May  2015, 9(2): 447-467. doi: 10.3934/ipi.2015.9.447

4D-CT reconstruction with unified spatial-temporal patch-based regularization

1. 

The Manchester X-ray Imaging Facility, School of Materials, The University of Manchester, Manchester, M13 9PL, United Kingdom, United Kingdom, United Kingdom, United Kingdom

2. 

School of Mathematics, The University of Manchester, Alan Turing Building, Manchester, M13 9PL, United Kingdom, United Kingdom

3. 

iMinds-Vision Lab, The University of Antwerp, Wilrijk, B-2610, Belgium

4. 

Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut (PSI), Villigen, 5232, Switzerland

Received  July 2013 Revised  April 2014 Published  March 2015

In this paper, we consider a limited data reconstruction problem for temporarily evolving computed tomography (CT), where some regions are static during the whole scan and some are dynamic (intensely or slowly changing). When motion occurs during a tomographic experiment one would like to minimize the number of projections used and reconstruct the image iteratively. To ensure stability of the iterative method spatial and temporal constraints are highly desirable. Here, we present a novel spatial-temporal regularization approach where all time frames are reconstructed collectively as a unified function of space and time. Our method has two main differences from the state-of-the-art spatial-temporal regularization methods. Firstly, all available temporal information is used to improve the spatial resolution of each time frame. Secondly, our method does not treat spatial and temporal penalty terms separately but rather unifies them in one regularization term. Additionally we optimize the temporal smoothing part of the method by considering the non-local patches which are most likely to belong to one intensity class. This modification significantly improves the signal-to-noise ratio of the reconstructed images and reduces computational time. The proposed approach is used in combination with golden ratio sampling of the projection data which allows one to find a better trade-off between temporal and spatial resolution scenarios.
Citation: 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 and Imaging, 2015, 9 (2) : 447-467. doi: 10.3934/ipi.2015.9.447
References:
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S. Bougleux, G. Peyre and L. Cohen, Non-local regularization of inverse problems, Inverse Probl. Imaging, 5 (2011), 511-530. doi: 10.3934/ipi.2011.5.511.

[2]

K. S. Brown, S. Schluter, A. Sheppard and D. Wildenschild, On the challenges of measuring interfacial characteristics of three-phase fluid flow with x-ray microtomography, Journal of Microscopy, 253 (2014), 171-182. doi: 10.1111/jmi.12106.

[3]

A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms with a new one, Multiscale Model. Simul., 4 (2005), 490-530. doi: 10.1137/040616024.

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C. L. Byrne, Applied Iterative Methods, Natick, MA: Peters, 2008. doi: 10.1201/b10651.

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P. L. Combettes and J.-C. Pesquet, Proximal splitting methods in signal processing, in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optim. Appl., {49}, Springer, New York, 2011, 185-212. doi: 10.1007/978-1-4419-9569-8_10.

[6]

H. Gao, J-F. Cai, Z. Shen and H. Zhao, Robust principal component analysis-based four-dimensional computed tomography, Phys Med. Biol., 56 (2011), 3181-3198. doi: 10.1088/0031-9155/56/11/002.

[7]

X. Jia, Y. Lou, B. Dong, Z. Tian and S. Jiang, 4D computed tomography reconstruction from few-projection data via temporal non-local regularization, in Medical Image Computing and Computer-Assisted Intervention - MICCAI 2010, Lecture Notes in Computer Science, 6361, Springer, Berlin-Heidelberg, 2010, 143-150. doi: 10.1007/978-3-642-15705-9_18.

[8]

A. P. Kaestner, S. Hartmann, G. Kuhne, G. Frei, C. Grunzweig, L. Josic, F. Schmid and E. H. Lehmann, The ICON beamline - a facility for cold neutron imaging at SINQ, Nuclear Instruments and Methods in Physics Research, 659 (2011), 387-393. doi: 10.1016/j.nima.2011.08.022.

[9]

A. P. Kaestner, B. Muench, P. Trtik and L. G. Butler, Spatio-temporal computed tomography of dynamic processes, SPIE Optical Engineering, 50 (2011), 1-10.

[10]

J. Kaipio and E. Somersalo, Statistical inverse problems: discretization, model reduction and inverse crimes, Journal of Computational and Applied Mathematics, 198 (2007), 493-504. doi: 10.1016/j.cam.2005.09.027.

[11]

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, IEEE Press, New York, 1998. doi: 10.1137/1.9780898719277.

[12]

D. Kazantsev, W. R. B. Lionheart, P. J. Withers and P. D. Lee, GPU accelerated 4D-CT reconstruction using higher order PDE regularization in spatial and temporal domains, in Proc. CMSSE, 3, Almeria, Spain, 2013, 843-852.

[13]

D. Kazantsev, S. R. Arridge, S. Pedemonte, A. Bousse, K. Erlandsson, B. F. Hutton and S. Ourselin, An anatomically driven anisotropic diffusion filtering method for 3D SPECT reconstruction, Phys Med. Biol., 57 (2012), 3793-3810. doi: 10.1088/0031-9155/57/12/3793.

[14]

T. Kohler, A projection access scheme for iterative reconstruction based on the golden section, in IEEE Symposium Conference Record Nuclear Science 2004, Vol. 6, IEEE, 2004, 3961-3965. doi: 10.1109/NSSMIC.2004.1466745.

[15]

D. S. Lalush and M. N. Wernick, Iterative image reconstruction, in Emission Tomography (ed. Mark T. Madsen), Academic Press, 2004, 443-472. doi: 10.1016/B978-012744482-6/50024-7.

[16]

D. S. Lalush and B. M. W. Tsui, Block-iterative techniques for fast 4D reconstruction using a priori motion models in gated cardiac SPECT, Phys Med. Biol., 43 (1998), 875-886. doi: 10.1088/0031-9155/43/4/015.

[17]

S. Z. Li, Markov Random Field Modeling in Image Analysis, Springer, 2009. doi: 10.1007/978-1-84800-279-1.

[18]

J. Nocedal and S. Wright, Numerical Optimization, Springer, 2006. doi: 10.1007/978-0-387-40065-5.

[19]

W. J. Palenstijn, K. J. Batenburg and J. Sijbers, Performance improvements for iterative electron tomography reconstruction using graphics processing units (GPUs), J. of Struct. Biol., 176 (2011), 250-253. doi: 10.1016/j.jsb.2011.07.017.

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, PB regularization package (open-source code),, , (). 

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K. Perlin, Improving noise, ACM T. Graphic., 21 (2002), 681-682. doi: 10.1145/566654.566636.

[22]

J. Qi and R. M. Leahy, Iterative reconstruction techniques in emission computed tomography, Phys Med. Biol., 51 (2006), R541-R578. doi: 10.1088/0031-9155/51/15/R01.

[23]

A. Rahmim, J. Tang and H. Zaidi, Four-dimensional (4D) image reconstruction strategies in dynamic PET: Beyond conventional independent frame reconstruction, Med. Phys., 36 (2009), 3654-3670. doi: 10.1118/1.3160108.

[24]

L. Ritschl, S. Sawall, M. Knaup, A. Hess and M. Kachelrieß, Iterative 4D cardiac micro-CT image reconstruction using an adaptive spatio-temporal sparsity prior, Phys Med. Biol., 57 (2012), 1517-1526. doi: 10.1088/0031-9155/57/6/1517.

[25]

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

[26]

M. Strobl, I. Manke, N. Kardjilov, A. Hilger, M. Dawson and J. Banhart, Advances in neutron radiography and tomography, J. Phys. D: Appl. Phys., 42 (2009), 1-21. doi: 10.1088/0022-3727/42/24/243001.

[27]

W. M. Thompson, W. R. Lionheart and E. J. Morton, Real-Time Imaging with a high speed X-Ray CT system, in Proc. 6th International Symposium on Process Tomography, 2012.

[28]

G. Van Eyndhoven, K. J. Batenburg and J. Sijbers, Region-based iterative reconstruction of structurally changing objects in CT, IEEE Trans. on Image Process, 23 (2014), 909-919. doi: 10.1109/TIP.2013.2297024.

[29]

H. Wu, A. Maier, R. Fahrig and J. Hornegger, Spatial-temporal total variation regularization (STTVR) for 4D-CT reconstruction, Proc. SPIE, 8313 (2012), 237-240. doi: 10.1117/12.911162.

[30]

G. Wang and J. Qi, Penalized likelihood PET image reconstruction using patch-based edge-preserving regularization, IEEE Transactions on Medical Imaging, 31 (2012), 2194-2204. doi: 10.1109/tmi.2012.2211378.

[31]

Z. Yang and M. Jacob, Nonlocal regularization of inverse problems: A unified variational framework, IEEE Trans. on Image Process, 22 (2013), 3192-3203. doi: 10.1109/tip.2012.2216278.

[32]

Z. Yang and M. Jacob, Robust non-local regularization framework for motion compensated dynamic imaging without explicit motion estimation, in 2012 9th IEEE International Symposium on Biomedical Imaging (ISBI), 2012, 1056-1059. doi: 10.1109/ISBI.2012.6235740.

show all references

References:
[1]

S. Bougleux, G. Peyre and L. Cohen, Non-local regularization of inverse problems, Inverse Probl. Imaging, 5 (2011), 511-530. doi: 10.3934/ipi.2011.5.511.

[2]

K. S. Brown, S. Schluter, A. Sheppard and D. Wildenschild, On the challenges of measuring interfacial characteristics of three-phase fluid flow with x-ray microtomography, Journal of Microscopy, 253 (2014), 171-182. doi: 10.1111/jmi.12106.

[3]

A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms with a new one, Multiscale Model. Simul., 4 (2005), 490-530. doi: 10.1137/040616024.

[4]

C. L. Byrne, Applied Iterative Methods, Natick, MA: Peters, 2008. doi: 10.1201/b10651.

[5]

P. L. Combettes and J.-C. Pesquet, Proximal splitting methods in signal processing, in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optim. Appl., {49}, Springer, New York, 2011, 185-212. doi: 10.1007/978-1-4419-9569-8_10.

[6]

H. Gao, J-F. Cai, Z. Shen and H. Zhao, Robust principal component analysis-based four-dimensional computed tomography, Phys Med. Biol., 56 (2011), 3181-3198. doi: 10.1088/0031-9155/56/11/002.

[7]

X. Jia, Y. Lou, B. Dong, Z. Tian and S. Jiang, 4D computed tomography reconstruction from few-projection data via temporal non-local regularization, in Medical Image Computing and Computer-Assisted Intervention - MICCAI 2010, Lecture Notes in Computer Science, 6361, Springer, Berlin-Heidelberg, 2010, 143-150. doi: 10.1007/978-3-642-15705-9_18.

[8]

A. P. Kaestner, S. Hartmann, G. Kuhne, G. Frei, C. Grunzweig, L. Josic, F. Schmid and E. H. Lehmann, The ICON beamline - a facility for cold neutron imaging at SINQ, Nuclear Instruments and Methods in Physics Research, 659 (2011), 387-393. doi: 10.1016/j.nima.2011.08.022.

[9]

A. P. Kaestner, B. Muench, P. Trtik and L. G. Butler, Spatio-temporal computed tomography of dynamic processes, SPIE Optical Engineering, 50 (2011), 1-10.

[10]

J. Kaipio and E. Somersalo, Statistical inverse problems: discretization, model reduction and inverse crimes, Journal of Computational and Applied Mathematics, 198 (2007), 493-504. doi: 10.1016/j.cam.2005.09.027.

[11]

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, IEEE Press, New York, 1998. doi: 10.1137/1.9780898719277.

[12]

D. Kazantsev, W. R. B. Lionheart, P. J. Withers and P. D. Lee, GPU accelerated 4D-CT reconstruction using higher order PDE regularization in spatial and temporal domains, in Proc. CMSSE, 3, Almeria, Spain, 2013, 843-852.

[13]

D. Kazantsev, S. R. Arridge, S. Pedemonte, A. Bousse, K. Erlandsson, B. F. Hutton and S. Ourselin, An anatomically driven anisotropic diffusion filtering method for 3D SPECT reconstruction, Phys Med. Biol., 57 (2012), 3793-3810. doi: 10.1088/0031-9155/57/12/3793.

[14]

T. Kohler, A projection access scheme for iterative reconstruction based on the golden section, in IEEE Symposium Conference Record Nuclear Science 2004, Vol. 6, IEEE, 2004, 3961-3965. doi: 10.1109/NSSMIC.2004.1466745.

[15]

D. S. Lalush and M. N. Wernick, Iterative image reconstruction, in Emission Tomography (ed. Mark T. Madsen), Academic Press, 2004, 443-472. doi: 10.1016/B978-012744482-6/50024-7.

[16]

D. S. Lalush and B. M. W. Tsui, Block-iterative techniques for fast 4D reconstruction using a priori motion models in gated cardiac SPECT, Phys Med. Biol., 43 (1998), 875-886. doi: 10.1088/0031-9155/43/4/015.

[17]

S. Z. Li, Markov Random Field Modeling in Image Analysis, Springer, 2009. doi: 10.1007/978-1-84800-279-1.

[18]

J. Nocedal and S. Wright, Numerical Optimization, Springer, 2006. doi: 10.1007/978-0-387-40065-5.

[19]

W. J. Palenstijn, K. J. Batenburg and J. Sijbers, Performance improvements for iterative electron tomography reconstruction using graphics processing units (GPUs), J. of Struct. Biol., 176 (2011), 250-253. doi: 10.1016/j.jsb.2011.07.017.

[20]

, PB regularization package (open-source code),, , (). 

[21]

K. Perlin, Improving noise, ACM T. Graphic., 21 (2002), 681-682. doi: 10.1145/566654.566636.

[22]

J. Qi and R. M. Leahy, Iterative reconstruction techniques in emission computed tomography, Phys Med. Biol., 51 (2006), R541-R578. doi: 10.1088/0031-9155/51/15/R01.

[23]

A. Rahmim, J. Tang and H. Zaidi, Four-dimensional (4D) image reconstruction strategies in dynamic PET: Beyond conventional independent frame reconstruction, Med. Phys., 36 (2009), 3654-3670. doi: 10.1118/1.3160108.

[24]

L. Ritschl, S. Sawall, M. Knaup, A. Hess and M. Kachelrieß, Iterative 4D cardiac micro-CT image reconstruction using an adaptive spatio-temporal sparsity prior, Phys Med. Biol., 57 (2012), 1517-1526. doi: 10.1088/0031-9155/57/6/1517.

[25]

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

[26]

M. Strobl, I. Manke, N. Kardjilov, A. Hilger, M. Dawson and J. Banhart, Advances in neutron radiography and tomography, J. Phys. D: Appl. Phys., 42 (2009), 1-21. doi: 10.1088/0022-3727/42/24/243001.

[27]

W. M. Thompson, W. R. Lionheart and E. J. Morton, Real-Time Imaging with a high speed X-Ray CT system, in Proc. 6th International Symposium on Process Tomography, 2012.

[28]

G. Van Eyndhoven, K. J. Batenburg and J. Sijbers, Region-based iterative reconstruction of structurally changing objects in CT, IEEE Trans. on Image Process, 23 (2014), 909-919. doi: 10.1109/TIP.2013.2297024.

[29]

H. Wu, A. Maier, R. Fahrig and J. Hornegger, Spatial-temporal total variation regularization (STTVR) for 4D-CT reconstruction, Proc. SPIE, 8313 (2012), 237-240. doi: 10.1117/12.911162.

[30]

G. Wang and J. Qi, Penalized likelihood PET image reconstruction using patch-based edge-preserving regularization, IEEE Transactions on Medical Imaging, 31 (2012), 2194-2204. doi: 10.1109/tmi.2012.2211378.

[31]

Z. Yang and M. Jacob, Nonlocal regularization of inverse problems: A unified variational framework, IEEE Trans. on Image Process, 22 (2013), 3192-3203. doi: 10.1109/tip.2012.2216278.

[32]

Z. Yang and M. Jacob, Robust non-local regularization framework for motion compensated dynamic imaging without explicit motion estimation, in 2012 9th IEEE International Symposium on Biomedical Imaging (ISBI), 2012, 1056-1059. doi: 10.1109/ISBI.2012.6235740.

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