Abstract
We study a well-known scalar quantity in differential geometry, the Ricci scalar, in the context of Diffusion Tensor Imaging (DTI). We explore the relation between the Ricci scalar and the two most popular scalar measures in DTI: Mean Diffusivity and Fractional Anisotropy. We discuss results of computing the Ricci scalar on synthetic as well as real DTI data.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
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.
References
Basser, P., Mattiello, J., Turner, R., Le Bihan, D.: Diffusion tensor echo-planar imaging of human brain. In: Proceedings of the SMRM, p. 584 (1993)
Basser, P.J., Mattiello, J., Lebihan, D.: Estimation of the effective self-diffusion tensor from the NMR spin echo. Journal of Magnetic Resonance Series B 103, 247–254 (1994)
Basser, P.J., Mattiello, J., Lebihan, D.: MR diffusion tensor spectroscopy and imaging. Biophysical Journal 66, 259–267 (1994)
Basser, P., Pierpaoli, C.: Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. Journal of Magnetic Resonance. Series B 111(3), 209–219 (1996)
Özarslan, E., Vemuri, B.C., Mareci, T.H.: Generalized scalar measures for diffusion MRI using trace, variance, and entropy. Magn. Reson. Med. 53(4), 866–876 (2005)
Saucan, E., Appleboim, E., Wolanski, G., Zeevi, Y.: Combinatorial ricci curvature for image processing. Presented at MICCAI 2008 Workshop Manifolds in Medical Imaging: Metrics, Learning and Beyond (2008)
Nakahara, M.: Geometry, Topology and Physics, 2nd edn. Graduate Student Series in Physics. Taylor & Francis, Abington (2003)
de Lara, M.C.: Geometric and symmetry properties of a nondegenerate diffusion process. The Annals of Probability 23(4), 1557–1604 (1995)
O’Donnell, L., Haker, S., Westin, C.-F.: New approaches to estimation of white matter connectivity in diffusion tensor MRI: Elliptic pDEs and geodesics in a tensor-warped space. In: Dohi, T., Kikinis, R. (eds.) MICCAI 2002. LNCS, vol. 2488, pp. 459–466. Springer, Heidelberg (2002)
Lenglet, C., Deriche, R., Faugeras, O.: Inferring white matter geometry from diffusion tensor MRI: Application to connectivity mapping. In: Pajdla, T., Matas, J.G. (eds.) ECCV 2004. LNCS, vol. 3024, pp. 127–140. Springer, Heidelberg (2004)
Prados, E., Lenglet, C., Pons, J.P., Wotawa, N., Deriche, R., Faugeras, O., Soatto, S.: Control theory and fast marching techniques for brain connectivity mapping. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 1076–1083. IEEE, New York (2006)
Astola, L., Florack, L.M.J., ter Haar Romeny, B.M.: Measures for pathway analysis in brain white matter using diffusion tensor images. In: Karssemeijer, N., Lelieveldt, B. (eds.) IPMI 2007. LNCS, vol. 4584, pp. 642–649. Springer, Heidelberg (2007)
Carroll, S.: Spacetime and Geometry: An Introduction to General Relativity. Addison-Wesley, Reading (2003)
Florack, L., Astola, L.: A multi-resolution framework for diffusion tensor images. In: CVPR Workshop on Tensors in Image Processing and Computer Vision, Anchorage, Alaska, The United States. CVPR, vol. 20, Springer, Heidelberg (2008)
ter Haar Romeny, B.M.: Front-End Vision and Multi-Scale Image Analysis: Multi-Scale Computer Vision Theory and Applications, written in Mathematica, vol. 27. Kluwer Academic Publisher, Dordrecht (2003)
Florack, L.M.J.: Image Structure, vol. 10. Kluwer Academic Publisher, Dordrecht (1997)
Faas, F., Van Vliet, L.: A crossing detector based on the structure tensor. In: Blanc-Talon, J., Philips, W., Popescu, D., Scheunders, P. (eds.) ACIVS 2007. LNCS, vol. 4678, pp. 212–220. Springer, Heidelberg (2007)
Brunenberg, E., Prckovska, V., Platel, B., Strijkers, G., ter Haar Romeny, B.M.: Untangling a fiber bundle knot: Preliminary results on STN connectivity using DTI and HARDI on rat brains. In: Proceedings of the 17th Meeting of the International Society for Magnetic Resonance in Medicine (ISMRM), Honolulu, Hawaii (2009)
Tuch, D., Reese, T., Wiegell, M., Makris, N., Belliveau, J., Wedeen, V.: High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity. Magnetic Resonance in Medicine 48(4), 577–582 (2002)
Alexander, D., Barker, G., Arridge, S.: Detection and modeling of non-gaussian apparent diffusion coefficient profiles in human brain data. Magnetic Resonance in Medicine 48, 331–340 (2002)
Deriche, R., Descoteaux, M.: Splitting tracking through crossing fibers: Multidirectional Q-ball tracking. In: Proceedings of the 4th International Symposium on Biomedical Imaging: From Nano to Macro (ISBI 2007), Arlington, Virginia, USA (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fuster, A., Astola, L., Florack, L. (2009). A Riemannian Scalar Measure for Diffusion Tensor Images. In: Jiang, X., Petkov, N. (eds) Computer Analysis of Images and Patterns. CAIP 2009. Lecture Notes in Computer Science, vol 5702. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03767-2_51
Download citation
DOI: https://doi.org/10.1007/978-3-642-03767-2_51
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03766-5
Online ISBN: 978-3-642-03767-2
eBook Packages: Computer ScienceComputer Science (R0)