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Numerical method for image registration model based on optimal mass transport
1. | Department of Applied Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada |
2. | David R. Cheriton School of Computer Science, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada |
This paper proposes a numerical method for solving a non-rigid image registration model based on optimal mass transport. The main contribution of this paper is to address two issues. One is that we impose a proper periodic boundary condition, such that when the reference and template images are related by translation, or a combination of translation and non-rigid deformation, the numerical solution gives the underlying transformation. The other is that we design a numerical scheme that converges to the optimal transformation between the two images. As an additional benefit, our approach can decompose the transformation into translation and non-rigid deformation. Our numerical results show that the numerical solutions yield good-quality transformations for non-rigid image registration problems.
References:
[1] |
G. Barles and P. E. Souganidis,
Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283.
|
[2] |
J. -D. Benamou, Y. Brenier and K. Guittet, The Monge-Kantorovitch mass transfer and its
computational fluid mechanics formulation, Internat. J. Numer. Methods Fluids, 40 (2002),
21-30, ICFD Conference on Numerical Methods for Fluid Dynamics (Oxford, 2001).
doi: 10.1002/fld.264. |
[3] |
J.-D. Benamou, B. D. Froese and A. M. Oberman,
Two numerical methods for the elliptic Monge-Ampère equation, M2AN Math. Model. Numer. Anal., 44 (2010), 737-758.
doi: 10.1051/m2an/2010017. |
[4] |
K. Böhmer,
On finite element methods for fully nonlinear elliptic equations of second order, SIAM J. Numer. Anal., 46 (2008), 1212-1249.
doi: 10.1137/040621740. |
[5] |
S. C. Brenner, T. Gudi, M. Neilan and L.-y. Sung,
$C^0$ penalty methods for the fully nonlinear Monge-Ampère equation, Math. Comp., 80 (2011), 1979-1995.
doi: 10.1090/S0025-5718-2011-02487-7. |
[6] | |
[7] |
L. G. Brown,
A survey of image registration techniques, ACM Computing Surveys (CSUR), 24 (1992), 325-376.
doi: 10.1145/146370.146374. |
[8] |
P. A. Browne, C. J. Budd, C. Piccolo and M. Cullen,
Fast three dimensional r-adaptive mesh redistribution, J. Comput. Phys., 275 (2014), 174-196.
doi: 10.1016/j.jcp.2014.06.009. |
[9] |
C. J. Budd and J. F. Williams,
Moving mesh generation using the parabolic Monge-Ampère equation, SIAM J. Sci. Comput., 31 (2009), 3438-3465.
doi: 10.1137/080716773. |
[10] |
L. A. Caffarelli,
Boundary regularity of maps with convex potentials. II, Ann. of Math.(2), 144 (1996), 453-496.
doi: 10.2307/2118564. |
[11] |
K. Y. Chan and J. W. Wan, Reconstruction of missing cells by a killing energy minimizing
nonrigid image registration, in Engineering in Medicine and Biology Society (EMBC), 2013
35th Annual International Conference of the IEEE, IEEE, 2013,3000-3003.
doi: 10.1109/EMBC.2013.6610171. |
[12] |
R. Chartrand, B. Wohlberg, K. R. Vixie and E. M. Bollt,
A gradient descent solution to the Monge-{K}antorovich problem, Appl. Math. Sci. (Ruse), 3 (2009), 1071-1080.
|
[13] |
Y. Chen and J. W. Wan, Monotone mixed narrow/wide stencil finite difference scheme for Monge-Ampère equation, arXiv preprint, arXiv: 1608.00644. |
[14] |
G. E. Christensen,
Deformable Shape Models for Anatomy, PhD thesis, Washington University Saint Louis, Mississippi, 1994. |
[15] |
M. G. Crandall, H. Ishii and P.-L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[16] |
M. G. Crandall and P.-L. Lions,
Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.
doi: 10.1090/S0002-9947-1983-0690039-8. |
[17] |
E. J. Dean and R. Glowinski,
Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type, Comput. Methods Appl. Mech. Engrg., 195 (2006), 1344-1386.
doi: 10.1016/j.cma.2005.05.023. |
[18] |
P. Dupuis, U. Grenander and M. I. Miller,
Variational problems on flows of diffeomorphisms for image matching, Quarterly of Applied Mathematics, 56 (1998), 587-600.
doi: 10.1090/qam/1632326. |
[19] |
X. Feng, R. Glowinski and M. Neilan,
Recent developments in numerical methods for fully nonlinear second order partial differential equations, SIAM Rev., 55 (2013), 205-267.
doi: 10.1137/110825960. |
[20] |
X. Feng and M. Neilan,
Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations, J. Sci. Comput., 38 (2009), 74-98.
doi: 10.1007/s10915-008-9221-9. |
[21] |
B. Fischer and J. Modersitzki,
Fast inversion of matrices arising in image processing, Numerical Algorithms, 22 (1999), 1-11.
doi: 10.1023/A:1019194421221. |
[22] |
P. A. Forsyth and G. Labahn, Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance,
Journal of Computational Finance, 11 (2007), 1pp.
doi: 10.21314/JCF.2007.163. |
[23] |
B. D. Froese,
A numerical method for the elliptic Monge-Ampère equation with transport boundary conditions, SIAM J. Sci. Comput., 34 (2012), A1432-A1459.
doi: 10.1137/110822372. |
[24] |
B. D. Froese and A. M. Oberman,
Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher, SIAM J. Numer. Anal., 49 (2011), 1692-1714.
doi: 10.1137/100803092. |
[25] |
S. K. Godunov,
A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. Sb. (N.S.), 47 (1959), 271-306.
|
[26] |
A. A. Goshtasby,
2-D and 3-D Image Registration: For Medical, Remote Sensing, and Industrial Applications, John Wiley & Sons, 2005. |
[27] |
S. Haker and A. Tannenbaum, Optimal mass transport and image registration, in Variational
and Level Set Methods in Computer Vision, 2001. Proceedings. IEEE Workshop on, IEEE,
2001, 29-36.
doi: 10.1109/VLSM.2001.938878. |
[28] |
S. Haker, L. Zhu, A. Tannenbaum and S. Angenent,
Optimal mass transport for registration and warping, International Journal of Computer Vision, 60 (2004), 225-240.
doi: 10.1023/B:VISI.0000036836.66311.97. |
[29] |
D. L. Hill, P. G. Batchelor, M. Holden and D. J. Hawkes, Medical image registration,
Physics in Medicine and Biology, 46 (2001), R1.
doi: 10.1088/0031-9155/46/3/201. |
[30] |
M. Irani and S. Peleg,
Improving resolution by image registration, CVGIP: Graphical Models and Image Processing, 53 (1991), 231-239.
doi: 10.1016/1049-9652(91)90045-L. |
[31] |
M. Knott and C. S. Smith,
On the optimal mapping of distributions, Journal of Optimization Theory and Applications, 43 (1984), 39-49.
doi: 10.1007/BF00934745. |
[32] |
N. V. Krylov,
The control of the solution of a stochastic integral equation, Teor. Verojatnost. i Primenen., 17 (1972), 111-128.
|
[33] |
K. Levenberg,
A method for the solution of certain non-linear problems in least squares, Quart. Appl. Math., 2 (1944), 164-168.
doi: 10.1090/qam/10666. |
[34] |
P. -L. Lions, Hamilton-Jacobi-Bellman equations and the optimal control of stochastic systems,
in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983),
PWN, Warsaw, 1984,1403-1417. |
[35] |
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. |
[36] |
D. W. Marquardt,
An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Indust. Appl. Math., 11 (1963), 431-441.
doi: 10.1137/0111030. |
[37] |
J. Modersitzki,
Numerical Methods for Image Registration, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2004, Oxford Science Publications. |
[38] |
J. Modersitzki,
FAIR: Flexible Algorithms for Image Registration, vol. 6 of Fundamentals of Algorithms, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009.
doi: 10.1137/1.9780898718843. |
[39] |
O. Museyko, M. Stiglmayr, K. Klamroth and G. Leugering,
On the application of the Monge-Kantorovich problem to image registration, SIAM J. Imaging Sci., 2 (2009), 1068-1097.
doi: 10.1137/080721522. |
[40] |
A. M. Oberman,
Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 221-238.
doi: 10.3934/dcdsb.2008.10.221. |
[41] |
V. I. Oliker and L. D. Prussner,
On the numerical solution of the equation $(\partial^ 2z/\partial x^ 2)(\partial^ 2z/\partial y^ 2)-((\partial^ 2z/\partial x\partial y))^ 2 = f$ and its discretizations. Ⅰ, Numer. Math., 54 (1988), 271-293.
doi: 10.1007/BF01396762. |
[42] |
S. Osher and C.-W. Shu,
High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal., 28 (1991), 907-922.
doi: 10.1137/0728049. |
[43] |
K. Rohr,
Landmark-based Image Analysis: Using Geometric and Intensity Models, vol. 21, Springer Science & Business Media, 2001.
doi: 10.1007/978-94-015-9787-6. |
[44] |
L.-P. Saumier, M. Agueh and B. Khouider,
An efficient numerical algorithm for the $L^ 2$
optimal transport problem with periodic densities, IMA J. Appl. Math., 80 (2015), 135-157.
doi: 10.1093/imamat/hxt032. |
[45] |
A. Sotiras, C. Davatzikos and N. Paragios,
Deformable medical image registration: A survey, IEEE Transactions on Medical Imaging, 32 (2013), 1153-1190.
doi: 10.1109/TMI.2013.2265603. |
[46] |
P. Thevenaz, U. E. Ruttimann and M. Unser,
A pyramid approach to subpixel registration based on intensity, IEEE Transactions on Image Processing, 7 (1998), 27-41.
doi: 10.1109/83.650848. |
[47] |
J.-P. Thirion,
Image matching as a diffusion process: An analogy with maxwell's demons, Medical Image Analysis, 2 (1998), 243-260.
doi: 10.1016/S1361-8415(98)80022-4. |
[48] |
E. F. Toro,
Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Third edition. Springer-Verlag, Berlin, 2009. |
[49] |
U. Trottenberg, C. W. Oosterlee and A. Schüller,
Multigrid, Academic Press, Inc., San Diego, CA, 2001, With contributions by A. Brandt, P. Oswald and K. Stüben. |
[50] |
A. Trouvé,
Diffeomorphisms groups and pattern matching in image analysis, International Journal of Computer Vision, 28 (1998), 213-221.
|
[51] |
P. Viola and W. M. Wells Ⅲ,
Alignment by maximization of mutual information, International Journal of Computer Vision, 24 (1997), 137-154.
|
show all references
References:
[1] |
G. Barles and P. E. Souganidis,
Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283.
|
[2] |
J. -D. Benamou, Y. Brenier and K. Guittet, The Monge-Kantorovitch mass transfer and its
computational fluid mechanics formulation, Internat. J. Numer. Methods Fluids, 40 (2002),
21-30, ICFD Conference on Numerical Methods for Fluid Dynamics (Oxford, 2001).
doi: 10.1002/fld.264. |
[3] |
J.-D. Benamou, B. D. Froese and A. M. Oberman,
Two numerical methods for the elliptic Monge-Ampère equation, M2AN Math. Model. Numer. Anal., 44 (2010), 737-758.
doi: 10.1051/m2an/2010017. |
[4] |
K. Böhmer,
On finite element methods for fully nonlinear elliptic equations of second order, SIAM J. Numer. Anal., 46 (2008), 1212-1249.
doi: 10.1137/040621740. |
[5] |
S. C. Brenner, T. Gudi, M. Neilan and L.-y. Sung,
$C^0$ penalty methods for the fully nonlinear Monge-Ampère equation, Math. Comp., 80 (2011), 1979-1995.
doi: 10.1090/S0025-5718-2011-02487-7. |
[6] | |
[7] |
L. G. Brown,
A survey of image registration techniques, ACM Computing Surveys (CSUR), 24 (1992), 325-376.
doi: 10.1145/146370.146374. |
[8] |
P. A. Browne, C. J. Budd, C. Piccolo and M. Cullen,
Fast three dimensional r-adaptive mesh redistribution, J. Comput. Phys., 275 (2014), 174-196.
doi: 10.1016/j.jcp.2014.06.009. |
[9] |
C. J. Budd and J. F. Williams,
Moving mesh generation using the parabolic Monge-Ampère equation, SIAM J. Sci. Comput., 31 (2009), 3438-3465.
doi: 10.1137/080716773. |
[10] |
L. A. Caffarelli,
Boundary regularity of maps with convex potentials. II, Ann. of Math.(2), 144 (1996), 453-496.
doi: 10.2307/2118564. |
[11] |
K. Y. Chan and J. W. Wan, Reconstruction of missing cells by a killing energy minimizing
nonrigid image registration, in Engineering in Medicine and Biology Society (EMBC), 2013
35th Annual International Conference of the IEEE, IEEE, 2013,3000-3003.
doi: 10.1109/EMBC.2013.6610171. |
[12] |
R. Chartrand, B. Wohlberg, K. R. Vixie and E. M. Bollt,
A gradient descent solution to the Monge-{K}antorovich problem, Appl. Math. Sci. (Ruse), 3 (2009), 1071-1080.
|
[13] |
Y. Chen and J. W. Wan, Monotone mixed narrow/wide stencil finite difference scheme for Monge-Ampère equation, arXiv preprint, arXiv: 1608.00644. |
[14] |
G. E. Christensen,
Deformable Shape Models for Anatomy, PhD thesis, Washington University Saint Louis, Mississippi, 1994. |
[15] |
M. G. Crandall, H. Ishii and P.-L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[16] |
M. G. Crandall and P.-L. Lions,
Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.
doi: 10.1090/S0002-9947-1983-0690039-8. |
[17] |
E. J. Dean and R. Glowinski,
Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type, Comput. Methods Appl. Mech. Engrg., 195 (2006), 1344-1386.
doi: 10.1016/j.cma.2005.05.023. |
[18] |
P. Dupuis, U. Grenander and M. I. Miller,
Variational problems on flows of diffeomorphisms for image matching, Quarterly of Applied Mathematics, 56 (1998), 587-600.
doi: 10.1090/qam/1632326. |
[19] |
X. Feng, R. Glowinski and M. Neilan,
Recent developments in numerical methods for fully nonlinear second order partial differential equations, SIAM Rev., 55 (2013), 205-267.
doi: 10.1137/110825960. |
[20] |
X. Feng and M. Neilan,
Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations, J. Sci. Comput., 38 (2009), 74-98.
doi: 10.1007/s10915-008-9221-9. |
[21] |
B. Fischer and J. Modersitzki,
Fast inversion of matrices arising in image processing, Numerical Algorithms, 22 (1999), 1-11.
doi: 10.1023/A:1019194421221. |
[22] |
P. A. Forsyth and G. Labahn, Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance,
Journal of Computational Finance, 11 (2007), 1pp.
doi: 10.21314/JCF.2007.163. |
[23] |
B. D. Froese,
A numerical method for the elliptic Monge-Ampère equation with transport boundary conditions, SIAM J. Sci. Comput., 34 (2012), A1432-A1459.
doi: 10.1137/110822372. |
[24] |
B. D. Froese and A. M. Oberman,
Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher, SIAM J. Numer. Anal., 49 (2011), 1692-1714.
doi: 10.1137/100803092. |
[25] |
S. K. Godunov,
A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. Sb. (N.S.), 47 (1959), 271-306.
|
[26] |
A. A. Goshtasby,
2-D and 3-D Image Registration: For Medical, Remote Sensing, and Industrial Applications, John Wiley & Sons, 2005. |
[27] |
S. Haker and A. Tannenbaum, Optimal mass transport and image registration, in Variational
and Level Set Methods in Computer Vision, 2001. Proceedings. IEEE Workshop on, IEEE,
2001, 29-36.
doi: 10.1109/VLSM.2001.938878. |
[28] |
S. Haker, L. Zhu, A. Tannenbaum and S. Angenent,
Optimal mass transport for registration and warping, International Journal of Computer Vision, 60 (2004), 225-240.
doi: 10.1023/B:VISI.0000036836.66311.97. |
[29] |
D. L. Hill, P. G. Batchelor, M. Holden and D. J. Hawkes, Medical image registration,
Physics in Medicine and Biology, 46 (2001), R1.
doi: 10.1088/0031-9155/46/3/201. |
[30] |
M. Irani and S. Peleg,
Improving resolution by image registration, CVGIP: Graphical Models and Image Processing, 53 (1991), 231-239.
doi: 10.1016/1049-9652(91)90045-L. |
[31] |
M. Knott and C. S. Smith,
On the optimal mapping of distributions, Journal of Optimization Theory and Applications, 43 (1984), 39-49.
doi: 10.1007/BF00934745. |
[32] |
N. V. Krylov,
The control of the solution of a stochastic integral equation, Teor. Verojatnost. i Primenen., 17 (1972), 111-128.
|
[33] |
K. Levenberg,
A method for the solution of certain non-linear problems in least squares, Quart. Appl. Math., 2 (1944), 164-168.
doi: 10.1090/qam/10666. |
[34] |
P. -L. Lions, Hamilton-Jacobi-Bellman equations and the optimal control of stochastic systems,
in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983),
PWN, Warsaw, 1984,1403-1417. |
[35] |
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. |
[36] |
D. W. Marquardt,
An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Indust. Appl. Math., 11 (1963), 431-441.
doi: 10.1137/0111030. |
[37] |
J. Modersitzki,
Numerical Methods for Image Registration, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2004, Oxford Science Publications. |
[38] |
J. Modersitzki,
FAIR: Flexible Algorithms for Image Registration, vol. 6 of Fundamentals of Algorithms, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009.
doi: 10.1137/1.9780898718843. |
[39] |
O. Museyko, M. Stiglmayr, K. Klamroth and G. Leugering,
On the application of the Monge-Kantorovich problem to image registration, SIAM J. Imaging Sci., 2 (2009), 1068-1097.
doi: 10.1137/080721522. |
[40] |
A. M. Oberman,
Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 221-238.
doi: 10.3934/dcdsb.2008.10.221. |
[41] |
V. I. Oliker and L. D. Prussner,
On the numerical solution of the equation $(\partial^ 2z/\partial x^ 2)(\partial^ 2z/\partial y^ 2)-((\partial^ 2z/\partial x\partial y))^ 2 = f$ and its discretizations. Ⅰ, Numer. Math., 54 (1988), 271-293.
doi: 10.1007/BF01396762. |
[42] |
S. Osher and C.-W. Shu,
High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal., 28 (1991), 907-922.
doi: 10.1137/0728049. |
[43] |
K. Rohr,
Landmark-based Image Analysis: Using Geometric and Intensity Models, vol. 21, Springer Science & Business Media, 2001.
doi: 10.1007/978-94-015-9787-6. |
[44] |
L.-P. Saumier, M. Agueh and B. Khouider,
An efficient numerical algorithm for the $L^ 2$
optimal transport problem with periodic densities, IMA J. Appl. Math., 80 (2015), 135-157.
doi: 10.1093/imamat/hxt032. |
[45] |
A. Sotiras, C. Davatzikos and N. Paragios,
Deformable medical image registration: A survey, IEEE Transactions on Medical Imaging, 32 (2013), 1153-1190.
doi: 10.1109/TMI.2013.2265603. |
[46] |
P. Thevenaz, U. E. Ruttimann and M. Unser,
A pyramid approach to subpixel registration based on intensity, IEEE Transactions on Image Processing, 7 (1998), 27-41.
doi: 10.1109/83.650848. |
[47] |
J.-P. Thirion,
Image matching as a diffusion process: An analogy with maxwell's demons, Medical Image Analysis, 2 (1998), 243-260.
doi: 10.1016/S1361-8415(98)80022-4. |
[48] |
E. F. Toro,
Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Third edition. Springer-Verlag, Berlin, 2009. |
[49] |
U. Trottenberg, C. W. Oosterlee and A. Schüller,
Multigrid, Academic Press, Inc., San Diego, CA, 2001, With contributions by A. Brandt, P. Oswald and K. Stüben. |
[50] |
A. Trouvé,
Diffeomorphisms groups and pattern matching in image analysis, International Journal of Computer Vision, 28 (1998), 213-221.
|
[51] |
P. Viola and W. M. Wells Ⅲ,
Alignment by maximization of mutual information, International Journal of Computer Vision, 24 (1997), 137-154.
|












1: Start with an initial guess |
2: Set |
3: for |
4: if |
5: |
6: |
7: end if |
8: Compute |
9: Compute |
10: Compute |
11: Solve |
$ \begin{array}{l} [\lambda I + (\mathbf{J}^{(k+\frac{1}{2})})^T \mathbf{J}^{(k+\frac{1}{2})}] E^{(k+\frac{1}{2})} \\ = -(\mathbf{J}^{(k+\frac{1}{2})})^T R^{(k+\frac{1}{2})}. \end{array} $ |
for |
12: |
13: end for |
1: Start with an initial guess |
2: Set |
3: for |
4: if |
5: |
6: |
7: end if |
8: Compute |
9: Compute |
10: Compute |
11: Solve |
$ \begin{array}{l} [\lambda I + (\mathbf{J}^{(k+\frac{1}{2})})^T \mathbf{J}^{(k+\frac{1}{2})}] E^{(k+\frac{1}{2})} \\ = -(\mathbf{J}^{(k+\frac{1}{2})})^T R^{(k+\frac{1}{2})}. \end{array} $ |
for |
12: |
13: end for |
Examples | Example 2 | Example 3 | Example 4 | Example 5 | Example 6 | Example 7 |
| 0.47 | 0.52 | 0.61 | 0.64 | 0.69 | 0.59 |
| 0 | | | | | |
Examples | Example 2 | Example 3 | Example 4 | Example 5 | Example 6 | Example 7 |
| 0.47 | 0.52 | 0.61 | 0.64 | 0.69 | 0.59 |
| 0 | | | | | |
net flow of mass/pixels | area change of a square element | change of mass/pixels intensity | morphing magnitude | color of a square element |
zero | invariance | invariance | | white |
inflow | compressed | increase | | red |
outflow | expanded | decrease | | blue |
net flow of mass/pixels | area change of a square element | change of mass/pixels intensity | morphing magnitude | color of a square element |
zero | invariance | invariance | | white |
inflow | compressed | increase | | red |
outflow | expanded | decrease | | blue |
Examples | Example 2 | Example 3 | Example 4 | Example 5 |
| Periodic: | Periodic: | Periodic: | Mass transport, periodic: |
Neumann: | Neumann: | Neumann: | Two-step empirical: |
Examples | Example 2 | Example 3 | Example 4 | Example 5 |
| Periodic: | Periodic: | Periodic: | Mass transport, periodic: |
Neumann: | Neumann: | Neumann: | Two-step empirical: |
Example | Example 3 | |||
Image size | 100x100 | 200x200 | 400x400 | 800x800 |
Number of steps for convergence | 5 | 3 | 3 | 3 |
CPU time for corrections of translation kernels (sec) | 1.0 | 4.6 | 30 | 259 |
CPU time for the primary nonlinear solver (sec) | 3.1 | 7.3 | 58 | 1083 |
Total CPU time (sec) | 4.1 | 11.9 | 88 | 1342 |
Example | Example 3 | |||
Image size | 100x100 | 200x200 | 400x400 | 800x800 |
Number of steps for convergence | 5 | 3 | 3 | 3 |
CPU time for corrections of translation kernels (sec) | 1.0 | 4.6 | 30 | 259 |
CPU time for the primary nonlinear solver (sec) | 3.1 | 7.3 | 58 | 1083 |
Total CPU time (sec) | 4.1 | 11.9 | 88 | 1342 |
Examples | Example 3 | Example 4 | Example 5 | Example 6 | Example 7 |
Image size | 600x600 | ||||
Number of steps for convergence | 3 | 3 | 10 | 10 | 19 |
CPU time for the primary nonlinear solver (sec) | 147 | 152 | 668 | 627 | 1613 |
Examples | Example 3 | Example 4 | Example 5 | Example 6 | Example 7 |
Image size | 600x600 | ||||
Number of steps for convergence | 3 | 3 | 10 | 10 | 19 |
CPU time for the primary nonlinear solver (sec) | 147 | 152 | 668 | 627 | 1613 |
The number of iteration | 1 | 2 | 3 | 4 | 5 |
Residual | 1382 | 195 | 2.32 | 1.71 | 0.131 |
The number of iteration | 6 | 7 | 8 | 9 | 10 |
Residual | 0.0236 | 0.00492 | 9.50x10 | 3.42x10 | 9.41x10 |
The number of iteration | 1 | 2 | 3 | 4 | 5 |
Residual | 1382 | 195 | 2.32 | 1.71 | 0.131 |
The number of iteration | 6 | 7 | 8 | 9 | 10 |
Residual | 0.0236 | 0.00492 | 9.50x10 | 3.42x10 | 9.41x10 |
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