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

  • * Corresponding author: A. Laghrib

    * Corresponding author: A. Laghrib 
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  • 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.

    Mathematics Subject Classification: 65M12, 65M06.

    Citation:

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

    Figure 2.  The Reference and Template images of human hands

    Figure 3.  The obtained Template image, using the fluid registration and the proposed approach, compared with the Original one for the human hands image

    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).

    Figure 5.  The time progression of the transformation applied to a rectangular grid with respect to the iterations (human hands).

    Figure 6.  Difference error between template and reference images using fluid registration (on the left) and the proposed approach (on the right)

    Figure 7.  The Reference and Template images of human brain 1.

    Figure 8.  The obtained Template image, using the fluid image registration and the proposed approach, compared with the Original one (human brain 1)

    Figure 9.  (a) The whole deformation grid and the Jacobian determinant of the transformation, where $\min (\det(J)) = 0.09$

    Figure 10.  The time progression of the transformation applied to a rectangular grid with respect to the iterations (human brain 1).

    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).

    Figure 12.  The Reference and Template images of human brain 2

    Figure 13.  The obtained Template image, using the fluid image registration and the proposed approach, compared with the Original one (human brain 2)

    Figure 14.  (a) The whole deformation grid. The whole deformation grid and the Jacobian determinant of the transformation, where $\min (\det(J)) = 0.0995$

    Figure 15.  The time progression of the transformation applied to a rectangular grid with respect to the iterations (human brain 2)

    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).

    Figure 17.  The Reference and Template images of human brain

    Figure 18.  The obtained Template image, using the fluid image registration and the proposed approach, compared with the Original one (human brain)

    Figure 19.  (a) The whole deformation grid. The whole deformation grid and the Jacobian determinant of the transformation, where $\min (\det(J)) = 0.0935$

    Figure 20.  The time progression of the transformation applied to a rectangular grid with respect to the iterations (human brain)

    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)

    Figure 22.  The Reference and Template images of EPI slice

    Figure 23.  The obtained Template image, using the fluid image registration and the proposed approach, compared with the Original one (EPI slice)

    Figure 24.  (a) The whole deformation grid and the Jacobian determinant of the transformation, where $\min (\det(J)) = 0.0965$

    Figure 25.  The time progression of the transformation applied to a rectangular grid with respect to the iterations (EPI slice)

    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)

    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

    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

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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$\bf{0.9480}$
     | Show Table
    DownLoad: CSV
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