Abstract
We propose a novel framework for regularization of symmetric positive-definite (SPD) tensors (e.g., diffusion tensors). This framework is based on a local differential geometric approach. The manifold of symmetric positive-definite (SPD) matrices, P n , is parameterized via the Iwasawa coordinate system. In this framework distances on P n are measured in terms of a natural GL(n)-invariant metric. Via the mathematical concept of fiber bundles, we describe the tensor-valued image as a section where the metric over the section is induced by the metric over P n . Then, a functional over the sections accompanied by a suitable data fitting term is defined. The variation of this functional with respect to the Iwasawa coordinates leads to a set of \(\frac{1}{2}n(n+1)\) coupled equations of motion. By means of the gradient descent method, these equations of motion define a Beltrami flow over P n . It turns out that the local coordinate approach via the Iwasawa coordinate system results in very simple numerics that leads to fast convergence of the algorithm. Regularization results as well as results of fibers tractography for DTI are presented.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Basser, P. J., & Pierpaoli, C. (1996). Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. Journal of Magnetic Resonance, 111, 209–219.
Basser, P. J., Mattielo, J., & LeBihan, D. (1994). MR diffusion tensor spectroscopy and imaging. Biophysical Journal, 66, 259–267.
Batchelor, P. G., Moakher, M., Atkinson, D., Calamante, F., & Connelly, A. (2005). A rigorous framework for diffusion tensor calculus. Magnetic Resonance in Medicine, 53, 221–225.
Blomgren, P., & Chan, T. F. (1998). Total variation methods for restoration of vector valued images. IEEE Transactions on Image processing, 7, 304–309.
Chefd’hotel, C., Tschumperlé, D., Deriche, R., & Faugeras, O. (2004). Regularizing flows for constrained matrix-valued images. Journal of Mathematical Imaging and Vision, 20, 147–162.
Coulon, O., Alexander, D. C., & Arridge, S. R. (2001). A regularization scheme for diffusion tensor magnetic resonance images. In IPMI ’01: Proceedings of the 17th international conference on information processing in medical imaging (pp. 92–105). London, UK. Berlin: Springer.
Feddern, C., Weickert, J., Burgeth, B., & Welk, M. (2006). Curvature-driven PDE methods for matrix-valued images. International Journal of Computer Vision, 69, 91–103.
Fillard, P., Arsigny, V., Pennec, X., & Ayache, N. (2007). Clinical DT-MRI estimation, smoothing and fiber tracking with Log-Euclidean metrics. IEEE Transactions on Medical Imaging, 26(11), 1472–1482.
Fletcher, P. T., & Joshi, S. (2007). Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Processing, 87, 250–262.
Gur, Y., & Sochen, N. (2007). Coordinate-free diffusion over compact Lie groups. In Lecture notes in computer science: Vol. 4485. Proceedings of the 1st international conference on scale-space and variational methods (pp. 580–591). Ischia Island, Italy, May 2007. Berlin: Springer.
Jorgenson, J., & Lang, S. (2005). Pos n (R) and Eisenstein series. Lecture notes in mathematics. Berlin: Springer.
Jost, J. (2001). Riemannian geometry and geometric analysis. New York: Springer.
Kimmel, R., Sochen, N., & Malladi, R. (1997). From high energy physics to low level vision. In SCALE-SPACE ’97: Proceedings of the first international conference on scale-space theory in computer vision (pp. 236–247). London, UK. Berlin: Springer.
Lang, S. (1999). Fundementals of differential geometry. New York: Springer.
Moakher, M. (2005). A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM Journal on Matrix Analysis and Applications, 26(3), 735–747.
Moakher, M., & Zerai, M. (2007). Riemannian curvature-driven flows for tensor-valued data. In Lecture notes in computer science: Vol. 4485. In proceedings of the 1st international conference on scale-space and variational methods (pp. 592–602). Ischia Island, Italy. Berlin: Springer.
Mori, S., Wakana, S., van Zijl, P. C. M., & Nagae-Poetscher, L. M. (2005). MRI atlas of human white matter. Amsterdam: Elsevier.
Pajevic, S., & Pierpaoli, C. (1999). 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, 526–540.
Pasternak, O., Sochen, N., Intrator, N., & Assaf, Y. (2008). Variational multiple-tensors fitting of fiber-ambiguous DW-MRI voxels. MRI, 26(8), 1133–1144.
Pennec, X., Fillard, P., & Ayache, N. (2006). A Riemmannian framework for tensor computing. International Journal of Computer Vision, 66, 41–66.
Pierpaoli, C., Jezzard, P., Basser, P. J., Barnett, A., & Chiro, G. D. (1996). Diffusion tensor MRI of the human brain. Radiology, 201, 637–648.
Polyakov, A. M. (1981). Quantum geometry of bosonic strings. Phys. Lett. B, 103, 207–210.
Rosenfeld, B. (1997). The geometry of Lie groups. Dordrecht: Kluwer Academic.
Shafrir, D., Sochen, N., & Deriche, R. (2005). Regularization of mappings between implicit manifolds of arbitrary dimension and codimension. In Proceedings of the 3rd IEEE workshop on variational, geometric and level-set methods (VLSM) in computer vision.
Sochen, N., Kimmel, R., & Malladi, R. (1998). A general framework for low level vision. IEEE Transactions in Image Processing, 7, 310–318. Special Issue on Geometry Driven Diffusion.
Terras, A. (1988). Harmonic analysis on symmetric spaces and applications (Vol. 2). New York: Springer.
Tschumperlé, D., & Deriche, R. (2002). Orthonormal vector sets regularization with PDE’s and applications. International Journal of Computer Vision, 50(3), 237–252.
Wang, Z., Vemuri, B. C., Chen, Y., & Mareci, T. H. (2004). A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI. IEEE Transactions on Medical Imaging, 23, 930–939.
Weickert, J. (1998). Anisotropic diffusion in image processing. Stuttgart: Teubner.
Weickert, J., & Hagen, H. (Eds.) (2005). Visualization and processing of tensor fields. Berlin: Springer.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gur, Y., Pasternak, O. & Sochen, N. Fast GL(n)-Invariant Framework for Tensors Regularization. Int J Comput Vis 85, 211–222 (2009). https://doi.org/10.1007/s11263-008-0196-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11263-008-0196-7