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Blow-up for the heat equation with a general memory boundary condition
Neuronal Fiber--tracking via optimal mass transportation
1. | Signal Processing Laboratory (LTS5), Ecole Polytechnique Federale de Lausanne (EPFL), ELD 232 - CH-1015 Lausanne, Swaziland |
2. | Department of Computer Sciences, University of Verona, Strada Le Grazie, 15 - I-37134 Verona, Italy, Italy, Italy |
References:
[1] |
Yves Achdou and Italo Capuzzo-Dolcetta, Mean field games: numerical methods, SIAM Journal on Numerical Analysis, 48 (2010), 1136-1162.
doi: 10.1137/090758477. |
[2] |
Andrei Agrachev and Paul Lee, Optimal transportation under nonholonomic constraints, Trans. Amer. Math. Soc., 361 (2009), 6019-6047.
doi: 10.1090/S0002-9947-09-04813-2. |
[3] |
D. C. Alexander, P. L. Hubbard, M. G. Hall, E. A. Moore, M. Ptito, G. J. Parker and T. B. Dyrby, Orientationally invariant indices of axon diameter and density from diffusion MRI, Neuroimage, (2010).
doi: 10.1016/j.neuroimage.2010.05.043. |
[4] |
Luigi Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields, in "Calculus of variations and nonlinear partial differential equations,'' Lecture Notes in Math., 1927 (2008), Springer, Berlin, 1-41.
doi: 10.1007/978-3-540-75914-0_1. |
[5] |
Luigi Ambrosio and Gianluca Crippa, Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields, in "Transport Equations and Multi-D Hyperbolic Conservation Laws,'' Lecture Notes of the Unione Matematica Italiana, 5 (2008).
doi: 10.1007/978-3-540-76781-7_1. |
[6] |
Luigi Ambrosio, Nicola Gigli and Giuseppe Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'' Second Ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. |
[7] |
Sigurd Angenent, Eric Pichon and Allen Tannenbaum, Mathematical methods in medical image processing, American Mathematical Society. Bulletin. New Series, 43 (2006), 365-396.
doi: 10.1090/S0273-0979-06-01104-9. |
[8] |
Y. Assaf, T. Blumenfeld-Katzir, Y. Yovel and P. J. Basser, AxCaliber: a method for measuring axon diameter distribution from diffusion MRI, Magn. Reson. Med., 59 (2008), 1347-1354.
doi: 10.1002/mrm.21577. |
[9] |
Jean-Pierre Aubin and Hélène Frankowska, "Set-valued Analysis,'' Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2009. |
[10] |
Martino Bardi and Italo Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,'' Birkhäuser Boston Inc., Boston, MA, 1997. |
[11] |
P. J. Basser, S. Pajevic, C. Pierpaoli, J. Duda and A Aldroubi, In vivo fiber tractography using DT-MR data, Magn. Reson. Med., 44 (2000), 625-632.
doi: 10.1002/1522-2594(200010)44:4<625::AID- ?MRM17>3.0.CO;2-O. |
[12] |
Jean-David Benamou and Yann Brenier, A numerical method for the optimal time-continuous mass transport problem and related problems, in "Monge Ampère Equation: Applications to Geometry and Optimization'' (Deerfield Beach, FL, 1997), Contemp. Math., 226 (1999), 1-11.
doi: 10.1090/conm/226. |
[13] |
J.-D. Benamou, Y. Brenier and K. Guittet, The Monge-Kantorovitch mass transfer and its computational fluid mechanics formulation, International Journal for Numerical Methods in Fluids, 40 (2002), 21-30.
doi: 10.1002/fld.264. |
[14] |
Stefano Bianchini and Fabio Cavalletti, The Monge problem for distance cost in geodesic spaces, to appear in "Nonlinear Conservation Laws and Applications," IMA Vol. Math. Appl., Springer, New York, 2010. |
[15] |
Piermarco Cannarsa and Carlo Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,'' Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston Inc., Boston, MA, 2004. |
[16] |
Piermarco Cannarsa and Carlo Sinestrari, Convexity properties of the minimum time function, Calculus of Variations and Partial Differential Equations, 3 (1995), 273-298.
doi: 10.1007/BF01189393. |
[17] |
F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,'' Graduate Texts in Mathematics, 178, Springer-Verlag, New York, 1998. |
[18] |
J. Coremans, R. Luypaert, F. Verhelle, T. Stadnik and M Osteaux, A method for myelin fiber orientation mapping using diffusion-weighted MR images, Magn Reson Imaging, 12 (1994), 443-454,
doi: 10.1016/0730-725X(94)92538-0. |
[19] |
J. Dauguet, S. Peled, V. Berezovskii, T. Delzescaux, S. K. Warfield, R. Born and C. F. Westin, Comparison of fiber tracts derived from in-vivo DTI tractography with 3D histological neural tract tracer reconstruction on a macaque brain, Neuroimage, 37 (2007), 530-538.
doi: 10.1016/j.neuroimage.2007.04.067. |
[20] |
T. B. Dyrby, L. V. Sogaard, G. J. Parker, D. C. Alexander, N. M. Lind, W. F. Baaré, A. Hay-Schmidt, N. Eriksen, B. Pakkenberg, O. B. Paulson and J. Jelsing, Validation of in vitro probabilistic tractography, Neuroimage, 37 (2007), 1267-1277.
doi: 10.1016/j.neuroimage.2007.06.022. |
[21] |
Mikhail Feldman and Robert J. McCann, Monge's transport problem on a Riemannian manifold, Transactions of the American Mathematical Society, 354 (2002), 1667-1697.
doi: 10.1090/S0002-9947-01-02930-0. |
[22] |
Albert Fathi and Antonio Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians, Calculus of Variations and Partial Differential Equations, 22 (2005), 185-228.
doi: 10.1007/s00526-004-0271-z. |
[23] |
P. Fillard and C. Poupon and J. F. Mangin, A novel global tractography algorithm based on an adaptive spin glass model, Med. Image Comput. Comput. Assist. Interv., 12 (2009), 927-934.
doi: 10.1007/978-3-642-04268-3_114. |
[24] |
Ola Friman and Gunnar Farneback and Carl-Fredrik Westin, A Bayesian approach for stochastic white matter tractography, IEEE transactions on medical imaging, 25 (2006), 965-978.
doi: 10.1109/TMI.2006.877093. |
[25] |
Wilfrid Gangbo and Robert J. McCann, The geometry of optimal transportation, Acta Mathematica, 177 (1996), 113-161.
doi: 10.1007/BF02392620. |
[26] |
P. Hagmann, L. Jonasson, P. Maeder, J. Thiran, V. Wedeen and R. Meuli, Understanding diffusion MR imaging techniques: from scalar diffusion-weighted imaging to diffusion tensor imaging and beyond, Radiographics, 26 (2006), S205-S223.
doi: 10.1148/rg.26si065510. |
[27] |
P. Hagmann, J. P. Thiran, L. Jonasson, P. Vandergheynst, S. Clarke, P. Maeder and R Meuli, DTI mapping of human brain connectivity: statistical fibre tracking and virtual dissection, Neuroimage, 19 (2003), 545-554.
doi: 10.1016/S1053-8119(03)00142-3. |
[28] |
Y. Iturria-Medina, E. J. Canales-Rodríguez, L. Melie-García, P. A. Valdés-Hernández, E. Martínez-Montes, Y. Alemán-Gómez and J M Sánchez-Bornot, Characterizing brain anatomical connections using diffusion weighted MRI and graph theory, Neuroimage, 36 (2007), 645-660.
doi: 10.1016/j.neuroimage.2007.02.012. |
[29] |
S. Jbabdi, M. W. Woolrich, J. L. Andersson and T. E. Behrens, A Bayesian framework for global tractography, Neuroimage, 37 (2007), 116-129.
doi: 10.1016/j.neuroimage.2007.04.039. |
[30] |
B. W. Kreher, I. Mader and V. G. Kiselev, Gibbs tracking: a novel approach for the reconstruction of neuronal pathways, Magn. Reson. Med., 60 (2008), 953-963.
doi: 10.1002/mrm.21749. |
[31] |
M. Lazar and A. L. Alexander, An error analysis of white matter tractography methods: synthetic diffusion tensor field simulations, Neuroimage, 20 (2003), 1140-1153.
doi: 10.1016/S1053-8119(03)00277-5. |
[32] |
M. Lazar, D. M. Weinstein, J. S. Tsuruda, K. M. Hasan, K. Arfanakis, M. E. Meyerand, B. Badie, H. A. Rowley, V. Haughton, A. Field and A. L. Alexander, White matter tractography using diffusion tensor deflection, Hum. Brain Mapp., 18 (2003), 306-321.
doi: 10.1002/hbm.10102. |
[33] |
C. P. Lin, V. J. Wedeen, J. H. Chen, C. Yao and W. Y. Tseng, Validation of diffusion spectrum magnetic resonance imaging with manganese-enhanced rat optic tracts and ex vivo phantoms, Neuroimage, 19 (2003), 482-495.
doi: 10.1016/S1053-8119(03)00154-X. |
[34] |
Y. Lu, A. Aldroubi, J. C. Gore, A. W. Anderson and Z. Ding, Improved fiber tractography with Bayesian tensor regularization, Neuroimage, 31 (2006), 1061-1074.
doi: 10.1016/j.neuroimage.2006.01.043. |
[35] |
G. J. Parker, H. A. Haroon and C. A. Wheeler-Kingshott, A framework for a streamline-based probabilistic index of connectivity (PICo) using a structural interpretation of MRI diffusion measurements, J. Magn. Reson. Imaging, 18 (2003), 242-254.
doi: 10.1002/jmri.10350. |
[36] |
Eric Pichon, Carl-Fredrik Westin and Allen Tannenbaum, A Hamilton-Jacobi-Bellman approach to high angular resolution diffusion tractography, in "Medical Image Computing and Computer-Assisted Intervention? MICCAI 2005", Lecture Notes in Computer Science 3759,Springer Berlin / Heidelberg, 2005, 180-187.
doi: 10.1007/11566465_23. |
[37] |
R. Tyrrell Rockafellar, "Convex Analysis,'' Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. |
[38] |
Antonio Siconolfi, Metric character of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 355 (2003), 1987-2009.
doi: 10.1090/S0002-9947-03-03237-9. |
[39] |
Antonio Siconolfi, Errata to: "Metric character of Hamilton-Jacobi equations'' [Trans. Amer. Math. Soc. 355 (2003), no. 5, 1987-2009 (electronic), Transactions of the American Mathematical Society, 355 (2003), 4265.
doi: 10.1090/S0002-9947-03-03410-X. |
[40] |
J. D. Tournier, F. Calamante, M. D. King, D. G. Gadian and A. Connelly, Limitations and requirements of diffusion tensor fiber tracking: an assessment using simulations, Magn. Reson. Med., 47 (2002), 701-708.
doi: 10.1002/mrm.10116. |
[41] |
J. D. Tournier, C. H. Yeh, F. Calamante, K. H. Cho, A. Connelly and C. P. Lin, Resolving crossing fibres using constrained spherical deconvolution: validation using diffusion-weighted imaging phantom data, Neuroimage, 42 (2008), 617-625.
doi: 10.1016/j.neuroimage.2008.05.002. |
[42] |
Cédric Villani, "Topics in Optimal Transportation,'' Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. |
[43] |
Cédric Villani, "Optimal Transport,'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9. |
[44] |
U. C. Wieshmann, C. A. Clark, M. R. Symms, F. Franconi, G. J. Barker and S. D. Shorvon, Reduced anisotropy of water diffusion in structural cerebral abnormalities demonstrated with diffusion tensor imaging, Magn. Reson. Imaging, 17 (1999), 1269-1274.
doi: 10.1016/S0730-725X(99)00082-X. |
[45] |
X. Wu, Q. Xu, L. Xu, J. Zhou, A. W. Anderson and Z. Ding, Genetic white matter fiber tractography with global optimization, J. Neurosci. Methods, 184 (2009), 375-379.
doi: 10.1016/j.jneumeth.2009.07.032. |
[46] |
A. Zalesky, DT-MRI fiber tracking: a shortest paths approach, IEEE Trans Med Imaging, 27 (2008), 1458-1471.
doi: 10.1109/TMI.2008.923644. |
show all references
References:
[1] |
Yves Achdou and Italo Capuzzo-Dolcetta, Mean field games: numerical methods, SIAM Journal on Numerical Analysis, 48 (2010), 1136-1162.
doi: 10.1137/090758477. |
[2] |
Andrei Agrachev and Paul Lee, Optimal transportation under nonholonomic constraints, Trans. Amer. Math. Soc., 361 (2009), 6019-6047.
doi: 10.1090/S0002-9947-09-04813-2. |
[3] |
D. C. Alexander, P. L. Hubbard, M. G. Hall, E. A. Moore, M. Ptito, G. J. Parker and T. B. Dyrby, Orientationally invariant indices of axon diameter and density from diffusion MRI, Neuroimage, (2010).
doi: 10.1016/j.neuroimage.2010.05.043. |
[4] |
Luigi Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields, in "Calculus of variations and nonlinear partial differential equations,'' Lecture Notes in Math., 1927 (2008), Springer, Berlin, 1-41.
doi: 10.1007/978-3-540-75914-0_1. |
[5] |
Luigi Ambrosio and Gianluca Crippa, Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields, in "Transport Equations and Multi-D Hyperbolic Conservation Laws,'' Lecture Notes of the Unione Matematica Italiana, 5 (2008).
doi: 10.1007/978-3-540-76781-7_1. |
[6] |
Luigi Ambrosio, Nicola Gigli and Giuseppe Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'' Second Ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. |
[7] |
Sigurd Angenent, Eric Pichon and Allen Tannenbaum, Mathematical methods in medical image processing, American Mathematical Society. Bulletin. New Series, 43 (2006), 365-396.
doi: 10.1090/S0273-0979-06-01104-9. |
[8] |
Y. Assaf, T. Blumenfeld-Katzir, Y. Yovel and P. J. Basser, AxCaliber: a method for measuring axon diameter distribution from diffusion MRI, Magn. Reson. Med., 59 (2008), 1347-1354.
doi: 10.1002/mrm.21577. |
[9] |
Jean-Pierre Aubin and Hélène Frankowska, "Set-valued Analysis,'' Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2009. |
[10] |
Martino Bardi and Italo Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,'' Birkhäuser Boston Inc., Boston, MA, 1997. |
[11] |
P. J. Basser, S. Pajevic, C. Pierpaoli, J. Duda and A Aldroubi, In vivo fiber tractography using DT-MR data, Magn. Reson. Med., 44 (2000), 625-632.
doi: 10.1002/1522-2594(200010)44:4<625::AID- ?MRM17>3.0.CO;2-O. |
[12] |
Jean-David Benamou and Yann Brenier, A numerical method for the optimal time-continuous mass transport problem and related problems, in "Monge Ampère Equation: Applications to Geometry and Optimization'' (Deerfield Beach, FL, 1997), Contemp. Math., 226 (1999), 1-11.
doi: 10.1090/conm/226. |
[13] |
J.-D. Benamou, Y. Brenier and K. Guittet, The Monge-Kantorovitch mass transfer and its computational fluid mechanics formulation, International Journal for Numerical Methods in Fluids, 40 (2002), 21-30.
doi: 10.1002/fld.264. |
[14] |
Stefano Bianchini and Fabio Cavalletti, The Monge problem for distance cost in geodesic spaces, to appear in "Nonlinear Conservation Laws and Applications," IMA Vol. Math. Appl., Springer, New York, 2010. |
[15] |
Piermarco Cannarsa and Carlo Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,'' Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston Inc., Boston, MA, 2004. |
[16] |
Piermarco Cannarsa and Carlo Sinestrari, Convexity properties of the minimum time function, Calculus of Variations and Partial Differential Equations, 3 (1995), 273-298.
doi: 10.1007/BF01189393. |
[17] |
F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,'' Graduate Texts in Mathematics, 178, Springer-Verlag, New York, 1998. |
[18] |
J. Coremans, R. Luypaert, F. Verhelle, T. Stadnik and M Osteaux, A method for myelin fiber orientation mapping using diffusion-weighted MR images, Magn Reson Imaging, 12 (1994), 443-454,
doi: 10.1016/0730-725X(94)92538-0. |
[19] |
J. Dauguet, S. Peled, V. Berezovskii, T. Delzescaux, S. K. Warfield, R. Born and C. F. Westin, Comparison of fiber tracts derived from in-vivo DTI tractography with 3D histological neural tract tracer reconstruction on a macaque brain, Neuroimage, 37 (2007), 530-538.
doi: 10.1016/j.neuroimage.2007.04.067. |
[20] |
T. B. Dyrby, L. V. Sogaard, G. J. Parker, D. C. Alexander, N. M. Lind, W. F. Baaré, A. Hay-Schmidt, N. Eriksen, B. Pakkenberg, O. B. Paulson and J. Jelsing, Validation of in vitro probabilistic tractography, Neuroimage, 37 (2007), 1267-1277.
doi: 10.1016/j.neuroimage.2007.06.022. |
[21] |
Mikhail Feldman and Robert J. McCann, Monge's transport problem on a Riemannian manifold, Transactions of the American Mathematical Society, 354 (2002), 1667-1697.
doi: 10.1090/S0002-9947-01-02930-0. |
[22] |
Albert Fathi and Antonio Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians, Calculus of Variations and Partial Differential Equations, 22 (2005), 185-228.
doi: 10.1007/s00526-004-0271-z. |
[23] |
P. Fillard and C. Poupon and J. F. Mangin, A novel global tractography algorithm based on an adaptive spin glass model, Med. Image Comput. Comput. Assist. Interv., 12 (2009), 927-934.
doi: 10.1007/978-3-642-04268-3_114. |
[24] |
Ola Friman and Gunnar Farneback and Carl-Fredrik Westin, A Bayesian approach for stochastic white matter tractography, IEEE transactions on medical imaging, 25 (2006), 965-978.
doi: 10.1109/TMI.2006.877093. |
[25] |
Wilfrid Gangbo and Robert J. McCann, The geometry of optimal transportation, Acta Mathematica, 177 (1996), 113-161.
doi: 10.1007/BF02392620. |
[26] |
P. Hagmann, L. Jonasson, P. Maeder, J. Thiran, V. Wedeen and R. Meuli, Understanding diffusion MR imaging techniques: from scalar diffusion-weighted imaging to diffusion tensor imaging and beyond, Radiographics, 26 (2006), S205-S223.
doi: 10.1148/rg.26si065510. |
[27] |
P. Hagmann, J. P. Thiran, L. Jonasson, P. Vandergheynst, S. Clarke, P. Maeder and R Meuli, DTI mapping of human brain connectivity: statistical fibre tracking and virtual dissection, Neuroimage, 19 (2003), 545-554.
doi: 10.1016/S1053-8119(03)00142-3. |
[28] |
Y. Iturria-Medina, E. J. Canales-Rodríguez, L. Melie-García, P. A. Valdés-Hernández, E. Martínez-Montes, Y. Alemán-Gómez and J M Sánchez-Bornot, Characterizing brain anatomical connections using diffusion weighted MRI and graph theory, Neuroimage, 36 (2007), 645-660.
doi: 10.1016/j.neuroimage.2007.02.012. |
[29] |
S. Jbabdi, M. W. Woolrich, J. L. Andersson and T. E. Behrens, A Bayesian framework for global tractography, Neuroimage, 37 (2007), 116-129.
doi: 10.1016/j.neuroimage.2007.04.039. |
[30] |
B. W. Kreher, I. Mader and V. G. Kiselev, Gibbs tracking: a novel approach for the reconstruction of neuronal pathways, Magn. Reson. Med., 60 (2008), 953-963.
doi: 10.1002/mrm.21749. |
[31] |
M. Lazar and A. L. Alexander, An error analysis of white matter tractography methods: synthetic diffusion tensor field simulations, Neuroimage, 20 (2003), 1140-1153.
doi: 10.1016/S1053-8119(03)00277-5. |
[32] |
M. Lazar, D. M. Weinstein, J. S. Tsuruda, K. M. Hasan, K. Arfanakis, M. E. Meyerand, B. Badie, H. A. Rowley, V. Haughton, A. Field and A. L. Alexander, White matter tractography using diffusion tensor deflection, Hum. Brain Mapp., 18 (2003), 306-321.
doi: 10.1002/hbm.10102. |
[33] |
C. P. Lin, V. J. Wedeen, J. H. Chen, C. Yao and W. Y. Tseng, Validation of diffusion spectrum magnetic resonance imaging with manganese-enhanced rat optic tracts and ex vivo phantoms, Neuroimage, 19 (2003), 482-495.
doi: 10.1016/S1053-8119(03)00154-X. |
[34] |
Y. Lu, A. Aldroubi, J. C. Gore, A. W. Anderson and Z. Ding, Improved fiber tractography with Bayesian tensor regularization, Neuroimage, 31 (2006), 1061-1074.
doi: 10.1016/j.neuroimage.2006.01.043. |
[35] |
G. J. Parker, H. A. Haroon and C. A. Wheeler-Kingshott, A framework for a streamline-based probabilistic index of connectivity (PICo) using a structural interpretation of MRI diffusion measurements, J. Magn. Reson. Imaging, 18 (2003), 242-254.
doi: 10.1002/jmri.10350. |
[36] |
Eric Pichon, Carl-Fredrik Westin and Allen Tannenbaum, A Hamilton-Jacobi-Bellman approach to high angular resolution diffusion tractography, in "Medical Image Computing and Computer-Assisted Intervention? MICCAI 2005", Lecture Notes in Computer Science 3759,Springer Berlin / Heidelberg, 2005, 180-187.
doi: 10.1007/11566465_23. |
[37] |
R. Tyrrell Rockafellar, "Convex Analysis,'' Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. |
[38] |
Antonio Siconolfi, Metric character of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 355 (2003), 1987-2009.
doi: 10.1090/S0002-9947-03-03237-9. |
[39] |
Antonio Siconolfi, Errata to: "Metric character of Hamilton-Jacobi equations'' [Trans. Amer. Math. Soc. 355 (2003), no. 5, 1987-2009 (electronic), Transactions of the American Mathematical Society, 355 (2003), 4265.
doi: 10.1090/S0002-9947-03-03410-X. |
[40] |
J. D. Tournier, F. Calamante, M. D. King, D. G. Gadian and A. Connelly, Limitations and requirements of diffusion tensor fiber tracking: an assessment using simulations, Magn. Reson. Med., 47 (2002), 701-708.
doi: 10.1002/mrm.10116. |
[41] |
J. D. Tournier, C. H. Yeh, F. Calamante, K. H. Cho, A. Connelly and C. P. Lin, Resolving crossing fibres using constrained spherical deconvolution: validation using diffusion-weighted imaging phantom data, Neuroimage, 42 (2008), 617-625.
doi: 10.1016/j.neuroimage.2008.05.002. |
[42] |
Cédric Villani, "Topics in Optimal Transportation,'' Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. |
[43] |
Cédric Villani, "Optimal Transport,'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9. |
[44] |
U. C. Wieshmann, C. A. Clark, M. R. Symms, F. Franconi, G. J. Barker and S. D. Shorvon, Reduced anisotropy of water diffusion in structural cerebral abnormalities demonstrated with diffusion tensor imaging, Magn. Reson. Imaging, 17 (1999), 1269-1274.
doi: 10.1016/S0730-725X(99)00082-X. |
[45] |
X. Wu, Q. Xu, L. Xu, J. Zhou, A. W. Anderson and Z. Ding, Genetic white matter fiber tractography with global optimization, J. Neurosci. Methods, 184 (2009), 375-379.
doi: 10.1016/j.jneumeth.2009.07.032. |
[46] |
A. Zalesky, DT-MRI fiber tracking: a shortest paths approach, IEEE Trans Med Imaging, 27 (2008), 1458-1471.
doi: 10.1109/TMI.2008.923644. |
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