Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. (English) Zbl 1196.62063
Summary: The statistical analysis of covariance matrix data is considered and, in particular a methodology is discussed which takes into account the non-Euclidean nature of the space of positive semi-definite symmetric matrices. The main motivation for this work is the analysis of diffusion tensors in medical image analysis. The primary focus is on estimation of a mean covariance matrix and, in particular, on the use of a Procrustes size-and-shape space. Comparisons are made with other estimation techniques, including using the matrix logarithm, matrix square root and Cholesky decomposition. Applications to diffusion tensor imaging are considered and, in particular, a new measure of fractional anisotropy called Procrustes anisotropy is discussed.
MSC:
| 62H12 | Estimation in multivariate analysis |
| 62H35 | Image analysis in multivariate analysis |
| 92C55 | Biomedical imaging and signal processing |
| 65C60 | Computational problems in statistics (MSC2010) |
Keywords:
anisotropy; Cholesky; geodesic; matrix logarithm; principal components; Procrustes; Riemannian; shape; size; WishartSoftware:
RReferences:
| [1] | Alexa, M. (2002). Linear combination of transformations. ACM Trans. Graph. 21 380-387. |
| [2] | Alexander, D. C. (2005). Multiple-fiber reconstruction algorithms for diffusion MRI. Ann. N. Y. Acad. Sci. 1064 113-133. |
| [3] | Aljabar, P., Bhatia, K. K., Murgasova, M., Hajnal, J. V., Boardman, J. P., Srinivasan, L., Rutherford, M. A., Dyet, L. E., Edwards, A. D. and Rueckert, D. (2008). Assessment of brain growth in early childhood using deformation-based morphometry. Neuroimage 39 348-358. |
| [4] | Amaral, G. J. A., Dryden, I. L. and Wood, A. T. A. (2007). Pivotal bootstrap methods for k -sample problems in directional statistics and shape analysis. J. Amer. Statist. Assoc. 102 695-707. · Zbl 1172.62313 · doi:10.1198/016214506000001400 |
| [5] | Arsigny, V., Fillard, P., Pennec, X. and Ayache, N. (2007). Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 29 328-347. · Zbl 1144.47015 · doi:10.1137/050637996 |
| [6] | Basser, P. J., Mattiello, J. and Le Bihan, D. (1994). Estimation of the effective self-diffusion tensor from the NMR spin echo. J. Magn. Reson. B 103 247-254. |
| [7] | Basu, S., Fletcher, P. T. and Whitaker, R. T. (2006). Rician noise removal in diffusion tensor MRI. In MICCAI (1) (R. Larsen, M. Nielsen and J. Sporring, eds.) Lecture Notes in Computer Science 4190 117-125. Springer, Berlin. |
| [8] | Bhattacharya, R. and Patrangenaru, V. (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds. I. Ann. Statist. 31 1-29. · Zbl 1020.62026 · doi:10.1214/aos/1046294456 |
| [9] | Bhattacharya, R. and Patrangenaru, V. (2005). Large sample theory of intrinsic and extrinsic sample means on manifolds. II. Ann. Statist. 33 1225-1259. · Zbl 1072.62033 · doi:10.1214/009053605000000093 |
| [10] | Bickel, P. J. and Levina, E. (2008). Regularized estimation of large covariance matrices. Ann. Statist. 36 199-227. · Zbl 1132.62040 · doi:10.1214/009053607000000758 |
| [11] | Bookstein, F. L. (1986). Size and shape spaces for landmark data in two dimensions (with discussion). Statist. Sci. 1 181-242. · Zbl 0614.62144 · doi:10.1214/ss/1177013696 |
| [12] | Chen, Z. and Dunson, D. B. (2003). Random effects selection in linear mixed models. Biometrics 59 762-769. · Zbl 1214.62027 · doi:10.1111/j.0006-341X.2003.00089.x |
| [13] | Daniels, M. J. and Kass, R. E. (2001). Shrinkage estimators for covariance matrices. Biometrics 57 1173-1184. · Zbl 1209.62132 · doi:10.1111/j.0006-341X.2001.01173.x |
| [14] | Daniels, M. J. and Pourahmadi, M. (2002). Bayesian analysis of covariance matrices and dynamic models for longitudinal data. Biometrika 89 553-566. · Zbl 1036.62019 · doi:10.1093/biomet/89.3.553 |
| [15] | Dryden, I. L. (2005). Statistical analysis on high-dimensional spheres and shape spaces. Ann. Statist. 33 1643-1665. · Zbl 1078.62058 · doi:10.1214/009053605000000264 |
| [16] | Dryden, I. L. and Mardia, K. V. (1998). Statistical Shape Analysis . Wiley, Chichester. · Zbl 0901.62072 |
| [17] | Fillard, P., Arsigny, V., Pennec, X. and Ayache, N. (2007). Clinical DT-MRI estimation, smoothing and fiber tracking with log-Euclidean metrics. IEEE Trans. Med. Imaging 26 1472-1482. |
| [18] | Fletcher, P. T. and Joshi, S. (2007). Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Process. 87 250-262. · Zbl 1186.94126 · doi:10.1016/j.sigpro.2005.12.018 |
| [19] | Fréchet, M. (1948). Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. H. Poincaré 10 215-310. · Zbl 0035.20802 |
| [20] | Goodall, C. R. and Mardia, K. V. (1992). The noncentral Bartlett decompositions and shape densities. J. Multivariate Anal. 40 94-108. · Zbl 0742.62053 · doi:10.1016/0047-259X(92)90092-T |
| [21] | Gower, J. C. (1975). Generalized Procrustes analysis. Psychometrika 40 33-50. · Zbl 0305.62038 · doi:10.1007/BF02291478 |
| [22] | James, W. and Stein, C. (1961). Estimation with quadratic loss. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I 361-379. Univ. California Press, Berkeley, CA. · Zbl 1281.62026 |
| [23] | Karcher, H. (1977). Riemannian center of mass and mollifier smoothing. Comm. Pure Appl. Math. 30 509-541. · Zbl 0354.57005 · doi:10.1002/cpa.3160300502 |
| [24] | Kendall, D. G. (1984). Shape manifolds, Procrustean metrics and complex projective spaces. Bull. London Math. Soc. 16 81-121. · Zbl 0579.62100 · doi:10.1112/blms/16.2.81 |
| [25] | Kendall, D. G. (1989). A survey of the statistical theory of shape. Statist. Sci. 4 87-120. · Zbl 0955.60507 · doi:10.1214/ss/1177012582 |
| [26] | Kendall, D. G., Barden, D., Carne, T. K. and Le, H. (1999). Shape and Shape Theory . Wiley, Chichester. · Zbl 0940.60006 |
| [27] | Kendall, W. S. (1990). Probability, convexity, and harmonic maps with small image. I. Uniqueness and fine existence. Proc. London Math. Soc. (3) 61 371-406. · Zbl 0675.58042 · doi:10.1112/plms/s3-61.2.371 |
| [28] | Kent, J. T. (1994). The complex Bingham distribution and shape analysis. J. Roy. Statist. Soc. Ser. B 56 285-299. · Zbl 0806.62040 |
| [29] | Le, H. (2001). Locating Fréchet means with application to shape spaces. Adv. in Appl. Probab. 33 324-338. · Zbl 0990.60008 · doi:10.1239/aap/999188316 |
| [30] | Le, H. and Small, C. G. (1999). Multidimensional scaling of simplex shapes. Pattern Recognition 32 1601-1613. |
| [31] | Le, H.-L. (1988). Shape theory in flat and curved spaces, and shape densities with uniform generators. Ph.D. thesis, Univ. Cambridge. |
| [32] | Le, H.-L. (1992). The shapes of non-generic figures, and applications to collinearity testing. Proc. Roy. Soc. London Ser. A 439 197-210. · Zbl 0764.60014 · doi:10.1098/rspa.1992.0144 |
| [33] | Le, H.-L. (1995). Mean size-and-shapes and mean shapes: A geometric point of view. Adv. in Appl. Probab. 27 44-55. · Zbl 0818.60011 · doi:10.2307/1428094 |
| [34] | Lenglet, C., Rousson, M. and Deriche, R. (2006). DTI segmentation by statistical surface evolution. IEEE Trans. Med. Imaging 25 685-700. |
| [35] | Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis . Academic Press, London. · Zbl 0432.62029 |
| [36] | Moakher, M. (2005). A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 26 735-747 (electronic). · Zbl 1079.47021 · doi:10.1137/S0895479803436937 |
| [37] | Pennec, X. (1999). Probabilities and statistics on Riemannian manifolds: Basic tools for geometric measurements. In Proceedings of IEEE Workshop on Nonlinear Signal and Image Processing (NSIP99) (A. Cetin, L. Akarun, A. Ertuzun, M. Gurcan and Y. Yardimci, eds.) 1 194-198. IEEE, Los Alamitos, CA. |
| [38] | Pennec, X., Fillard, P. and Ayache, N. (2006). A Riemannian framework for tensor computing. Int. J. Comput. Vision 66 41-66. · Zbl 1287.53031 |
| [39] | Pourahmadi, M. (2007). Cholesky decompositions and estimation of a covariance matrix: Orthogonality of variance correlation parameters. Biometrika 94 1006-1013. · Zbl 1156.62043 · doi:10.1093/biomet/asm073 |
| [40] | R Development Core Team (2007). R: A Language and Environment for Statistical Computing . R Foundation for Statistical Computing, Vienna, Austria. |
| [41] | Schwartzman, A. (2006). Random ellipsoids and false discovery rates: Statistics for diffusion tensor imaging data. Ph.D. thesis, Stanford Univ. |
| [42] | Schwartzman, A., Dougherty, R. F. and Taylor, J. E. (2008). False discovery rate analysis of brain diffusion direction maps. Ann. Appl. Statist. 2 153-175. · Zbl 1137.62033 · doi:10.1214/07-AOAS133 |
| [43] | Wang, Z., Vemuri, B., Chen, Y. and Mareci, T. (2004). A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI. IEEE Trans. Med. Imaging 23 930-939. |
| [44] | Zhou, D., Dryden, I. L., Koloydenko, A. and Bai, L. (2008). A Bayesian method with reparameterisation for diffusion tensor imaging. In Proceedings, SPIE conference. Medical Imaging 2008: Image Processing (J. M. Reinhardt and J. P. W. Pluim, eds.) 69142J. SPIE, Bellingham, WA. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
