Abstract
Motivated by the problem of analyzing shapes of fiber tracts in DT-MRI data, we present a geometric framework for studying shapes of open curves in ℝ3. We start with a space of unit-length curves and define the shape space to be its quotient space modulo rotation and re-parametrization groups. Thus, the resulting shape analysis is invariant to parameterizations of curves. Furthermore, a Riemannian structure on this quotient shape space allows us to compute geodesic paths between given curves and helps develop algorithms for: (i) computing statistical summaries of a collection of curves using means and covariances, and (ii) clustering a given set of curves into clusters of similar shapes. Examples using fiber tracts, extracted as parameterized curves from DT-MRI images, are presented to demonstrate this framework.
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References
Sebastian, T.B., Klein, P.N., Kimia, B.B.: On aligning curves. IEEE Transactions on Pattern Analysis and Machine Intelligence 25(1), 116–125 (2003)
Klassen, E., Srivastava, A., Mio, W., Joshi, S.: Analysis of planar shapes using geodesic paths on shape spaces. IEEE Patt. Analysis and Machine Intell. 26(3), 372–383 (2004)
Mio, W., Srivastava, A., Joshi, S.H.: On Shape of Plane Elastic Curves. International Journal of Computer Vision (2007)
Michor, P.W., Mumford, D.: Riemannian geometries on spaces of plane curves. Journal of the European Mathematical Society 8, 1–48 (2006)
Younes, L.: Computable elastic distance between shapes. SIAM Journal of Applied Mathematics 58, 565–586 (1998)
Miller, M.I., Younes, L.: Group Actions, Homeomorphisms, and Matching: A General Framework. International Journal of Computer Vision 8, 1–48 (2002)
Batchelor, P.G., Calamante, F., Tournier, J.-D., Atkinson, D., Hill, D.L.G., Connelly, A.: Quantification of the shape of fiber tracts. Magnetic Resonance in Medicine 55, 894–903 (2006)
Sherbondy, A.: Shape analysis of fiber tractography in the human brain (2006), Website http://www.stanford.edu/sherbond/pres.pdf
Davies, R.H., Twining, C.J., Allen, P.D., Cootes, T.F., Taylor, C.J.: Building optimal 2D statistical shape models. Image and Vision Computing 21, 82–1171 (2003)
Srivastava, A., Joshi, S., Mio, W., Liu, X.: Statistical Shape Analysis: Clustering, Learning and Testing. IEEE Transactions on Pattern Analysis and Machine Intelligence 27(4), 590–602 (2005)
Bakircioglu, M., Grenander, U., Khaneja, N., Miller, M.: Curve matching on brain surfaces using induced fr’enet distances matrices. Special issue of Human brain mapping (2000)
Corouge, I., Gouttard, S., Gerig, G.: Towards a Shape Model Of White Matter Fiber Bundles Using Diffusion Tensor MRI. In: International Symposium on Biomedical Imaging (2004)
Fillard, P., Gilmore, J., Piven, J., Lin, W., Gerig, G.: Quantitative Analysis of White Matter Fiber Properties along Geodesic Paths. Medical Image Computing and Computer-Assisted Intervention (2003)
Srivastava, A., Jermyn, I., Joshi, S.: Riemannian Analysis of Probability Density Functions with Applications in Vision. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2007)
Le, H.L., Kendall, D.G.: The Riemannian Structure of Euclidean shape spaces: a novel environment for Statistics. Annals of Statistics (1993)
O’Donnell, L., Kubicki, M., Shenton, M.E., Dreusicke, M.E., Grimson, W.E.L., Westin, C.-F.: A Method for Clustering White Matter Fiber Tracts. American Journal of Neuroradiology (AJNR) (2006)
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Balov, N., Srivastava, A., Li, C., Ding, Z. (2007). Shape Analysis of Open Curves in ℝ3 with Applications to Study of Fiber Tracts in DT-MRI Data. In: Yuille, A.L., Zhu, SC., Cremers, D., Wang, Y. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2007. Lecture Notes in Computer Science, vol 4679. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74198-5_31
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DOI: https://doi.org/10.1007/978-3-540-74198-5_31
Publisher Name: Springer, Berlin, Heidelberg
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