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. AlexanderC. PierpaoliP. J. Basser and J. C. Gee, Spatial transformations of diffusion tensor magnetic resonance images, IEEE Transaction on Medical imaging, 20 (2001), 1131-1139.   Google Scholar

[2]

M. F. BegM. I. MillerA. Trouve and L. Younes, Computing large deformation metric mappings via geodesic flows of diffeomorphisms, International Journal of Computer Vision, 61 (2005), 139-157.   Google Scholar

[3]

M. BruverisF. Gay-BalmazD. 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.  Google Scholar

[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.  Google Scholar

[5]

P. DupuisU. 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.  Google Scholar

[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.  Google Scholar

[7]

L. C. Evans, Partial differential equations, American Mathematical Society, (1997), 251-308.   Google Scholar

[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.  Google Scholar

[9]

H. Han and H. Zhou, Spectral representation of solution of a variational model in diffusion tensor images registration, preprint. Google Scholar

[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.   Google Scholar

[11]

C. R. JohnsonK. 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.  Google Scholar

[12]

J. LiY. ShiG. TranI. DinovD. 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.   Google Scholar

[13]

R. LiS. 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.  Google Scholar

[14]

F. O'Sullivan, The Analysis of Some Penalized Likelihood Schemes, Statistics Department Technical Report No.726, University of Wisconsin, 1983. Google Scholar

[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.   Google Scholar

[16]

G. Teschl, Ordinary differential equations and Dynamical systems, American Mathematical Society, (2012), 50-230.  doi: 10.1090/gsm/140.  Google Scholar

[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.  Google Scholar

[18]

T. YeoT. VercauterenP. FiclardJ. PeyratX. PennecP. GollandN Ayache and O. Clatz, DTREFinD: Diffusion tensor registration with exact finite-strain differential, IEEE Transaction on Medical imaging, 28 (2009), 1914-1928.   Google Scholar

[19]

S. Zhan, On the determinantal inequalities, Journal of Inequalities in Pure and Applied Mathematics, 6 (2005), Article 105, 7 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

D. C. AlexanderC. PierpaoliP. J. Basser and J. C. Gee, Spatial transformations of diffusion tensor magnetic resonance images, IEEE Transaction on Medical imaging, 20 (2001), 1131-1139.   Google Scholar

[2]

M. F. BegM. I. MillerA. Trouve and L. Younes, Computing large deformation metric mappings via geodesic flows of diffeomorphisms, International Journal of Computer Vision, 61 (2005), 139-157.   Google Scholar

[3]

M. BruverisF. Gay-BalmazD. 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.  Google Scholar

[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.  Google Scholar

[5]

P. DupuisU. 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.  Google Scholar

[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.  Google Scholar

[7]

L. C. Evans, Partial differential equations, American Mathematical Society, (1997), 251-308.   Google Scholar

[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.  Google Scholar

[9]

H. Han and H. Zhou, Spectral representation of solution of a variational model in diffusion tensor images registration, preprint. Google Scholar

[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.   Google Scholar

[11]

C. R. JohnsonK. 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.  Google Scholar

[12]

J. LiY. ShiG. TranI. DinovD. 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.   Google Scholar

[13]

R. LiS. 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.  Google Scholar

[14]

F. O'Sullivan, The Analysis of Some Penalized Likelihood Schemes, Statistics Department Technical Report No.726, University of Wisconsin, 1983. Google Scholar

[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.   Google Scholar

[16]

G. Teschl, Ordinary differential equations and Dynamical systems, American Mathematical Society, (2012), 50-230.  doi: 10.1090/gsm/140.  Google Scholar

[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.  Google Scholar

[18]

T. YeoT. VercauterenP. FiclardJ. PeyratX. PennecP. GollandN Ayache and O. Clatz, DTREFinD: Diffusion tensor registration with exact finite-strain differential, IEEE Transaction on Medical imaging, 28 (2009), 1914-1928.   Google Scholar

[19]

S. Zhan, On the determinantal inequalities, Journal of Inequalities in Pure and Applied Mathematics, 6 (2005), Article 105, 7 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  One slice of $T(\cdot)$ and $D(\cdot)$
Figure 2.  $a$ and ${\rm Re-SSD}$ change with differential order $\alpha$
Figure 3.  $a$ and ${\rm Re-SSD}$ change with time $s$ in iteration process
Figure 4.  The 22th slice of $T\diamond h(\cdot)$
[1]

Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011

[2]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355

[3]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[4]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[5]

Jiahao Qiu, Jianjie Zhao. Maximal factors of order $ d $ of dynamical cubespaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 601-620. doi: 10.3934/dcds.2020278

[6]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255

[7]

Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265

[8]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268

[9]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[10]

Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054

[11]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[12]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

[13]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[14]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

[15]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[16]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164

[17]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[18]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[19]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[20]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

2019 Impact Factor: 1.373

Metrics

  • PDF downloads (210)
  • HTML views (306)
  • Cited by (5)

Other articles
by authors

[Back to Top]