Fiber orientation distribution estimation using a Peaceman-Rachford splitting method. (English) Zbl 1346.92038
Summary: In diffusion-weighted magnetic resonance imaging, the estimation of the orientations of multiple nerve fibers in each voxel (the fiber orientation distribution (FOD)) is a critical issue for exploring the connection of cerebral tissue. In this paper, we establish a convex semidefinite programming (CSDP) model for the FOD estimation. One feature of the new model is that it can ensure the statistical meaning of FOD since as a probability density function, FOD must be nonnegative and have a unit mass. To construct such a statistically meaningful FOD, we consider its approximation by a sum of squares (SOS) polynomial and impose the unit-mass by a linear constraint. Another feature of the new model is that it introduces a new regularization based on the sparsity of nerve fibers. Due to the sparsity of the orientations of nerve fibers in cerebral white matter, a heuristic regularization is raised, which is inspired by the Z-eigenvalue of a symmetric tensor that closely relates to the SOS polynomial. To solve the CSDP efficiently, we propose a new Peaceman-Rachford splitting method and prove its global convergence. Numerical experiments on synthetic and real-world human brain data show that, when compared with some existing approaches for fiber estimations, the new method gives a sharp and smooth FOD. Further, the proposed Peaceman-Rachford splitting method is shown to have good numerical performances comparing several existing methods.
MSC:
| 92C55 | Biomedical imaging and signal processing |
| 65H17 | Numerical solution of nonlinear eigenvalue and eigenvector problems |
| 65K05 | Numerical mathematical programming methods |
| 90C22 | Semidefinite programming |
| 90C90 | Applications of mathematical programming |
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
Keywords:
fiber orientation distribution; magnetic resonance imaging; Peaceman-Rachford splitting method; positive semidefinite tensor; semidefinite programming; sum of squares polynomialReferences:
| [1] | I. Aganj, C. Lenglet, G. Sapiro, E. Yacoub, K. Ugurbil, and N. Harel, {\it Reconstruction of the orientation distribution function in single- and multiple-shell Q-ball imaging within constant solid angle}, Magn. Reson. Med., 64 (2010), pp. 554-566. |
| [2] | D. C. Alexander, {\it Maximum entropy spherical deconvolution for diffusion MRI}, Inf. Process. Med. Imaging, 3565 (2005), pp. 76-87. |
| [3] | D. C. Alexander, G. J. Barker, and S. R. Arridge, {\it Detection and modeling of non-Gaussian apparent diffusion coefficient profiles in human brain data}, Magn. Reson. Med., 48 (2002), pp. 331-340. |
| [4] | A. Barmpoutis, J. Ho, and B. C. Vemuri, {\it Approximating symmetric positive semidefinite tensors of even order}, SIAM J. Imaging Sci., 5 (2012), pp. 434-464. · Zbl 1250.65056 |
| [5] | P. J. Basser, J. Mattiello, and D. LeBihan, {\it Estimation of the effective self-diffusion tensor from the NMR spin echo}, J. Magn. Reson., 103 (1994), pp. 247-254. |
| [6] | T. E. J. Behrens, H. J. Berg, S. Jbabdi, M. F. S. Rushworth, and M. W. Woolrich, {\it Probabilistic diffusion tractography with multiple fibre orientations: What can we gain?}, Neuroimage, 34 (2007), pp. 144-155. |
| [7] | S. Boyd and L. Vandenberghe, {\it Convex Optimization}, Cambridge University Press, Cambridge, UK, 2004. · Zbl 1058.90049 |
| [8] | E. J. Canales-Rodríguez, C. Lin, Y. Iturria-Medina, C. Yeh, K. Cho, and L. Melie-García, {\it Diffusion orientation transform revisited}, Neuroimage, 49 (2010), pp. 1326-1339. |
| [9] | E. J. Candés, J. Romberg, and T. Tao, {\it Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information}, IEEE Trans. Inform. Theory, 52 (2006), pp. 489-509. · Zbl 1231.94017 |
| [10] | E. J. Candés, J. K. Romberg, and T. Tao, {\it Stable signal recovery from incomplete and inaccurate measurements}, Comm. Pure Appl. Math., 59 (2006), pp. 1207-1223. · Zbl 1098.94009 |
| [11] | G. Casella and R. L. Berger, {\it Statistical Inference}, 2nd ed., Duxbury, Pacific Grove, CA, 2002. · Zbl 0699.62001 |
| [12] | Y. Chen, Y. Dai, D. Han, and W. Sun, {\it Positive semidefinite generalized diffusion tensor imaging via quadratic semidefinite programming}, SIAM J. Imaging Sci., 6 (2013), pp. 1531-1552. · Zbl 1283.65057 |
| [13] | J. Cheng, R. Deriche, T. Jiang, D. Shen, and P. T. Yap, {\it Non-negative spherical deconvolution (NNSD) for estimation of fiber orientation distribution function in single-/multi-shell diffusion MRI}, Neuroimage, 101 (2014), pp. 750-764. |
| [14] | O. Ciccarelli, M. Catani, H. Johansen-Berg, C. Clark, and A. Thompson, {\it Diffusion-based tractography in neurological disorders: Concepts, applications, and future developments}, LANCET Neurol., 7 (2008), pp. 715-727. |
| [15] | C. Cui, Y. Dai, and J. Nie, {\it All real eigenvalues of symmetric tensors}, SIAM J. Matrix Anal. Appl., 35 (2014), pp. 1582-1601. · Zbl 1312.65053 |
| [16] | F. Dell’Acqua, G. Rizzo, P. Scifo, R. A. Clarke, G. Scotti, and F. Fazio, {\it A model-based deconvolution approach to solve fiber crossing in diffusion-weighted MR imaging}, IEEE Trans. Biomed. Eng., 54 (2007), pp. 462-472. |
| [17] | F. Dell’Acqua, P. Scifo, G. Rizzo, M. Catani, A. Simmons, G. Scotti, and F. Fazio, {\it A modified damped Richardson-Lucy algorithm to reduce isotropic background effects in spherical deconvolution}, Neuroimage, 49 (2010), pp. 1446-1458. |
| [18] | M. Descoteaux, E. Angelino, S. Fitzgibbons, and R. Deriche, {\it Apparent diffusion coefficients from high angular resolution diffusion imaging: Estimation and applications}, Magn. Reson. Med., 56 (2006), pp. 395-410. |
| [19] | M. Descoteaux, E. Angelino, S. Fitzgibbons, and R. Deriche, {\it Regularized, fast, and robust analytical Q-ball imaging}, Magn. Reson. Med., 58 (2007), pp. 497-510. |
| [20] | M. Descoteaux, R. Deriche, T. R. Knösche, and A. Anwander, {\it Deterministic and probabilistic tractography based on complex fibre orientation distributions}, IEEE Trans. Med. Imaging, 28 (2009), pp. 269-286. |
| [21] | D. L. Donoho, {\it For most large underdetermined systems of linear equations the minimal \(ℓ_1\)-norm solution is also the sparsest solution}, Comm. Pure Appl. Math., 59 (2006), pp. 797-829. · Zbl 1113.15004 |
| [22] | J. Douglas and H. H. Rachford, {\it On the numerical solution of heat conduction problems in two and three space variables}, Trans. Amer. Math. Soc., 82 (1956), pp. 421-439. · Zbl 0070.35401 |
| [23] | D. Gabay, {\it Applications of the method of multipliers to variational inequalities}, in Augmented Lagrange Methods: Applications to the Solution of Boundary-Valued Problems, M. Fortin and R. Glowinski, eds., North-Holland, Amsterdam, 1983, pp. 59-82. |
| [24] | R. Glowinski, {\it On alternating direction methods of multipliers: A historical perspective}, in Modeling, Simulation and Optimization for Science and Technology, William Fitzgibbon, Yuri A. Kuznetsov, Pekka Neittaanmäki, and Olivier Pironneau, eds., Comput. Methods Appl. Sci. 34, Springer, Dordrecht, Netherlands, 2014, pp. 59-82. · Zbl 1320.65098 |
| [25] | R. Glowinski and A. Marroco, {\it Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problémes de Dirichlet non linéaires}, Revue Française d’Automatique, Informatique, Recherche Opérationnelle, Analyse numérique, 9 (1975), pp. 41-76. · Zbl 0368.65053 |
| [26] | A. Goh, C. Lenglet, P. M. Thompson, and R. Vidal, {\it Estimating orientation distribution functions with probability density constraints and spatial regularity}, in Medical Image Computing and Computer-Assisted Intervention-MICCAI 2009, Lecture Notes in Comput. Sci. 5761, Springer-Verlag, Berlin, Heidelberg, 2009, pp. 877-885. |
| [27] | D. Han, L. Qi, and Ed X. Wu, {\it Extreme diffusion values for non-Gaussian diffusions}, Optim. Methods Softw., 23 (2008), pp. 703-716. · Zbl 1154.60342 |
| [28] | B. He, H. Liu, Z. Wang, and X. Yuan, {\it A strictly contractive Peaceman-Rachford splitting method for convex programming}, SIAM J. Optim., 24 (2014), pp. 1011-1040. · Zbl 1327.90210 |
| [29] | B. He, M. Tao, and X. Yuan, {\it Alternating direction method with Gaussian back substitution for separable convex programming}, SIAM J. Optim., 22 (2012), pp. 313-340. · Zbl 1273.90152 |
| [30] | R. A. Horn and C. R. Johnson, {\it Topics in Matrix Analysis}, Cambridge University Press, Cambridge, UK, 1991. · Zbl 0729.15001 |
| [31] | S. Hu, Z. Huang, H. Ni, and L Qi, {\it Positive definiteness of diffusion kurtosis imaging}, Inverse Probl. Imaging, 6 (2012), pp. 57-75. · Zbl 1242.90164 |
| [32] | J. H. Jensen, J. A. Helpern, A. Ramani, H. Lu, and K. Kaczynski, {\it Diffusional kurtosis imaging: The quantification of non-Gaussian water diffusion by means of magnetic resonance imaging}, Magn. Reson. Med., 53 (2005), pp. 1432-1440. |
| [33] | B. Jian and B. C. Vemuri, {\it A unified computational framework for deconvolution to reconstruct multiple fibers from diffusion weighted MRI}, IEEE Trans. Med. Imaging, 26 (2007), pp. 1464-1471. |
| [34] | B. Jian, B. C. Vemuri, E. Özarslan, P. R. Carney, and T. H. Mareci, {\it A novel tensor distribution model for the diffusion-weighted MR signal}, NeuroIimage, 37 (2007), pp. 164-176. |
| [35] | H. Johansen-Berg and M. F. S. Rushworth, {\it Using diffusion imaging to study human connectional anatomy}, Annu. Rev. Neurosci., 32 (2009), pp. 75-94. |
| [36] | E. Kaden, T. R. Knösche, and A. Anwander, {\it Parametric spherical deconvolution: Inferring anatomical connectivity using diffusion MR imaging}, Neuroimage, 37 (2007), pp. 474-488. |
| [37] | J. B. Lasserre, {\it A sum of squares approximation of nonnegative polynomials}, SIAM J. Optim., 16 (2006), pp. 751-765. · Zbl 1129.12003 |
| [38] | V. I. Lebedev and D. N. Laikov, {\it A quadrature formula for the sphere of the \(131\) st algebraic order of accuracy}, Dokl. Math., 59 (1999), pp. 477-481. · Zbl 0960.65029 |
| [39] | C. Lin, W. I. Tseng, H. Cheng, and J. Chen, {\it Validation of diffusion tensor magnetic resonance axonal fiber imaging with registered manganese-enhanced optic tracts}, Neuroimage, 14 (2001), pp. 1035-1047. |
| [40] | R. Neji, N. Azzabou, N. Paragios, and G. Fleury, {\it A convex semi-definite positive framework for DTI estimation and regularization}, in Advances in Visual Computing, Lecture Notes in Comput. Sci. 4841, Springer-Verlag, Berlin, Heidelberg, 2007, pp. 220-229. |
| [41] | J. Nocedal and S. J. Wright, {\it Numerical Optimization}, 2nd ed., Springer Ser. Oper. Res. Financ. Eng. Springer, New York, 2006. · Zbl 1104.65059 |
| [42] | Y. Ouyang, Y. Chen, Y. Wu, and H. Zhou, {\it Total variation and wavelet regularization of orientation distribution functions in diffusion MRI}, Inverse Probl. Imaging, 7 (2013), pp. 565-583. · Zbl 1336.92046 |
| [43] | P. A. Parrilo, {\it Semidefinite programming relaxations for semialgebraic problems}, Math. Program., 96 (2003), pp. 293-320. · Zbl 1043.14018 |
| [44] | V. Patel, Y. Shi, P. M. Thompson, and A. W. Toga, {\it Mesh-based spherical deconvolution: A flexible approach to reconstruction of non-negative fiber orientation distributions}, Neuroimage, 51 (2010), pp. 1071-1081. |
| [45] | D. W. Peaceman and H. H. Rachford, Jr., {\it The numerical solution of parabolic and elliptic differential equations}, J. Soc. Ind. Appl. Math., 3 (1955), pp. 28-41. · Zbl 0067.35801 |
| [46] | L. Qi, {\it Eigenvalues of a real supersymmetric tensor}, J. Symbolic Comput., 40 (2005), pp. 1302-1324. · Zbl 1125.15014 |
| [47] | L. Qi, D. Han, and Ed X. Wu, {\it Principal invariants and inherent parameters of diffusion kurtosis tensors}, J. Math. Anal. Appl., 349 (2009), pp. 165-180. · Zbl 1158.15029 |
| [48] | L. Qi, Y. Wang, and Ed X. Wu, {\it D-eigenvalues of diffusion kurtosis tensors}, J. Comput. Appl. Math., 221 (2008), pp. 150-157. · Zbl 1176.65046 |
| [49] | L. Qi, G. Yu, and Y. Xu, {\it Nonnegative diffusion orientation distribution function}, J. Math. Imaging Vis., 45 (2013), pp. 103-113. · Zbl 1276.94004 |
| [50] | A. Ramirez-Manzanares, M. Rivera, B. C. Vemuri, P. Carney, and T. Mareci, {\it Diffusion basis functions decomposition for estimating white matter intravoxel fiber geometry}, IEEE Trans. Med. Imaging, 26 (2007), pp. 1091-1102. |
| [51] | E. Schwab, B. Afsari, and R. Vidal, {\it Estimation of non-negative ODFs using the eigenvalue distribution of spherical functions}, in Medical Image Computing and Computer-Assisted Intervention-MICCAI 2012, Springer-Verlag, Berlin, Heidelberg, 2012, pp. 322-330. |
| [52] | J. D. Tournier, F. Calamante, and A. Connelly, {\it Robust determination of the fibre orientation distribution in diffusion MRI: Non-negativity constrained super-resolved spherical deconvolution}, Neuroimage, 35 (2007), pp. 1459-1472. |
| [53] | J. D. Tournier, F. Calamante, D. G. Gadian, and A. Connelly, {\it Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution}, Neuroimage, 23 (2004), pp. 1176-1185. |
| [54] | A. Tristän-Vega, C. F. Westin, and S. Aja-Fernández, {\it Estimation of fiber orientation probability density functions in high angular resolution diffusion imaging}, Neuroimage, 47 (2009), pp. 638-650. |
| [55] | A. Tristán-Vega, C. F. Westin, and S. Aja-Fernández, {\it A new methodology for the estimation of fiber populations in the white matter of the brain with the Funk-Radon transform}, Neuroimage, 49 (2010), pp. 1301-1315. |
| [56] | D. S. Tuch, {\it Q-ball imaging}, Magn. Reson. Med., 52 (2004), pp. 1358-1372. |
| [57] | D. S. Tuch, T. G. Reese, M. R. Wiegell, N. Makris, J. W. Belliveau, and V. J. Wedeen, {\it High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity}, Magn. Reson. Med., 48 (2002), pp. 577-582. |
| [58] | V. J. Wedeen, P. Hagmann, W. I. Tseng, T. G. Reese, and R. M. Weisskoff, {\it Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging}, Magn. Reson. Med., 54 (2005), pp. 1377-1386. |
| [59] | Y. T. Weldeselassie, A. Barmpoutis, and M. S. Atkins, {\it Symmetric positive semi-definite cartesian tensor fiber orientation distributions (CT-FOD)}, Med. Image Anal., 16 (2012), pp. 1121-1129. |
| [60] | S. Wolfers, E. Schwab, and R. Vidal, {\it Nonnegative ODF estimation via optimal constraint selection}, in 2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI), IEEE, Piscataway, NJ, 2014, pp. 734-737. |
| [61] | C. Ye and X. Yuan, {\it A descent method for structured monotone variational inequalities}, Optim. Methods Softw., 22 (2007), pp. 329-338. · Zbl 1196.90118 |
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