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

Article Contents

2018, Volume 12, Issue 5: 1055-1081. Doi: 10.3934/ipi.2018044

This issue Previous Article Next Article Mitigating the influence of the boundary on PDE-based covariance operators

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

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

* Corresponding author: A. Laghrib
* Corresponding author: A. Laghrib

Received: August 31, 2016

Revised: May 31, 2018

Published: September 2018

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Abstract

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:

Mathematics Subject Classification: 65M12, 65M06.

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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

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

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Table 2. The parameters choice


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

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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

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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