An improved SIMPLEC scheme for fluid registration
Abstract
The image registration is always a strongly ill-posed problem, a stable numerical approach is then desired to better approximate the deformation vectors. This paper introduces an efficient numerical implementation of the Navier Stokes equation in the fluid image registration context. Although fluid registration approaches have succeeded in handling large image deformations, the numerical results are sometimes inconsistent and unexpected. This is related, in fact, to the used numerical scheme which does not take into consideration the different properties of the continuous operators. To take into account these properties, we use a robust numerical scheme based on finite volume with pressure correction. This scheme, which is called by the Semi-Implicit Method for Pressure-Linked Equation-Consistent (SIMPLEC), is known for its stability and consistency in fluid dynamics context. The experimental results demonstrate that the proposed method is more efficient and stable, visually and quantitatively, compared to some classical registration methods.
Keyword : image registration, fluid registration, SIMPLEC, Navier Stokes equations
How to Cite
Alahyane, M., Hakim, A., Laghrib, A., & Raghay, S. (2023). An improved SIMPLEC scheme for fluid registration. Mathematical Modelling and Analysis, 28(1), 71–90. https://doi.org/10.3846/mma.2023.15482
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
M. Alahyane, A. Hakim, A. Laghrib and S. Raghay. Fluid image registration using a finite volume scheme of the incompressible Navier Stokes equation. Inverse Problems and Imaging, 12(5):1055–1081, 2018. https://doi.org/10.3934/ipi.2018044
M. Alahyane, A. Hakim, A. Laghrib and S. Raghay. A fast approach of nonparametric elastic image registration problem. Mathematical Methods in the Applied Sciences, 42(18):7059–7075, 2019. https://doi.org/10.1002/mma.5810
M. Alahyane, A. Hakim, A. Laghrib and S. Raghay. A lattice Boltzmann method applied to the fluid image registration. Applied Mathematics and Computation, 349:421–438, 2019. https://doi.org/10.1016/j.amc.2018.12.051
C. Broit. Optimal registration of deformed images. University of Pennsylvania, 1981.
M. Burger, J. Modersitzki and L. Ruthotto. A hyperelastic regularization energy for image registration. SIAM Journal on Scientific Computing, 35(1):B132– B148, 2013. https://doi.org/10.1137/110835955
G.E. Christensen. Deformable shape models for anatomy, 1994.
G.E. Christensen. WE-H-202-04: Advanced medical image registration techniques. Medical Physics, 43(6):3845–3845, 2016. https://doi.org/10.1118/1.4958005
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, pp. 211–216. Citeseer, 1993.
G.E. Christensen, R.D. Rabbitt and M.I. Miller. Deformable templates using large deformation kinematics. IEEE transactions on image processing, 5(10):1435–1447, 1996. https://doi.org/10.1109/83.536892
N. Chumchob and K. Chen. A variational approach for discontinuity-preserving image registration. Proceedings of ICMA-CU, pp. 266–282, 2010.
E. D’Agostino, F. Maes, D. Vandermeulen and P. Suetens. A viscous fluid model for multimodal non-rigid image registration using mutual information. Medical image analysis, 7(4):565–575, 2003. https://doi.org/10.1016/S13618415(03)00039-2
J.P. Van Doormaal and G.D. Raithby. Enhancements of the simple method for predicting incompressible fluid flows. Numerical heat transfer, 7(2):147–163, 1984. https://doi.org/10.1080/01495728408961817
R. Eymard, T. Gallou¨et and R. Herbin. Finite volume methods, volume 7. Elsevier, 2000.
J.H. Ferziger and M. Peric. Computational methods for fluid dynamics. Springer Science & Business Media, 2012.
B. Fischer and J. Modersitzki. Ill-posed medicine-an introduction to image registration. Inverse Problems, 24(3):034008, 2008. https://doi.org/10.1088/02665611/24/3/034008
C. Foias, O. Manley, R. Rosa and R. Temam. Navier-Stokes equations and turbulence, volume 83. Cambridge University Press, 2001.
C. Frohn-Schauf, S. Henn and K. Witsch. Multigrid based total variation image registration. Computing and Visualization in Science, 11(2):101–113, 2008. https://doi.org/10.1007/s00791-007-0060-2
V. Girault and P.-A. Raviart. Finite element methods for Navier-Stokes equations: theory and algorithms, volume 5. Springer Science & Business Media, 2012.
E. Haber and J. Modersitzki. Numerical methods for volume preserving image registration. Inverse problems, 20(5):1621, 2004. https://doi.org/10.1088/02665611/20/5/018
H. Han. A fractional-order decomposition model of image registration and its numerical algorithm. Computational and Applied Mathematics, 39(2):1–19, 2020. https://doi.org/10.1007/s40314-020-1066-3
A. Laghrib and A. Hakim S. Raghay M. El Rhabi. Robust super resolution of images with non-parametric deformations using an elastic registration. Appl. Math. Sci, 8:8897–8907, 2014. https://doi.org/10.12988/ams.2014.49751
A. Laghrib, A. Ghazdali, A. Hakim and S. Raghay. A multi-frame superresolution using diffusion registration and a nonlocal variational image restoration. Computers & Mathematics with Applications, 72(9):2535–2548, 2016. https://doi.org/10.1016/j.camwa.2016.09.013
A. Mang and G. Biros. An inexact Newton–Krylov algorithm for constrained diffeomorphic image registration. SIAM journal on imaging sciences, 8(2):1030– 1069, 2015. https://doi.org/10.1137/140984002
T. Mansi, X. Pennec, M. Sermesant, H. Delingette and N. Ayache. iLogDemons: A demons-based registration algorithm for tracking incompressible elastic biological tissues. International journal of computer vision, 92(1):92–111, 2011. https://doi.org/10.1007/s11263-010-0405-z
J. Modersitzki. FAIR: flexible algorithms for image registration, volume 6. SIAM, 2009.
S. Patankar. Numerical heat transfer and fluid flow. CRC press, 1980.
S.V. Patankar. A calculation procedure for two-dimensional elliptic situations. Numerical Heat Transfer, 4(4):409–425, 1981. https://doi.org/10.1080/01495728108961801
L.I. Rudin, S. Osher and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, 60(1-4):259–268, 1992.
R. Temam. Navier-Stokes equations: theory and numerical analysis, volume 343. American Mathematical Soc., 2001.
J. Zhang, K. Chen and B. Yu. An improved discontinuity-preserving image registration model and its fast algorithm. Applied Mathematical Modelling, 40(2324):10740–10759, 2016. https://doi.org/10.1016/j.apm.2016.08.009
W. Zhou, A.C. Bovik, H.R. Sheikh and E.P. Simoncelli. Image quality assessment: from error visibility to structural similarity. IEEE Transactions on Image Processing, 13:600–612, April 2004. https://doi.org/10.1109/TIP.2003.819861
M. Alahyane, A. Hakim, A. Laghrib and S. Raghay. A fast approach of nonparametric elastic image registration problem. Mathematical Methods in the Applied Sciences, 42(18):7059–7075, 2019. https://doi.org/10.1002/mma.5810
M. Alahyane, A. Hakim, A. Laghrib and S. Raghay. A lattice Boltzmann method applied to the fluid image registration. Applied Mathematics and Computation, 349:421–438, 2019. https://doi.org/10.1016/j.amc.2018.12.051
C. Broit. Optimal registration of deformed images. University of Pennsylvania, 1981.
M. Burger, J. Modersitzki and L. Ruthotto. A hyperelastic regularization energy for image registration. SIAM Journal on Scientific Computing, 35(1):B132– B148, 2013. https://doi.org/10.1137/110835955
G.E. Christensen. Deformable shape models for anatomy, 1994.
G.E. Christensen. WE-H-202-04: Advanced medical image registration techniques. Medical Physics, 43(6):3845–3845, 2016. https://doi.org/10.1118/1.4958005
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, pp. 211–216. Citeseer, 1993.
G.E. Christensen, R.D. Rabbitt and M.I. Miller. Deformable templates using large deformation kinematics. IEEE transactions on image processing, 5(10):1435–1447, 1996. https://doi.org/10.1109/83.536892
N. Chumchob and K. Chen. A variational approach for discontinuity-preserving image registration. Proceedings of ICMA-CU, pp. 266–282, 2010.
E. D’Agostino, F. Maes, D. Vandermeulen and P. Suetens. A viscous fluid model for multimodal non-rigid image registration using mutual information. Medical image analysis, 7(4):565–575, 2003. https://doi.org/10.1016/S13618415(03)00039-2
J.P. Van Doormaal and G.D. Raithby. Enhancements of the simple method for predicting incompressible fluid flows. Numerical heat transfer, 7(2):147–163, 1984. https://doi.org/10.1080/01495728408961817
R. Eymard, T. Gallou¨et and R. Herbin. Finite volume methods, volume 7. Elsevier, 2000.
J.H. Ferziger and M. Peric. Computational methods for fluid dynamics. Springer Science & Business Media, 2012.
B. Fischer and J. Modersitzki. Ill-posed medicine-an introduction to image registration. Inverse Problems, 24(3):034008, 2008. https://doi.org/10.1088/02665611/24/3/034008
C. Foias, O. Manley, R. Rosa and R. Temam. Navier-Stokes equations and turbulence, volume 83. Cambridge University Press, 2001.
C. Frohn-Schauf, S. Henn and K. Witsch. Multigrid based total variation image registration. Computing and Visualization in Science, 11(2):101–113, 2008. https://doi.org/10.1007/s00791-007-0060-2
V. Girault and P.-A. Raviart. Finite element methods for Navier-Stokes equations: theory and algorithms, volume 5. Springer Science & Business Media, 2012.
E. Haber and J. Modersitzki. Numerical methods for volume preserving image registration. Inverse problems, 20(5):1621, 2004. https://doi.org/10.1088/02665611/20/5/018
H. Han. A fractional-order decomposition model of image registration and its numerical algorithm. Computational and Applied Mathematics, 39(2):1–19, 2020. https://doi.org/10.1007/s40314-020-1066-3
A. Laghrib and A. Hakim S. Raghay M. El Rhabi. Robust super resolution of images with non-parametric deformations using an elastic registration. Appl. Math. Sci, 8:8897–8907, 2014. https://doi.org/10.12988/ams.2014.49751
A. Laghrib, A. Ghazdali, A. Hakim and S. Raghay. A multi-frame superresolution using diffusion registration and a nonlocal variational image restoration. Computers & Mathematics with Applications, 72(9):2535–2548, 2016. https://doi.org/10.1016/j.camwa.2016.09.013
A. Mang and G. Biros. An inexact Newton–Krylov algorithm for constrained diffeomorphic image registration. SIAM journal on imaging sciences, 8(2):1030– 1069, 2015. https://doi.org/10.1137/140984002
T. Mansi, X. Pennec, M. Sermesant, H. Delingette and N. Ayache. iLogDemons: A demons-based registration algorithm for tracking incompressible elastic biological tissues. International journal of computer vision, 92(1):92–111, 2011. https://doi.org/10.1007/s11263-010-0405-z
J. Modersitzki. FAIR: flexible algorithms for image registration, volume 6. SIAM, 2009.
S. Patankar. Numerical heat transfer and fluid flow. CRC press, 1980.
S.V. Patankar. A calculation procedure for two-dimensional elliptic situations. Numerical Heat Transfer, 4(4):409–425, 1981. https://doi.org/10.1080/01495728108961801
L.I. Rudin, S. Osher and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, 60(1-4):259–268, 1992.
R. Temam. Navier-Stokes equations: theory and numerical analysis, volume 343. American Mathematical Soc., 2001.
J. Zhang, K. Chen and B. Yu. An improved discontinuity-preserving image registration model and its fast algorithm. Applied Mathematical Modelling, 40(2324):10740–10759, 2016. https://doi.org/10.1016/j.apm.2016.08.009
W. Zhou, A.C. Bovik, H.R. Sheikh and E.P. Simoncelli. Image quality assessment: from error visibility to structural similarity. IEEE Transactions on Image Processing, 13:600–612, April 2004. https://doi.org/10.1109/TIP.2003.819861