Abstract
This paper introduces eigenvalue derivatives as a fundamental tool to discern the different types of edges present in matrix-valued images. It reviews basic results from perturbation theory, which allow one to compute such derivatives, and shows how they can be used to obtain novel edge detectors for matrix-valued images. It is demonstrated that previous methods for edge detection in matrix-valued images are simplified by considering them in terms of eigenvalue derivatives. Moreover, eigenvalue derivatives are used to analyze and refine the recently proposed Log-Euclidean edge detector. Application examples focus on data from diffusion tensor magnetic resonance imaging (DT-MRI).
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References
Basser, P.J., Mattiello, J., Bihan, D.L.: Estimation of the effective self-diffusion tensor from the NMR spin echo. Journal of Magnetic Resonance B(103), 247–254 (1994)
Pajevic, S., Aldroubi, A., Basser, P.J.: A continuous tensor field approximation of discrete DT-MRI data for extracting microstructural and architectural features of tissue. Journal of Magnetic Resonance 154, 85–100 (2002)
Kindlmann, G., Ennis, D., Whitaker, R., Westin, C.F.: Diffusion tensor analysis with invariant gradients and rotation tangents. IEEE Transactions on Medical Imaging 26(11), 1483–1499 (2007)
Kindlmann, G.: Visualization and Analysis of Diffusion Tensor Fields. PhD thesis, School of Computing, University of Utah (September 2004)
Schultz, T., Burgeth, B., Weickert, J.: Flexible segmentation and smoothing of DT-MRI fields through a customizable structure tensor. In: Bebis, G., Boyle, R., Parvin, B., Koracin, D., Remagnino, P., Nefian, A.V., Gopi, M., Pascucci, V., Zara, J., Molineros, J., Theisel, H., Malzbender, T. (eds.) ISVC 2006. LNCS, vol. 4291, pp. 455–464. Springer, Heidelberg (2006)
Kato, T.: Perturbation theory for linear operators. 2nd edn. Volume 132 of Die Grundlehren der mathematischen Wissenschaften. Springer (1976)
Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Log-euclidean metrics for fast and simple calculus on diffusion tensors. Magnetic Resonance in Medicine 56(2), 411–421 (2006)
Anderson, A.W.: Theoretical analysis of the effects of noise on diffusion tensor imaging. Magnetic Resonance in Medicine 46(6), 1174–1188 (2001)
O’Donnell, L., Grimson, W.E.L., Westin, C.F.: Interface detection in diffusion tensor MRI. In: Barillot, C., Haynor, D.R., Hellier, P. (eds.) MICCAI 2004. LNCS, vol. 3216, pp. 360–367. Springer, Heidelberg (2004)
Kindlmann, G., Tricoche, X., Westin, C.F.: Delineating white matter structure in diffusion tensor MRI with anisotropy creases. Medical Image Analysis 11(5), 492–502 (2007)
Basser, P.J., Pierpaoli, C.: Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. Journal of Magnetic Resonance B(111), 209–219 (1996)
Westin, C.F., Peled, S., Gudbjartsson, H., Kikinis, R., Jolesz, F.A.: Geometrical diffusion measures for MRI from tensor basis analysis. In: International Society for Magnetic Resonance in Medicine 1997, Vancouver, Canada, p. 1742 (1997)
Pajevic, S., Pierpaoli, C.: Color schemes to represent the orientation of anisotropic tissues from diffusion tensor data: application to white matter fiber tract mapping in the human brain. Magnetic Resonance in Medicine 42(3), 526–540 (1999)
Bahn, M.M.: Invariant and orthonormal scalar measures derived from magnetic resonance diffusion tensor imaging. Journal of Magnetic Resonance 141, 68–77 (1999)
Ennis, D.B., Kindlmann, G.: Orthogonal tensor invariants and the analysis of diffusion tensor magnetic resonance images. Magnetic Resonance in Medicine 55(1), 136–146 (2006)
Fillard, P., Pennec, X., Arsigny, V., Ayache, N.: Clinical DT-MRI estimation, smoothing and fiber tracking with log-euclidean metrics. IEEE Transactions on Medical Imaging 26(11), 1472–1482 (2007)
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Schultz, T., Seidel, HP. (2008). Using Eigenvalue Derivatives for Edge Detection in DT-MRI Data. In: Rigoll, G. (eds) Pattern Recognition. DAGM 2008. Lecture Notes in Computer Science, vol 5096. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69321-5_20
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DOI: https://doi.org/10.1007/978-3-540-69321-5_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69320-8
Online ISBN: 978-3-540-69321-5
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