Abstract
In many engineering applications that use tensor analysis, such as tensor imaging, the underlying tensors have the characteristic of being positive definite. It might therefore be more appropriate to use techniques specially adapted to such tensors. We will describe the geometry and calculus on the Riemannian symmetric space of positive-definite tensors. First, we will explain why the geometry, constructed by Emile Cartan, is a natural geometry on that space. Then, we will use this framework to present formulas for means and interpolations specific to positive-definite tensors.
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References
Amari, S. (1985) Differential-Geometrical Methods in Statistics. Springer-Verlag, Berlin Heidelberg.
Basser, P. and Pierpaoli, C. (1996) Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI, J. Magn. Reson. B 111/3, pp. 209–219.
Batchelor, P. G., Moakher, M., Atkinson, D., Calamante, F., Connelly, A. (2004) A rigorous framework for diffusion tensor analysis using Riemannian geometry. Magn. Reson. Med., in press.
Helgason, S. (1978) Differential Geometry, Lie Groups, and Symmetric spaces. Academic Press, New York.
Kullback, S. L. (1959) Information Theory and Statistics. Wiley, New York.
Lang, S. (1999) Fundamentals of Differential Geometry. Springer-Verlag, New York.
Lenglet, C., Rousson, M., Deriche, R. (2004) Segmentation of 3D Probability Density Fields by Surface Evolution: Application to Diffusion MRI, in: MICCAI 2004, Part I, C. Barillot, D. R. Haynor, P. Hellier, eds. Lecture Notes in Computer Science, Vol. 3216, Springer-Verlag, Berlin, pp. 18–25.
Moakher, M. (2003) A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl., in press.
Moakher, M. (2004) On the averaging of symmetric positive-definite tensors, submitted to: J. Elasticity.
Papadakis, N. G., Xing, D., Houston, G. C., Smith, J.M., Smith, M. I., James, M. F., Parsons, A. A., Huang, C. L.-H., Hall, L. D., Carpenter, T. A. (1999) A study of rotationally invariant and symmetric indices of diffusion anisotropy, Magn. Reson. Imag., 17/6, pp. 881–892.
Pierpaoli, C. and Basser, P. J. (1996) Toward a quantitative assessment of diffusion anisotropy, Magn. Reson. Med., 36/6, pp. 893–906.
Smith, S. T. (1993) Geometric optimization methods for adaptive filtering. Ph.D. thesis, Harvard University, Cambridge, Massachussetts.
Terras, A. (1988) Harmonic Analysis on Symmetric Spaces and Applications II. Springer-Verlag, New York.
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Moakher, M., Batchelor, P.G. (2006). Symmetric Positive-Definite Matrices: From Geometry to Applications and Visualization. In: Weickert, J., Hagen, H. (eds) Visualization and Processing of Tensor Fields. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31272-2_17
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DOI: https://doi.org/10.1007/3-540-31272-2_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25032-6
Online ISBN: 978-3-540-31272-7
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