American Institute of Mathematical Sciences

Advanced
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 & 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. Broit, Optimal registration of deformed images. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[9]

G. Christensen, We-h-202-04: Advanced medical image registration techniques, Medical Physics, 43 (2016), 3845-3845.  doi: 10.1118/1.4958005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[14]

G. E. Christensen, Deformable shape models for anatomy. Google Scholar

[15]

T. Chung, Computational Fluid Dynamics, Cambridge university press, Cambridge, 2002. doi: 10.1017/CBO9780511606205.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

A. Goshtasby, Image Registration: Principles, Tools and Methods, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4471-2458-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, vol. 31, Cambridge university press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

J. Modersitzki, Numerical Methods for Image Registration, Oxford university press, 2004.  Google Scholar

[32]

J. Modersitzki, FAIR: Flexible Algorithms for Image Registration, vol. 6, SIAM, 2009. doi: 10.1137/1.9780898718843.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

S. Patankar, Numerical Heat Transfer And Fluid Flow CRC press, 1980. Google Scholar

[36]

S. V. Patankar, A calculation procedure for two-dimensional elliptic situations, Numerical Heat Transfer, 4 (1981), 409-425.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

J. S. Suri, D. Wilson and S. Laxminarayan, Handbook of Biomedical Image Analysis, vol. 2, Springer Science & Business Media, 2005. Google Scholar

[41]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, vol. 66, Siam, 1995. doi: 10.1137/1.9781611970050.  Google Scholar

[42]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, vol. 343, American Mathematical Soc., 2001. doi: 10.1090/chel/343.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. Broit, Optimal registration of deformed images. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[9]

G. Christensen, We-h-202-04: Advanced medical image registration techniques, Medical Physics, 43 (2016), 3845-3845.  doi: 10.1118/1.4958005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[14]

G. E. Christensen, Deformable shape models for anatomy. Google Scholar

[15]

T. Chung, Computational Fluid Dynamics, Cambridge university press, Cambridge, 2002. doi: 10.1017/CBO9780511606205.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

A. Goshtasby, Image Registration: Principles, Tools and Methods, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4471-2458-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, vol. 31, Cambridge university press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

J. Modersitzki, Numerical Methods for Image Registration, Oxford university press, 2004.  Google Scholar

[32]

J. Modersitzki, FAIR: Flexible Algorithms for Image Registration, vol. 6, SIAM, 2009. doi: 10.1137/1.9780898718843.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

S. Patankar, Numerical Heat Transfer And Fluid Flow CRC press, 1980. Google Scholar

[36]

S. V. Patankar, A calculation procedure for two-dimensional elliptic situations, Numerical Heat Transfer, 4 (1981), 409-425.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

J. S. Suri, D. Wilson and S. Laxminarayan, Handbook of Biomedical Image Analysis, vol. 2, Springer Science & Business Media, 2005. Google Scholar

[41]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, vol. 66, Siam, 1995. doi: 10.1137/1.9781611970050.  Google Scholar

[42]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, vol. 343, American Mathematical Soc., 2001. doi: 10.1090/chel/343.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Control volume of $v$ and $p$
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Figure 2.  The Reference and Template images of human hands
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Figure 3.  The obtained Template image, using the fluid registration and the proposed approach, compared with the Original one for the human hands image
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Figure 4.  (a) The whole deformation grid. The whole deformation grid and the Jacobian determinant of the transformation, where $\min (\det(J)) = 0.0923$ (human hands image).
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Figure 5.  The time progression of the transformation applied to a rectangular grid with respect to the iterations (human hands).
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Figure 6.  Difference error between template and reference images using fluid registration (on the left) and the proposed approach (on the right)
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Figure 7.  The Reference and Template images of human brain 1.
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Figure 8.  The obtained Template image, using the fluid image registration and the proposed approach, compared with the Original one (human brain 1)
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Figure 9.  (a) The whole deformation grid and the Jacobian determinant of the transformation, where $\min (\det(J)) = 0.09$
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Figure 10.  The time progression of the transformation applied to a rectangular grid with respect to the iterations (human brain 1).
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Figure 11.  Difference error between template and reference images of human brain 1 using fluid registration (on the left) and the proposed approach (on the right).
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Figure 12.  The Reference and Template images of human brain 2
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Figure 13.  The obtained Template image, using the fluid image registration and the proposed approach, compared with the Original one (human brain 2)
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Figure 14.  (a) The whole deformation grid. The whole deformation grid and the Jacobian determinant of the transformation, where $\min (\det(J)) = 0.0995$
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Figure 15.  The time progression of the transformation applied to a rectangular grid with respect to the iterations (human brain 2)
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Figure 16.  Difference error between template and reference images of human brain 2 using fluid registration (on the left) and the proposed approach (on the right).
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Figure 17.  The Reference and Template images of human brain
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Figure 18.  The obtained Template image, using the fluid image registration and the proposed approach, compared with the Original one (human brain)
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Figure 19.  (a) The whole deformation grid. The whole deformation grid and the Jacobian determinant of the transformation, where $\min (\det(J)) = 0.0935$
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Figure 20.  The time progression of the transformation applied to a rectangular grid with respect to the iterations (human brain)
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Figure 21.  Difference error between template and reference images of human brain using fluid registration (on the left) and the proposed approach (on the right)
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Figure 22.  The Reference and Template images of EPI slice
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Figure 23.  The obtained Template image, using the fluid image registration and the proposed approach, compared with the Original one (EPI slice)
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Figure 24.  (a) The whole deformation grid and the Jacobian determinant of the transformation, where $\min (\det(J)) = 0.0965$
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Figure 25.  The time progression of the transformation applied to a rectangular grid with respect to the iterations (EPI slice)
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Figure 26.  Difference error between template and reference images of EPI slice using fluid registration (on the left) and the proposed approach (on the right)
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Figure 27.  The obtained Template image of the human hands by the proposed approach using different values of Reynold's number $R_{e}$ with the associated Jacobian determinant
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Figure 28.  The obtained Template image of the human brain 1 by the proposed approach using different values of Reynold's number $R_{e}$ with the associated Jacobian determinant
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Table 1.  The analogy between the incompressible Newtonian fluid and the image registration
QuantitiesNavier-StokesImage Registration
$u$fluid displacementpixel displacement
$v$fluid velocitypixel velocity
$p$pressurethe effect of each region
$\nu$fluid viscosityfactor of diffusion
$\nabla\cdot v=0$incompressible fluidthe pixels are not condensable
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QuantitiesNavier-StokesImage Registration
$u$fluid displacementpixel displacement
$v$fluid velocitypixel velocity
$p$pressurethe effect of each region
$\nu$fluid viscosityfactor of diffusion
$\nabla\cdot v=0$incompressible fluidthe pixels are not condensable
Table 2.  The parameters choice
ImageParameters
Reynold number $R_{e}$Iteration number $N$Time-step $\tau$
Human hands5004000.01
Human brain 110005000.05
Human brain 210005000.01
Human head1004000.01
EPI slice1003000.01
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ImageParameters
Reynold number $R_{e}$Iteration number $N$Time-step $\tau$
Human hands5004000.01
Human brain 110005000.05
Human brain 210005000.01
Human head1004000.01
EPI slice1003000.01
Table 3.  PSNR and SSIM results obtained using the fluid image registration and proposed approach to the benchmark images. In bold the highest value of each row is shown
ImageMethod
Image sizeMetricFluid image registrationproposed
Human hands $128 \times 128$PSNR27.0273$\bf{28.0153}$
SSIM0.8265$\bf{0.8342}$
Human brain 1 $128 \times 128$PSNR24.1694$\bf{25.4103}$
SSIM0.8860$\bf{0.8995}$
Human brain 2 $256 \times 256$PSNR26.5761$\bf{27.7770}$
SSIM0.8181$\bf{0.8461}$
Human head $400 \times 400$PSNR23.6568$\bf{24.9482}$
SSIM0.7963$\bf{0.8268}$
EPI slice $256 \times 256$PSNR31.4214$\bf{32.8485}$
SSIM0.9278$\bf{0.9480}$
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ImageMethod
Image sizeMetricFluid image registrationproposed
Human hands $128 \times 128$PSNR27.0273$\bf{28.0153}$
SSIM0.8265$\bf{0.8342}$
Human brain 1 $128 \times 128$PSNR24.1694$\bf{25.4103}$
SSIM0.8860$\bf{0.8995}$
Human brain 2 $256 \times 256$PSNR26.5761$\bf{27.7770}$
SSIM0.8181$\bf{0.8461}$
Human head $400 \times 400$PSNR23.6568$\bf{24.9482}$
SSIM0.7963$\bf{0.8268}$
EPI slice $256 \times 256$PSNR31.4214$\bf{32.8485}$
SSIM0.9278$\bf{0.9480}$
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