# American Institute of Mathematical Sciences

December 2018, 12(6): 1263-1291. doi: 10.3934/ipi.2018053

## 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

* Corresponding author: Huan Han

Received  December 2016 Revised  August 2018 Published  October 2018

Fund Project: The first author is supported by NSFC under grant No.11471331 and partially supported by National Center for Mathematics and Interdisciplinary Sciences

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.

Citation: Huan Han. A variational model with fractional-order regularization term arising in registration of diffusion tensor image. Inverse Problems & Imaging, 2018, 12 (6) : 1263-1291. doi: 10.3934/ipi.2018053
##### 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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##### 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.
One slice of $T(\cdot)$ and $D(\cdot)$
$a$ and ${\rm Re-SSD}$ change with differential order $\alpha$
$a$ and ${\rm Re-SSD}$ change with time $s$ in iteration process
The 22th slice of $T\diamond h(\cdot)$
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