Abstract
In previous work we studied left-invariant diffusion on the 2D Euclidean motion group for crossing-preserving coherence-enhancing diffusion on 2D images. In this paper we study the equivalent three-dimensional case. This is particularly useful for processing High Angular Resolution Diffusion Imaging (HARDI) data, which can be considered as 3D orientation scores directly. A complicating factor in 3D is that all practical 3D orientation scores are functions on a coset space of the 3D Euclidean motion group instead of on the entire group. We show that, conceptually, we can still apply operations on the entire group by requiring the operations to be α-right-invariant. Subsequently, we propose to describe the local structure of the 3D orientation score using left-invariant derivatives and we smooth 3D orientation scores using left-invariant diffusion. Finally, we show a number of results for linear diffusion on artificial HARDI data.
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References
Weickert, J.A.: Coherence-enhancing diffusion filtering. International Journal of Computer Vision 31(2/3), 111–127 (1999)
Franken, E., Duits, R., ter Haar Romeny, B.M.: Nonlinear diffusion on the 2D Euclidean motion group. In: Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol. 4485, pp. 461–472. Springer, Heidelberg (2007)
Franken, E., Duits, R.: Crossing-preserving coherence-enhancing diffusion on invertible orientation scores. International Journal of Computer Vision (IJCV) (to appear, 2009)
Özarslan, E., Mareci, T.H.: Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution imaging. Magnetic Resonance in Medicine 50, 955–965 (2003)
Franken, E.: Enhancement of Crossing Elongated Structures in Images. PhD thesis, Eindhoven University of Technology, Department of Biomedical Engineering, Eindhoven, The Netherlands (2008)
Özarslan, E., Shepherd, T.M., Vemuri, B.C., Blackband, S.J., Mareci, T.H.: Resolution of complex tissue microarchitecture using the diffusion orientation transform (DOT). NeuroImage 31, 1086–1103 (2006)
Descoteaux, M., Angelino, E., Fitzgibbons, S., Deriche, R.: Regularized, fast, and robust analytical Q-ball imaging. Magnetic Resonance in Medicine 58(3), 497–510 (2007)
Jian, B., Vemuri, B.C., Özarslan, E., Carney, P.R., Mareci, T.H.: A novel tensor distribution model for the diffusion-weighted MR signal. NeuroImage 37, 164–176 (2007)
Florack, L.: Codomain scale space and regularization for high angular resolution diffusion imaging. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, 2008. CVPR Workshops 2008, June 2008, pp. 1–6 (2008)
Varadarajan, V.: Lie groups, Lie algebras, and their representations. Prentice-Hall, Englewood Cliffs (1974)
Kin, G., Sato, M.: Scale space filtering on spherical pattern. In: Proc. 11th international conference on Pattern Recognition, vol. C, pp. 638–641 (1992)
Macovski, A.: Noise in MRI. Magnetic Resonance in Medicine 36(3), 494–497 (1996)
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© 2009 Springer-Verlag Berlin Heidelberg
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Franken, E., Duits, R. (2009). Scale Spaces on the 3D Euclidean Motion Group for Enhancement of HARDI Data. In: Tai, XC., Mørken, K., Lysaker, M., Lie, KA. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2009. Lecture Notes in Computer Science, vol 5567. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02256-2_68
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DOI: https://doi.org/10.1007/978-3-642-02256-2_68
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