# 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 and 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.

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.
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 $T\diamond h(\cdot)$
 [1] Mujibur Rahman Chowdhury, Jun Zhang, Jing Qin, Yifei Lou. Poisson image denoising based on fractional-order total variation. Inverse Problems and Imaging, 2020, 14 (1) : 77-96. doi: 10.3934/ipi.2019064 [2] Jaydeep Swarnakar. Discrete-time realization of fractional-order proportional integral controller for a class of fractional-order system. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 309-320. doi: 10.3934/naco.2021007 [3] Fangfang Dong, Yunmei Chen. A fractional-order derivative based variational framework for image denoising. Inverse Problems and Imaging, 2016, 10 (1) : 27-50. doi: 10.3934/ipi.2016.10.27 [4] Zeng-bao Wu, Yun-zhi Zou, Nan-jing Huang. A new class of global fractional-order projective dynamical system with an application. Journal of Industrial and Management Optimization, 2020, 16 (1) : 37-53. doi: 10.3934/jimo.2018139 [5] Shaowen Yan, Guoxi Ni, Tieyong Zeng. Nonconvex model for mixing noise with fractional-order regularization. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022041 [6] Denis R. Akhmetov, Renato Spigler. $L^1$-estimates for the higher-order derivatives of solutions to parabolic equations subject to initial values of bounded total variation. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1051-1074. doi: 10.3934/cpaa.2007.6.1051 [7] Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023 [8] Muhammad Mansha Ghalib, Azhar Ali Zafar, Zakia Hammouch, Muhammad Bilal Riaz, Khurram Shabbir. Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 683-693. doi: 10.3934/dcdss.2020037 [9] Kolade M. Owolabi. Dynamical behaviour of fractional-order predator-prey system of Holling-type. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 823-834. doi: 10.3934/dcdss.2020047 [10] Udhayakumar Kandasamy, Rakkiyappan Rajan. Hopf bifurcation of a fractional-order octonion-valued neural networks with time delays. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2537-2559. doi: 10.3934/dcdss.2020137 [11] Guy Joseph Eyebe, Betchewe Gambo, Alidou Mohamadou, Timoleon Crepin Kofane. Melnikov analysis of the nonlocal nanobeam resting on fractional-order softening nonlinear viscoelastic foundations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2213-2228. doi: 10.3934/dcdss.2020252 [12] Karthikeyan Rajagopal, Serdar Cicek, Akif Akgul, Sajad Jafari, Anitha Karthikeyan. Chaotic cuttlesh: king of camouage with self-excited and hidden flows, its fractional-order form and communication designs with fractional form. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1001-1013. doi: 10.3934/dcdsb.2019205 [13] Nasser Sweilam, Fathalla Rihan, Seham AL-Mekhlafi. A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2403-2424. doi: 10.3934/dcdss.2020120 [14] Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 709-722. doi: 10.3934/dcdss.2020039 [15] Boris Anicet Guimfack, Conrad Bertrand Tabi, Alidou Mohamadou, Timoléon Crépin Kofané. Stochastic dynamics of the FitzHugh-Nagumo neuron model through a modified Van der Pol equation with fractional-order term and Gaussian white noise excitation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2229-2243. doi: 10.3934/dcdss.2020397 [16] Fahd Jarad, Thabet Abdeljawad. Variational principles in the frame of certain generalized fractional derivatives. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 695-708. doi: 10.3934/dcdss.2020038 [17] Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control and Related Fields, 2021, 11 (3) : 653-679. doi: 10.3934/mcrf.2021017 [18] Harun Karsli, Purshottam Narain Agrawal. Rate of convergence of Stancu type modified $q$-Gamma operators for functions with derivatives of bounded variation. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2022002 [19] Zbigniew Gomolka, Boguslaw Twarog, Jacek Bartman. Improvement of image processing by using homogeneous neural networks with fractional derivatives theorem. Conference Publications, 2011, 2011 (Special) : 505-514. doi: 10.3934/proc.2011.2011.505 [20] Shakir Sh. Yusubov, Elimhan N. Mahmudov. Optimality conditions of singular controls for systems with Caputo fractional derivatives. Journal of Industrial and Management Optimization, 2023, 19 (1) : 246-264. doi: 10.3934/jimo.2021182

2021 Impact Factor: 1.483