Advertisement

Efficient Recursive Algorithms for Computing the Mean Diffusion Tensor and Applications to DTI Segmentation

  • Guang Cheng
  • Hesamoddin Salehian
  • Baba C. Vemuri
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7578)

Abstract

Computation of the mean of a collection of symmetric positive definite (SPD) matrices is a fundamental ingredient of many algorithms in diffusion tensor image (DTI) processing. For instance, in DTI segmentation, clustering, etc. In this paper, we present novel recursive algorithms for computing the mean of a set of diffusion tensors using several distance/divergence measures commonly used in DTI segmentation and clustering such as the Riemannian distance and symmetrized Kullback-Leibler divergence. To the best of our knowledge, to date, there are no recursive algorithms for computing the mean using these measures in literature. Recursive algorithms lead to a gain in computation time of several orders in magnitude over existing non-recursive algorithms. The key contributions of this paper are: (i) we present novel theoretical results on a recursive estimator for Karcher expectation in the space of SPD matrices, which in effect is a proof of the law of large numbers (with some restrictions) for the manifold of SPD matrices. (ii) We also present a recursive version of the symmetrized KL-divergence for computing the mean of a collection of SPD matrices. (iii) We present comparative timing results for computing the mean of a group of SPD matrices (diffusion tensors) depicting the gains in compute time using the proposed recursive algorithms over existing non-recursive counter parts. Finally, we also show results on gains in compute times obtained by applying these recursive algorithms to the task of DTI segmentation.

Keywords

Segmentation Result Active Contour Structure Tensor Recursive Algorithm Recursive Estimator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Wang, Z., Vemuri, B.C.: Tensor Field Segmentation Using Region Based Active Contour Model. In: Pajdla, T., Matas, J(G.) (eds.) ECCV 2004. LNCS, vol. 3024, pp. 304–315. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  2. 2.
    Chan, T., Vese, L.: Active contours without edges. IEEE Trans. on Image Proc. 10(2), 266–277 (2001)zbMATHCrossRefGoogle Scholar
  3. 3.
    Tsai, A., Yezzi, A.J., Willsky, A.: Curve Evolution Implementation of the Mumford-Shah Functional for Image Segmentation, Denoising, Interpolation, and Magnification. IEEE Trans. on Image Proc. 10(8), 1169–1186 (2001)zbMATHCrossRefGoogle Scholar
  4. 4.
    Feddern, C., Weickert, J., Burgeth, B.: Level-set Methods for Tensor-valued Images. In: Proc. 2nd IEEE Workshop on Variational, Geometric and Level Set Methods in Comp. Vis., pp. 65–72 (2003)Google Scholar
  5. 5.
    Malladi, R., Sethian, J., Vemuri, B.C.: Shape Modeling with Front Propagation: A Level Set Approach. IEEE Trans. on PAMI 17(2), 158–175 (1995)CrossRefGoogle Scholar
  6. 6.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic Active Contours. Intl. Journ. of Compu. Vision 22(1), 61–79 (1997)zbMATHCrossRefGoogle Scholar
  7. 7.
    Lenglet, C., Rousson, M., Deriche, R.: Dti segmentation by statistical surface evolution. IEEE Trans. on Medical Imaging 25(6), 685–700 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Goh, A., Vidal, R.: Segmenting Fiber Bundles in Diffusion Tensor Images. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part III. LNCS, vol. 5304, pp. 238–250. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    Vemuri, B., Liu, M., Amari, S., Nielsen, F.: Total bregman divergence and its applications to dti analysis. IEEE Trans. on Medical Imaging 30(2), 475–483 (2011)CrossRefGoogle Scholar
  10. 10.
    Moakher, M., Batchelor, P.G.: Symmetric Positive-Definite Matrices: From Geometry to Applications and Visualization. Visualization and Processing of Tensor Fields. Springer (2006)Google Scholar
  11. 11.
    Pennec, X., Fillard, P., Ayache, N.: A riemannian framework for tensor computing. International Journal of Computer Vision 66(1), 41–66 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Barmpoutis, A., Vemuri, B.C., Shepherd, T.M., Forder, J.R.: Tensor splines for interpolation and approximation of DT-MRI with applications to segmentation of isolated rat hippocampi. IEEE Trans. Med. Imag. 26 (2007)Google Scholar
  13. 13.
    Wang, Z., Vemuri, B.: Dti segmentation using an information theoretic tensor dissimilarity measure. IEEE Trans. on Medical Imaging 24(10), 1267–1277 (2005)CrossRefGoogle Scholar
  14. 14.
    Ziyan, U., Tuch, D., Westin, C.-F.: Segmentation of Thalamic Nuclei from DTI Using Spectral Clustering. In: Larsen, R., Nielsen, M., Sporring, J. (eds.) MICCAI 2006. LNCS, vol. 4191, pp. 807–814. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Weldeselassie, Y., Hamarneh, G.: Dt-mri segmentation using graph cuts. In: SPIE Medical Imaging, vol. 6512 (2007)Google Scholar
  16. 16.
    Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn. Reson. Med. 56, 411–421 (2006)CrossRefGoogle Scholar
  17. 17.
    Wu, Y., Wang, J., Lu, H.: Real-time visual tracking via incremental covariance model update on log-euclidean riemannian manifold. In: Proc. IEEE Chinese Conference on Pattern Recognition, CCPR (2009)Google Scholar
  18. 18.
    Karcher, H.: Riemannian center of mass and mollifier smoothing. Comm. Pure Appl. Math. 30, 509–541 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Lurie, J.: Lecture notes on the theory of hadamard spaces (metric spaces of nonpositive curvature), http://www.math.harvard.edu/~lurie/papers/hadamard.pdf
  20. 20.
    Ballmann, W.: Manifolds of non positive curvature. In: Hirzebruch, F., Schwermer, J., Suter, S. (eds.) Arbeitstagung Bonn 1984. Lecture Notes in Mathematics, vol. 1111, pp. 261–268. Springer, Heidelberg (1985) 10.1007/BFb0084594CrossRefGoogle Scholar
  21. 21.
    Schwartzman, A.: Random ellipsoids and false discovery rates: Statistics for diffusion tensor imaging data. PhD thesis, Stanford University (2006)Google Scholar
  22. 22.
    Söderman, O., Jönsson, B.: Restricted diffusion in cylindrical geometry. Journal of Magnetic Resonance. Series A 117(1), 94–97 (1995)CrossRefGoogle Scholar
  23. 23.
    Barmpoutis, A., Vemuri, B.C.: A unified framework for estimating diffusion tensors of any order with symmetric positive-definite constraints. In: Proceedings of ISBI 2010: IEEE International Symposium on Biomedical Imaging, pp. 1385–1388 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Guang Cheng
    • 1
  • Hesamoddin Salehian
    • 1
  • Baba C. Vemuri
    • 1
  1. 1.Department of Computer and Information Science and EngineeringUniversity of FloridaGainesvilleUSA

Personalised recommendations