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A variational model with fractional-order regularization term arising in registration of diffusion tensor image
School of Mathematics and Physics, China University of Geosciences, Wuhan 430076, China |
In this paper, a new variational model with fractional-order regularization term arising in registration of diffusion tensor image(DTI) is presented. Moreover, the existence of its solution is proved to ensure that there is a regular solution for this model. Furthermore, three numerical tests are also performed to show the effectiveness of this model.
References:
[1] |
D. C. Alexander, C. Pierpaoli, P. J. Basser and J. C. Gee,
Spatial transformations of diffusion tensor magnetic resonance images, IEEE Transaction on Medical imaging, 20 (2001), 1131-1139.
|
[2] |
M. F. Beg, M. I. Miller, A. Trouve and L. Younes,
Computing large deformation metric mappings via geodesic flows of diffeomorphisms, International Journal of Computer Vision, 61 (2005), 139-157.
|
[3] |
M. Bruveris, F. Gay-Balmaz, D. D. Holm and T. S. Ratiu,
The momentum map representation of images, Journal of Nonlinear Science, 21 (2011), 115-150.
doi: 10.1007/s00332-010-9079-5. |
[4] |
F. Demengel and G. Demengel,
Functional spaces for the theory of elliptic partial differential equations, Springer, (2011), 219-224.
doi: 10.1007/978-1-4471-2807-6. |
[5] |
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. |
[6] |
V. J. Ervin and J. P. Roop,
Variational formulation for the stationary fractional advection dispersion equation, Numerical Method for Partial Differential Equations, 22 (2006), 558-576.
doi: 10.1002/num.20112. |
[7] |
L. C. Evans,
Partial differential equations, American Mathematical Society, (1997), 251-308.
|
[8] |
H. Han and H. Zhou,
A variational problem arising in registration of diffusion tensor image, Acta Mathematica Scientia, 37 (2017), 539-554.
doi: 10.1016/S0252-9602(17)30020-6. |
[9] |
H. Han and H. Zhou, Spectral representation of solution of a variational model in diffusion tensor images registration, preprint. |
[10] |
W. V. Hecke and A. Leemans,
Nonrigid coregistration of diffusion tensor images using a viscous fluid model and mutual information, IEEE Transaction on Medical Imaging, 26 (2007), 1598-1612.
|
[11] |
C. R. Johnson, K. Okubo and R. Reams,
Uniqueness of matrix square roots and application, Linear Algebra and it Applications, 323 (2001), 51-60.
doi: 10.1016/S0024-3795(00)00243-3. |
[12] |
J. Li, Y. Shi, G. Tran, I. Dinov, D. Wang and A. Toga,
Fast local trust region for diffusion tensor registration using exact reorientation and regularization, IEEE Transaction on Medical Imaging, 33 (2014), 1-43.
|
[13] |
R. Li, S. Zhong and C. Swartz,
An improvement of the Arzela-Ascoli theorem, Topology and Its Applications, 159 (2012), 2058-2061.
doi: 10.1016/j.topol.2012.01.014. |
[14] |
F. O'Sullivan,
The Analysis of Some Penalized Likelihood Schemes, Statistics Department Technical Report No.726, University of Wisconsin, 1983. |
[15] |
I. Podlubny,
Fractional Differential Equations: An introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, Math. Sci. Eng. Elservier Science, (1999), 50-90.
|
[16] |
G. Teschl,
Ordinary differential equations and Dynamical systems, American Mathematical Society, (2012), 50-230.
doi: 10.1090/gsm/140. |
[17] |
H. Wang and N. Du,
Fast solution methods for space-fractional diffusion equations, Journal of Computational and Applied Mathematics, 255 (2014), 376-383.
doi: 10.1016/j.cam.2013.06.002. |
[18] |
T. Yeo, T. Vercauteren, P. Ficlard, J. Peyrat, X. Pennec, P. Golland, N Ayache and O. Clatz,
DTREFinD: Diffusion tensor registration with exact finite-strain differential, IEEE Transaction on Medical imaging, 28 (2009), 1914-1928.
|
[19] |
S. Zhan, On the determinantal inequalities,
Journal of Inequalities in Pure and Applied Mathematics, 6 (2005), Article 105, 7 pp. |
[20] |
J. Zhang and K. Chen,
Variational image registration by a total fractional-order variation model, Journal of Computational Physics, 293 (2015), 442-461.
doi: 10.1016/j.jcp.2015.02.021. |
[21] |
Y. Zhang and Z. Sun,
Error analysis of a compact ADI scheme for the 2D fractional subdiffusion equation, Journal of Scientific Computing, 59 (2014), 104-128.
doi: 10.1007/s10915-013-9756-2. |
show all references
References:
[1] |
D. C. Alexander, C. Pierpaoli, P. J. Basser and J. C. Gee,
Spatial transformations of diffusion tensor magnetic resonance images, IEEE Transaction on Medical imaging, 20 (2001), 1131-1139.
|
[2] |
M. F. Beg, M. I. Miller, A. Trouve and L. Younes,
Computing large deformation metric mappings via geodesic flows of diffeomorphisms, International Journal of Computer Vision, 61 (2005), 139-157.
|
[3] |
M. Bruveris, F. Gay-Balmaz, D. D. Holm and T. S. Ratiu,
The momentum map representation of images, Journal of Nonlinear Science, 21 (2011), 115-150.
doi: 10.1007/s00332-010-9079-5. |
[4] |
F. Demengel and G. Demengel,
Functional spaces for the theory of elliptic partial differential equations, Springer, (2011), 219-224.
doi: 10.1007/978-1-4471-2807-6. |
[5] |
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. |
[6] |
V. J. Ervin and J. P. Roop,
Variational formulation for the stationary fractional advection dispersion equation, Numerical Method for Partial Differential Equations, 22 (2006), 558-576.
doi: 10.1002/num.20112. |
[7] |
L. C. Evans,
Partial differential equations, American Mathematical Society, (1997), 251-308.
|
[8] |
H. Han and H. Zhou,
A variational problem arising in registration of diffusion tensor image, Acta Mathematica Scientia, 37 (2017), 539-554.
doi: 10.1016/S0252-9602(17)30020-6. |
[9] |
H. Han and H. Zhou, Spectral representation of solution of a variational model in diffusion tensor images registration, preprint. |
[10] |
W. V. Hecke and A. Leemans,
Nonrigid coregistration of diffusion tensor images using a viscous fluid model and mutual information, IEEE Transaction on Medical Imaging, 26 (2007), 1598-1612.
|
[11] |
C. R. Johnson, K. Okubo and R. Reams,
Uniqueness of matrix square roots and application, Linear Algebra and it Applications, 323 (2001), 51-60.
doi: 10.1016/S0024-3795(00)00243-3. |
[12] |
J. Li, Y. Shi, G. Tran, I. Dinov, D. Wang and A. Toga,
Fast local trust region for diffusion tensor registration using exact reorientation and regularization, IEEE Transaction on Medical Imaging, 33 (2014), 1-43.
|
[13] |
R. Li, S. Zhong and C. Swartz,
An improvement of the Arzela-Ascoli theorem, Topology and Its Applications, 159 (2012), 2058-2061.
doi: 10.1016/j.topol.2012.01.014. |
[14] |
F. O'Sullivan,
The Analysis of Some Penalized Likelihood Schemes, Statistics Department Technical Report No.726, University of Wisconsin, 1983. |
[15] |
I. Podlubny,
Fractional Differential Equations: An introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, Math. Sci. Eng. Elservier Science, (1999), 50-90.
|
[16] |
G. Teschl,
Ordinary differential equations and Dynamical systems, American Mathematical Society, (2012), 50-230.
doi: 10.1090/gsm/140. |
[17] |
H. Wang and N. Du,
Fast solution methods for space-fractional diffusion equations, Journal of Computational and Applied Mathematics, 255 (2014), 376-383.
doi: 10.1016/j.cam.2013.06.002. |
[18] |
T. Yeo, T. Vercauteren, P. Ficlard, J. Peyrat, X. Pennec, P. Golland, N Ayache and O. Clatz,
DTREFinD: Diffusion tensor registration with exact finite-strain differential, IEEE Transaction on Medical imaging, 28 (2009), 1914-1928.
|
[19] |
S. Zhan, On the determinantal inequalities,
Journal of Inequalities in Pure and Applied Mathematics, 6 (2005), Article 105, 7 pp. |
[20] |
J. Zhang and K. Chen,
Variational image registration by a total fractional-order variation model, Journal of Computational Physics, 293 (2015), 442-461.
doi: 10.1016/j.jcp.2015.02.021. |
[21] |
Y. Zhang and Z. Sun,
Error analysis of a compact ADI scheme for the 2D fractional subdiffusion equation, Journal of Scientific Computing, 59 (2014), 104-128.
doi: 10.1007/s10915-013-9756-2. |
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