# American Institute of Mathematical Sciences

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

 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.

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:

show all references

##### References:
Control volume of $v$ and $p$
The Reference and Template images of human hands
The obtained Template image, using the fluid registration and the proposed approach, compared with the Original one for the human hands image
(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).
The time progression of the transformation applied to a rectangular grid with respect to the iterations (human hands).
Difference error between template and reference images using fluid registration (on the left) and the proposed approach (on the right)
The Reference and Template images of human brain 1.
The obtained Template image, using the fluid image registration and the proposed approach, compared with the Original one (human brain 1)
(a) The whole deformation grid and the Jacobian determinant of the transformation, where $\min (\det(J)) = 0.09$
The time progression of the transformation applied to a rectangular grid with respect to the iterations (human brain 1).
Difference error between template and reference images of human brain 1 using fluid registration (on the left) and the proposed approach (on the right).
The Reference and Template images of human brain 2
The obtained Template image, using the fluid image registration and the proposed approach, compared with the Original one (human brain 2)
(a) The whole deformation grid. The whole deformation grid and the Jacobian determinant of the transformation, where $\min (\det(J)) = 0.0995$
The time progression of the transformation applied to a rectangular grid with respect to the iterations (human brain 2)
Difference error between template and reference images of human brain 2 using fluid registration (on the left) and the proposed approach (on the right).
The Reference and Template images of human brain
The obtained Template image, using the fluid image registration and the proposed approach, compared with the Original one (human brain)
(a) The whole deformation grid. The whole deformation grid and the Jacobian determinant of the transformation, where $\min (\det(J)) = 0.0935$
The time progression of the transformation applied to a rectangular grid with respect to the iterations (human brain)
Difference error between template and reference images of human brain using fluid registration (on the left) and the proposed approach (on the right)
The Reference and Template images of EPI slice
The obtained Template image, using the fluid image registration and the proposed approach, compared with the Original one (EPI slice)
(a) The whole deformation grid and the Jacobian determinant of the transformation, where $\min (\det(J)) = 0.0965$
The time progression of the transformation applied to a rectangular grid with respect to the iterations (EPI slice)
Difference error between template and reference images of EPI slice using fluid registration (on the left) and the proposed approach (on the right)
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
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
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
 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
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
 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
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 $\bf{0.9480}$
 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 $\bf{0.9480}$
 [1] Christiane Pöschl, Jan Modersitzki, Otmar Scherzer. A variational setting for volume constrained image registration. Inverse Problems & Imaging, 2010, 4 (3) : 505-522. doi: 10.3934/ipi.2010.4.505 [2] Dana Paquin, Doron Levy, Eduard Schreibmann, Lei Xing. Multiscale Image Registration. Mathematical Biosciences & Engineering, 2006, 3 (2) : 389-418. doi: 10.3934/mbe.2006.3.389 [3] Marcus Wagner. A direct method for the solution of an optimal control problem arising from image registration. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 487-510. doi: 10.3934/naco.2012.2.487 [4] Angel Angelov, Marcus Wagner. Multimodal image registration by elastic matching of edge sketches via optimal control. Journal of Industrial & Management Optimization, 2014, 10 (2) : 567-590. doi: 10.3934/jimo.2014.10.567 [5] Zhao Yi, Justin W. L. Wan. An inviscid model for nonrigid image registration. Inverse Problems & Imaging, 2011, 5 (1) : 263-284. doi: 10.3934/ipi.2011.5.263 [6] Michele Coti Zelati. Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2829-2838. doi: 10.3934/cpaa.2013.12.2829 [7] Dana Paquin, Doron Levy, Lei Xing. Hybrid multiscale landmark and deformable image registration. Mathematical Biosciences & Engineering, 2007, 4 (4) : 711-737. doi: 10.3934/mbe.2007.4.711 [8] Enrique Fernández-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021 [9] Yangang Chen, Justin W. L. Wan. Numerical method for image registration model based on optimal mass transport. Inverse Problems & Imaging, 2018, 12 (2) : 401-432. doi: 10.3934/ipi.2018018 [10] Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3421-3438. doi: 10.3934/dcdsb.2017173 [11] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [12] Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149 [13] Yann Brenier. Approximation of a simple Navier-Stokes model by monotonic rearrangement. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1285-1300. doi: 10.3934/dcds.2014.34.1285 [14] Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations & Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495 [15] Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349 [16] Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433 [17] Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747 [18] C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403 [19] Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319 [20] Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717

2018 Impact Factor: 1.469