Fluid image registration using a finite volume scheme of the incompressible Navier Stokes equation

Mohamed Alahyane; Abdelilah Hakim; Amine Laghrib; Said Raghay

doi:10.3934/ipi.2018044

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October 2018, 12(5): 1055-1081. doi: 10.3934/ipi.2018044

Fluid image registration using a finite volume scheme of the incompressible Navier Stokes equation

Mohamed Alahyane ^1, , Abdelilah Hakim ^1, , Amine Laghrib ^2,, and Said Raghay ^1,

1.	LAMAI, FST Marrakech, Université Cadi Ayyad, Maroc
2.	LMA FST Béni-Mellal, Université Sultan Moulay Slimane, Maroc

* Corresponding author: A. Laghrib

Received September 2016 Revised June 2018 Published July 2018

This paper proposes a stable numerical implementation of the Navier-Stokes equations for fluid image registration, based on a finite volume scheme. Although fluid registration methods have succeeded in handling large deformations in various applications, they still suffer from perturbed solutions due to the choice of the numerical implementation. Thus, a robust numerical scheme in the optimization step is required to enhance the quality of the registration. A key challenge is the use of a finite volume-based scheme, since we have to deal with a hyperbolic equation type. We propose the classical Patankar scheme based on pressure correction, which is called Semi-Implicit Method for Pressure-Linked Equation (SIMPLE). The performance of the proposed algorithm was tested on magnetic resonance images of the human brain and hands, and compared with the classical implementation of the fluid image registration [13], in which the authors used a successive overrelaxation in the spatial domain with Euler integration in time to handle the nonlinear viscous. The obtained results demonstrate the efficiency of the proposed approach, visually and quantitatively, using the differences between images criteria, PSNR and SSIM measures.

Keywords: Image registration, Patankar scheme, fluid registration, SIMPLE, control volume, Navier-Stokes equations.

Mathematics Subject Classification: 65M12, 65M06.

Citation: Mohamed Alahyane, Abdelilah Hakim, Amine Laghrib, Said Raghay. Fluid image registration using a finite volume scheme of the incompressible Navier Stokes equation. Inverse Problems and Imaging, 2018, 12 (5) : 1055-1081. doi: 10.3934/ipi.2018044

References:

[1]	Y. Amit, A nonlinear variational problem for image matching, SIAM Journal on Scientific Computing, 15 (1994), 207-224. doi: 10.1137/0915014.
[2]	B. Berkels, I. Cabrilo, S. Haller, M. Rumpf and K. Schaller, Co-registration of intra-operative brain surface photographs and pre-operative mr images, International Journal of Computer Assisted Radiology and Surgery, 9 (2014), 387-400.
[3]	M. Bertalmio, A. L. Bertozzi and G. Sapiro, Navier-stokes, fluid dynamics, and image and video inpainting, in Computer Vision and Pattern Recognition, 2001. CVPR 2001. Proceedings of the 2001 IEEE Computer Society Conference on, vol. 1, IEEE, 2001, Ⅰ–Ⅰ. doi: 10.1109/CVPR.2001.990497.
[4]	M. Bro-Nielsen and C. Gramkow, Fast fluid registration of medical images, in Visualization in Biomedical Computing, Springer, 1996,265–276. doi: 10.1007/BFb0046964.
[5]	C. Broit, Optimal registration of deformed images.
[6]	M. Burger, J. Modersitzki and L. Ruthotto, A hyperelastic regularization energy for image registration, SIAM Journal on Scientific Computing, 35 (2013), B132-B148. doi: 10.1137/110835955.
[7]	Y. Chen, J. Shi, M. Rao and J.-S. Lee, Deformable multi-modal image registration by maximizing rényi's statistical dependence measure, Inverse Problems & Imaging, 9 (2015), 79-103. doi: 10.3934/ipi.2015.9.79.
[8]	M.-C. Chiang, A. D. Leow, A. D. Klunder, R. A. Dutton, M. Barysheva, S. E. Rose, K. L. McMahon, G. I. De Zubicaray, A. W. Toga and P. M. Thompson, Fluid registration of diffusion tensor images using information theory, IEEE Transactions on Medical Imaging, 27 (2008), 442-456.
[9]	G. Christensen, We-h-202-04: Advanced medical image registration techniques, Medical Physics, 43 (2016), 3845-3845. doi: 10.1118/1.4958005.
[10]	G. E. Christensen, X. Geng, J. G. Kuhl, J. Bruss, T. J. Grabowski, I. A. Pirwani, M. W. Vannier, J. S. Allen and H. Damasio, Introduction to the non-rigid image registration evaluation project (nirep), in International Workshop on Biomedical Image Registration, Springer Berlin Heidelberg, 2006,128–135. doi: 10.1007/11784012_16.
[11]	G. E. Christensen and H. J. Johnson, Consistent image registration, IEEE Transactions on Medical Imaging, 20 (2001), 568-582. doi: 10.1109/42.932742.
[12]	G. E. Christensen, R. D. Rabbitt and M. I. Miller, A deformable neuroanatomy textbook based on viscous fluid mechanics, in 27th Ann. Conf. on Inf. Sciences and Systems, Citeseer, 1993,211–216.
[13]	G. E. Christensen, R. D. Rabbitt and M. I. Miller, Deformable templates using large deformation kinematics, IEEE Transactions on Image Processing, 5 (1996), 1435-1447. doi: 10.1109/83.536892.
[14]	G. E. Christensen, Deformable shape models for anatomy.
[15]	T. Chung, Computational Fluid Dynamics, Cambridge university press, Cambridge, 2002. doi: 10.1017/CBO9780511606205.
[16]	J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics, Second revised edition. Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-642-98037-4.
[17]	B. Fischer and J. Modersitzki, Ill-posed medicine–An introduction to image registration, Inverse Problems, 24 (2008), 034008, 16pp. doi: 10.1088/0266-5611/24/3/034008.
[18]	C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, vol. 83, Cambridge University Press, 2001. doi: 10.1017/CBO9780511546754.
[19]	A. Goshtasby, Image Registration: Principles, Tools and Methods, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4471-2458-0.
[20]	E. Haber and J. Modersitzki, Numerical methods for volume preserving image registration, Inverse Problems, 20 (2004), 1621-1638. doi: 10.1088/0266-5611/20/5/018.
[21]	F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Physics of Fluids, 8 (1965), 2182-2189. doi: 10.1063/1.1761178.
[22]	J. A. Iglesias, M. Rumpf and O. Scherzer, Shape-aware matching of implicit surfaces based on thin shell energies, Foundations of Computational Mathematics, (2017), 1-37. doi: 10.1007/s10208-017-9357-9.
[23]	A. Klein, J. Andersson, B. A. Ardekani, J. Ashburner, B. Avants, M.-C. Chiang, G. E. Christensen, D. L. Collins, J. Gee and P. Hellier, Evaluation of 14 nonlinear deformation algorithms applied to human brain mri registration, Neuroimage, 46 (2009), 786-802. doi: 10.1016/j.neuroimage.2008.12.037.
[24]	S. K. Kyriacou, C. Davatzikos, S. J. Zinreich and R. N. Bryan, Nonlinear elastic registration of brain images with tumor pathology using a biomechanical model [mri], IEEE Transactions on Medical Imaging, 18 (1999), 580-592. doi: 10.1109/42.790458.
[25]	A. Laghrib, A. Hakim, S. Raghay and M. El Rhabi, Robust super resolution of images with non-parametric deformations using an elastic registration, Appl. Math. Sci, 8 (2014), 8897-8907. doi: 10.12988/ams.2014.49751.
[26]	C. Le Guyader and L. A. Vese, A combined segmentation and registration framework with a nonlinear elasticity smoother, Scale Space and Variational Methods in Computer Vision, (2009), 600-611. doi: 10.1007/978-3-642-02256-2_50.
[27]	H. Lester, S. R. Arridge, K. M. Jansons, L. Lemieux, J. V. Hajnal and A. Oatridge, Non-linear registration with the variable viscosity fluid algorithm, in Information Processing in Medical Imaging, Springer, 1999,238–251. doi: 10.1007/3-540-48714-X_18.
[28]	R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, vol. 31, Cambridge university press, 2002. doi: 10.1017/CBO9780511791253.
[29]	B. Likar and F. Pernuš, A hierarchical approach to elastic registration based on mutual information, Image and Vision Computing, 19 (2001), 33-44. doi: 10.1016/S0262-8856(00)00053-6.
[30]	J. A. Maintz and M. A. Viergever, A survey of medical image registration, Medical Image Analysis, 2 (1998), 1-36. doi: 10.1016/S1361-8415(01)80026-8.
[31]	J. Modersitzki, Numerical Methods for Image Registration, Oxford university press, 2004.
[32]	J. Modersitzki, FAIR: Flexible Algorithms for Image Registration, vol. 6, SIAM, 2009. doi: 10.1137/1.9780898718843.
[33]	S. Ozere, C. Gout and C. Le Guyader, Joint segmentation/registration model by shape alignment via weighted total variation minimization and nonlinear elasticity, SIAM Journal on Imaging Sciences, 8 (2015), 1981-2020. doi: 10.1137/140990620.
[34]	S. Ozere and C. Le Guyader, Topology preservation for image-registration-related deformation fields, Communications in Mathematical Sciences, 13 (2015), 1135-1161. doi: 10.4310/CMS.2015.v13.n5.a4.
[35]	S. Patankar, Numerical Heat Transfer And Fluid Flow CRC press, 1980.
[36]	S. V. Patankar, A calculation procedure for two-dimensional elliptic situations, Numerical Heat Transfer, 4 (1981), 409-425.
[37]	S. V. Patankar and D. B. Spalding, A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Numerical Prediction of Flow, Heat Transfer, Turbulence and Combustion, (1983), 54-73. doi: 10.1016/B978-0-08-030937-8.50013-1.
[38]	W. Peckar, C. Schnörr, K. Rohr and H. S. Stiehl, Parameter-free elastic deformation approach for 2d and 3d registration using prescribed displacements, Journal of Mathematical Imaging and Vision, 10 (1999), 143-162. doi: 10.1023/A:1008375006703.
[39]	D. Rueckert and J. A. Schnabel, Medical image registration, in Biomedical Image Processing, Springer, 2010,131–154. doi: 10.1007/978-3-642-15816-2_5.
[40]	J. S. Suri, D. Wilson and S. Laxminarayan, Handbook of Biomedical Image Analysis, vol. 2, Springer Science & Business Media, 2005.
[41]	R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, vol. 66, Siam, 1995. doi: 10.1137/1.9781611970050.
[42]	R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, vol. 343, American Mathematical Soc., 2001. doi: 10.1090/chel/343.
[43]	Z. Wang, A. Bovik, H. Sheikh and E. 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.
[44]	Z. Yi and J. W. Wan, An inviscid model for nonrigid image registration, Inverse Problems and Imaging (IPI), 5 (2011), 263-284. doi: 10.3934/ipi.2011.5.263.
[45]	J. Zhang, K. Chen and B. Yu, An improved discontinuity-preserving image registration model and its fast algorithm, Applied Mathematical Modelling, 40 (2016), 10740-10759. doi: 10.1016/j.apm.2016.08.009.

show all references

References:

[1]	Y. Amit, A nonlinear variational problem for image matching, SIAM Journal on Scientific Computing, 15 (1994), 207-224. doi: 10.1137/0915014.
[2]	B. Berkels, I. Cabrilo, S. Haller, M. Rumpf and K. Schaller, Co-registration of intra-operative brain surface photographs and pre-operative mr images, International Journal of Computer Assisted Radiology and Surgery, 9 (2014), 387-400.
[3]	M. Bertalmio, A. L. Bertozzi and G. Sapiro, Navier-stokes, fluid dynamics, and image and video inpainting, in Computer Vision and Pattern Recognition, 2001. CVPR 2001. Proceedings of the 2001 IEEE Computer Society Conference on, vol. 1, IEEE, 2001, Ⅰ–Ⅰ. doi: 10.1109/CVPR.2001.990497.
[4]	M. Bro-Nielsen and C. Gramkow, Fast fluid registration of medical images, in Visualization in Biomedical Computing, Springer, 1996,265–276. doi: 10.1007/BFb0046964.
[5]	C. Broit, Optimal registration of deformed images.
[6]	M. Burger, J. Modersitzki and L. Ruthotto, A hyperelastic regularization energy for image registration, SIAM Journal on Scientific Computing, 35 (2013), B132-B148. doi: 10.1137/110835955.
[7]	Y. Chen, J. Shi, M. Rao and J.-S. Lee, Deformable multi-modal image registration by maximizing rényi's statistical dependence measure, Inverse Problems & Imaging, 9 (2015), 79-103. doi: 10.3934/ipi.2015.9.79.
[8]	M.-C. Chiang, A. D. Leow, A. D. Klunder, R. A. Dutton, M. Barysheva, S. E. Rose, K. L. McMahon, G. I. De Zubicaray, A. W. Toga and P. M. Thompson, Fluid registration of diffusion tensor images using information theory, IEEE Transactions on Medical Imaging, 27 (2008), 442-456.
[9]	G. Christensen, We-h-202-04: Advanced medical image registration techniques, Medical Physics, 43 (2016), 3845-3845. doi: 10.1118/1.4958005.
[10]	G. E. Christensen, X. Geng, J. G. Kuhl, J. Bruss, T. J. Grabowski, I. A. Pirwani, M. W. Vannier, J. S. Allen and H. Damasio, Introduction to the non-rigid image registration evaluation project (nirep), in International Workshop on Biomedical Image Registration, Springer Berlin Heidelberg, 2006,128–135. doi: 10.1007/11784012_16.
[11]	G. E. Christensen and H. J. Johnson, Consistent image registration, IEEE Transactions on Medical Imaging, 20 (2001), 568-582. doi: 10.1109/42.932742.
[12]	G. E. Christensen, R. D. Rabbitt and M. I. Miller, A deformable neuroanatomy textbook based on viscous fluid mechanics, in 27th Ann. Conf. on Inf. Sciences and Systems, Citeseer, 1993,211–216.
[13]	G. E. Christensen, R. D. Rabbitt and M. I. Miller, Deformable templates using large deformation kinematics, IEEE Transactions on Image Processing, 5 (1996), 1435-1447. doi: 10.1109/83.536892.
[14]	G. E. Christensen, Deformable shape models for anatomy.
[15]	T. Chung, Computational Fluid Dynamics, Cambridge university press, Cambridge, 2002. doi: 10.1017/CBO9780511606205.
[16]	J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics, Second revised edition. Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-642-98037-4.
[17]	B. Fischer and J. Modersitzki, Ill-posed medicine–An introduction to image registration, Inverse Problems, 24 (2008), 034008, 16pp. doi: 10.1088/0266-5611/24/3/034008.
[18]	C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, vol. 83, Cambridge University Press, 2001. doi: 10.1017/CBO9780511546754.
[19]	A. Goshtasby, Image Registration: Principles, Tools and Methods, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4471-2458-0.
[20]	E. Haber and J. Modersitzki, Numerical methods for volume preserving image registration, Inverse Problems, 20 (2004), 1621-1638. doi: 10.1088/0266-5611/20/5/018.
[21]	F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Physics of Fluids, 8 (1965), 2182-2189. doi: 10.1063/1.1761178.
[22]	J. A. Iglesias, M. Rumpf and O. Scherzer, Shape-aware matching of implicit surfaces based on thin shell energies, Foundations of Computational Mathematics, (2017), 1-37. doi: 10.1007/s10208-017-9357-9.
[23]	A. Klein, J. Andersson, B. A. Ardekani, J. Ashburner, B. Avants, M.-C. Chiang, G. E. Christensen, D. L. Collins, J. Gee and P. Hellier, Evaluation of 14 nonlinear deformation algorithms applied to human brain mri registration, Neuroimage, 46 (2009), 786-802. doi: 10.1016/j.neuroimage.2008.12.037.
[24]	S. K. Kyriacou, C. Davatzikos, S. J. Zinreich and R. N. Bryan, Nonlinear elastic registration of brain images with tumor pathology using a biomechanical model [mri], IEEE Transactions on Medical Imaging, 18 (1999), 580-592. doi: 10.1109/42.790458.
[25]	A. Laghrib, A. Hakim, S. Raghay and M. El Rhabi, Robust super resolution of images with non-parametric deformations using an elastic registration, Appl. Math. Sci, 8 (2014), 8897-8907. doi: 10.12988/ams.2014.49751.
[26]	C. Le Guyader and L. A. Vese, A combined segmentation and registration framework with a nonlinear elasticity smoother, Scale Space and Variational Methods in Computer Vision, (2009), 600-611. doi: 10.1007/978-3-642-02256-2_50.
[27]	H. Lester, S. R. Arridge, K. M. Jansons, L. Lemieux, J. V. Hajnal and A. Oatridge, Non-linear registration with the variable viscosity fluid algorithm, in Information Processing in Medical Imaging, Springer, 1999,238–251. doi: 10.1007/3-540-48714-X_18.
[28]	R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, vol. 31, Cambridge university press, 2002. doi: 10.1017/CBO9780511791253.
[29]	B. Likar and F. Pernuš, A hierarchical approach to elastic registration based on mutual information, Image and Vision Computing, 19 (2001), 33-44. doi: 10.1016/S0262-8856(00)00053-6.
[30]	J. A. Maintz and M. A. Viergever, A survey of medical image registration, Medical Image Analysis, 2 (1998), 1-36. doi: 10.1016/S1361-8415(01)80026-8.
[31]	J. Modersitzki, Numerical Methods for Image Registration, Oxford university press, 2004.
[32]	J. Modersitzki, FAIR: Flexible Algorithms for Image Registration, vol. 6, SIAM, 2009. doi: 10.1137/1.9780898718843.
[33]	S. Ozere, C. Gout and C. Le Guyader, Joint segmentation/registration model by shape alignment via weighted total variation minimization and nonlinear elasticity, SIAM Journal on Imaging Sciences, 8 (2015), 1981-2020. doi: 10.1137/140990620.
[34]	S. Ozere and C. Le Guyader, Topology preservation for image-registration-related deformation fields, Communications in Mathematical Sciences, 13 (2015), 1135-1161. doi: 10.4310/CMS.2015.v13.n5.a4.
[35]	S. Patankar, Numerical Heat Transfer And Fluid Flow CRC press, 1980.
[36]	S. V. Patankar, A calculation procedure for two-dimensional elliptic situations, Numerical Heat Transfer, 4 (1981), 409-425.
[37]	S. V. Patankar and D. B. Spalding, A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Numerical Prediction of Flow, Heat Transfer, Turbulence and Combustion, (1983), 54-73. doi: 10.1016/B978-0-08-030937-8.50013-1.
[38]	W. Peckar, C. Schnörr, K. Rohr and H. S. Stiehl, Parameter-free elastic deformation approach for 2d and 3d registration using prescribed displacements, Journal of Mathematical Imaging and Vision, 10 (1999), 143-162. doi: 10.1023/A:1008375006703.
[39]	D. Rueckert and J. A. Schnabel, Medical image registration, in Biomedical Image Processing, Springer, 2010,131–154. doi: 10.1007/978-3-642-15816-2_5.
[40]	J. S. Suri, D. Wilson and S. Laxminarayan, Handbook of Biomedical Image Analysis, vol. 2, Springer Science & Business Media, 2005.
[41]	R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, vol. 66, Siam, 1995. doi: 10.1137/1.9781611970050.
[42]	R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, vol. 343, American Mathematical Soc., 2001. doi: 10.1090/chel/343.
[43]	Z. Wang, A. Bovik, H. Sheikh and E. 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.
[44]	Z. Yi and J. W. Wan, An inviscid model for nonrigid image registration, Inverse Problems and Imaging (IPI), 5 (2011), 263-284. doi: 10.3934/ipi.2011.5.263.
[45]	J. Zhang, K. Chen and B. Yu, An improved discontinuity-preserving image registration model and its fast algorithm, Applied Mathematical Modelling, 40 (2016), 10740-10759. doi: 10.1016/j.apm.2016.08.009.

Figure 1. Control volume of $v$ and $p$

Quantities	Navier-Stokes	Image Registration

$u$	fluid displacement	pixel displacement
$v$	fluid velocity	pixel velocity
$p$	pressure	the effect of each region
$\nu$	fluid viscosity	factor of diffusion
$\nabla\cdot v=0$	incompressible fluid	the pixels are not condensable


Image	Parameters
	Reynold number $R_{e}$	Iteration number $N$	Time-step $\tau$
Human hands	500	400	0.01
Human brain 1	1000	500	0.05
Human brain 2	1000	500	0.01
Human head	100	400	0.01
EPI slice	100	300	0.01

Image			Method
	Image size	Metric	Fluid image registration	proposed
Human hands	$128 \times 128$	PSNR	27.0273	$\bf{28.0153}$
		SSIM	0.8265	$\bf{0.8342}$
Human brain 1	$128 \times 128$	PSNR	24.1694	$\bf{25.4103}$
		SSIM	0.8860	$\bf{0.8995}$
Human brain 2	$256 \times 256$	PSNR	26.5761	$\bf{27.7770}$
		SSIM	0.8181	$\bf{0.8461}$
Human head	$400 \times 400$	PSNR	23.6568	$\bf{24.9482}$
		SSIM	0.7963	$\bf{0.8268}$
EPI slice	$256 \times 256$	PSNR	31.4214	$\bf{32.8485}$
		SSIM	0.9278	$\bf{0.9480}$

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Fluid image registration using a finite volume scheme of the incompressible Navier Stokes equation

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