Comput. Vis. 2017, LNCS 10302 (2017), 221–231, URL http://link.springer.com/10.1007/978-3-642-24785-9. [16] R. D. Kongskov, Y. Dong and K. Knudsen, Directional total generalized variation regularization, arXiv: 1701.02675 [17] V. P. Krishnan and E. T. Quinto, Handbook of Mathematical Methods in Imaging, Springer Science + Business Media, 2015. [18] J. Li, C. Miao, Z. Shen, G. Wang and H. Yu, Robust frame based X-ray CT reconstruction, J. Comput. Math., 34 (2016), 683-704.  doi: 10.4208/jcm.1608-m2016-0499. [19] Y. Lou, T. Zeng, S. Osher and J. Xin, A weighted difference of anisotropic and isotropic total variation model for image processing, SIAM J. Imaging Sci., 8 (2015), 1798-1823.  doi: 10.1137/14098435X. [20] Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, vol. 22, American Mathematical Society, 2001. doi: 10.1090/ulect/022. [21] W. J. Palenstijn, K. J. Batenburg and J. Sijbers, Performance improvements for iterative electron tomography reconstruction using graphics processing units (GPUs), J. Struct. Biol., 176 (2011), 250-253. [22] E. T. Quinto, Singularities of the X-ray transform and limited data tomography in $\mathbb{R}^2$ and $\mathbb{R}^3$, SIAM J. Math. Anal., 24 (1993), 1215-1225.  doi: 10.1137/0524069. [23] J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Akad. Wiss., 69 (1917), 262-277. [24] J. E. Rouse, Characterisation of Impact Damage in Carbon Fibre Reinforced Plastics by 3D X-Ray Tomography, PhD thesis, University of Manchester, 2012. [25] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D Nonlinear Phenom., 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F. [26] S. R. Sandoghchi, G. T. Jasion, N. V. Wheeler, S. Jain, Z. Lian, J. P. Wooler, R. P. Boardman, N. K. Baddela, Y. Chen, J. R. Hayes, E. N. Fokoua, T. Bradley, D. R. Gray, S. M. Mousavi, M. N. Petrovich, F. Poletti and D. J. Richardson, X-ray tomography for structural analysis of microstructured and multimaterial optical fibers and preforms, Opt. Express, 22 (2014), 26181. [27] O. Scherzer, Handbook of Mathematical Methods in Imaging, Springer Science + Business Media, 2010. [28] E. Y. Sidky and X. Pan, Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Phys. Med. Biol., 53 (2008), 4777-4807. [29] J. L. Starck, M. Elad and D. L. Donoho, Image decomposition via the combination of sparse representation and a variational approach, IEEE Trans. Image Process., 14 (2005), 1570-1582.  doi: 10.1109/TIP.2005.852206. [30] W. van Aarle, W. J. Palenstijn, J. Cant, E. Janssens, F. Bleichrodt, A. Dabravolski, J. De Beenhouwer, K. J. Batenburg and J. Sijbers, Fast and flexible X-ray tomography using the ASTRA toolbox, Opt. Express, 24 (2016), 25129-25147. [31] W. van Aarle, W. J. Palenstijn, J. De Beenhouwer, T. Altantzis, S. Bals, K. J. Batenburg and J. Sijbers, The ASTRA Toolbox: A platform for advanced algorithm development in electron tomography, Ultramicroscopy, 157 (2015), 35-47. [32] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612. [33] J. Weickert, Anisotropic Diffusion in Image Processing, Teubner-Verlag, Stuttgart, Germany, 1998. [34] J. Weickert, Coherence-enhancing diffusion filtering, Int. J. Comput. Vis., 31 (1999), 111-127.
Comparison on simulated CT reconstruction problem. Regularization parameters in the $\ell_2$-TV and $\ell_2$-DTV models are tuned to maximize the PSNR values. The parameters in DTV are chosen as $a = 0.15$ and $\theta = 20^\circ$
Left: the objects with estimated direction from the noise-free sinogram. Right: the noise-free sinogram overlaid with the plot of the sum of the magnitudes
Left: fibre-crack phantom with fibres along the direction $20^\circ$ and cracks in a circular pattern. Right: simulated noise-free sinogram
Influence of the parameter $\alpha$ in the decomposition model (4) on the results. The SSIM values of $u+w$ are 0.9478, 0.9526, and 0.9513, respectively
Influence of the parameter $\beta$ in the decomposition model (4) on the results. Here, we set $\alpha = 0.7$ and $\lambda = 0.0038$. The SSIM values of $u+w$ are 0.9526 and 0.9545, respectively
Comparison the sinogram splitting method by applying FBP and the variational method, which are introduced in Section 3
Influence of $K$ in the sinogram splitting method on the results by applying the variational method
Carbon fibre sample from [24]
Comparison of the sinogram splitting method with the image decomposition method on a real fibre sample
Direction estimation results for the phantom and the real object shown in Figure 2. Note that the correct main direction for the phantom is 20$^\circ$
 $\rho$ (%) 0 1 3 5 10 20 30 40 Phantom 20.1 20.1 20.1 20.1 20.1 20.1 20.1 31.7 Real 81.5 81.7 81.5 80.9 81.7 79.5 -1.1 -34.9
 $\rho$ (%) 0 1 3 5 10 20 30 40 Phantom 20.1 20.1 20.1 20.1 20.1 20.1 20.1 31.7 Real 81.5 81.7 81.5 80.9 81.7 79.5 -1.1 -34.9
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