Fast GL(n)-Invariant framework for tensors regularization

International Journal of Computer Vision

Volumn 85, Issue 3, 2009, Pages 211-222

  Gur, Yaniv ^a   Pasternak, Ofer ^a   Sochen, Nir ^a  

^a TEL AVIV UNIVERSITY   (Israel)

Author keywords

Beltrami framework; Differential geometry; Diffusion tensor imaging; Pdes; Tensor valued image

Indexed keywords

BELTRAMI FLOW; BELTRAMI FRAMEWORK; CO-ORDINATE SYSTEM; COUPLED EQUATION; DATA FITTINGS; DIFFERENTIAL GEOMETRIC APPROACH; DIFFERENTIAL GEOMETRY; DIFFUSION TENSOR; DIFFUSION TENSOR IMAGING; FAST CONVERGENCE; FIBER BUNDLES; GRADIENT DESCENT METHOD; LOCAL COORDINATE; MATHEMATICAL CONCEPTS; PARAMETERIZED; PDES; TRACTOGRAPHY;

CONVERGENCE OF NUMERICAL METHODS; COORDINATE MEASURING MACHINES; DIFFUSION; EQUATIONS OF MOTION; MECHANICS; OPEN CHANNEL FLOW;

TENSORS;

EID: 70349680864     PISSN: 09205691     EISSN: 15731405     Source Type: Journal    
DOI: 10.1007/s11263-008-0196-7     Document Type: Conference Paper

References (30)

1. P. J. Basser C. Pierpaoli 1996 Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI Journal of Magnetic Resonance 111 209 219
2. P. J. Basser J. Mattielo D. LeBihan 1994 MR diffusion tensor spectroscopy and imaging Biophysical Journal 66 259 267 10.1016/S0006-3495(94)80775-1
3. P. G. Batchelor M. Moakher D. Atkinson F. Calamante A. Connelly 2005 A rigorous framework for diffusion tensor calculus Magnetic Resonance in Medicine 53 221 225 10.1002/mrm.20334 (Pubitemid 40110496)
4. P. Blomgren T. F. Chan 1998 Total variation methods for restoration of vector valued images IEEE Transactions on Image processing 7 304 309 10.1109/83.661180
5. C. Chefd'hotel D. Tschumperlé R. Deriche O. Faugeras 2004 Regularizing flows for constrained matrix-valued images Journal of Mathematical Imaging and Vision 20 147 162 10.1023/B:JMIV.0000011324.14508.fb 2049787
6. Coulon, O., Alexander, D. C., and 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.
7. C. Feddern J. Weickert B. Burgeth M. Welk 2006 Curvature-driven PDE methods for matrix-valued images International Journal of Computer Vision 69 91 103 10.1007/s11263-006-6854-8 (Pubitemid 43787403)
8. P. Fillard V. Arsigny X. Pennec N. Ayache 2007 Clinical DT-MRI estimation, smoothing and fiber tracking with Log-Euclidean metrics IEEE Transactions on Medical Imaging 26 11 1472 1482 10.1109/TMI.2007.899173 (Pubitemid 350025485)
9. P. T. Fletcher S. Joshi 2007 Riemannian geometry for the statistical analysis of diffusion tensor data Signal Processing 87 250 262 10.1016/j.sigpro.2005.12.018 (Pubitemid 44602252)
10. Gur, Y., and 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.
11. n (R) and Eisenstein series. Lecture notes in mathematics. Berlin: Springer.
12. Jost, J. (2001). Riemannian geometry and geometric analysis. New York: Springer.
13. Kimmel, R., Sochen, N., and 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.
14. Lang, S. (1999). Fundementals of differential geometry. New York: Springer.
15. M. Moakher 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 1079.47021 10.1137/S0895479803436937 2137480 (Pubitemid 41049842)
16. Moakher, M., and 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.
17. Mori, S., Wakana, S., van Zijl, P. C. M., and Nagae-Poetscher, L. M. (2005). MRI atlas of human white matter. Amsterdam: Elsevier.
18. S. Pajevic C. Pierpaoli 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 10.1002/(SICI)1522-2594(199909)42:3<526::AID-MRM15>3.0.CO;2-J (Pubitemid 29416214)
19. O. Pasternak N. Sochen N. Intrator Y. Assaf 2008 Variational multiple-tensors fitting of fiber-ambiguous DW-MRI voxels MRI 26 8 1133 1144 10.1016/j.mri.2008.01.006
20. X. Pennec P. Fillard N. Ayache 2006 A Riemmannian framework for tensor computing International Journal of Computer Vision 66 41 66 10.1007/s11263-005- 3222-z (Pubitemid 43004671)
21. C. Pierpaoli P. Jezzard P. J. Basser A. Barnett G. D. Chiro 1996 Diffusion tensor MRI of the human brain Radiology 201 637 648
22. A. M. Polyakov 1981 Quantum geometry of bosonic strings Phys. Lett. B 103 207 210 10.1016/0370-2693(81)90743-7 623209
23. Rosenfeld, B. (1997). The geometry of Lie groups. Dordrecht: Kluwer Academic.
24. Shafrir, D., Sochen, N., and 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.
25. N. Sochen R. Kimmel R. Malladi 1998 A general framework for low level vision IEEE Transactions in Image Processing 7 310 318 0973.94502 10.1109/83.661181 1669540 Special Issue on Geometry Driven Diffusion
26. Terras, A. (1988). Harmonic analysis on symmetric spaces and applications (Vol. 2). New York: Springer.
27. D. Tschumperlé R. Deriche 2002 Orthonormal vector sets regularization with PDE's and applications International Journal of Computer Vision 50 3 237 252 1012.68786 10.1023/A:1020870207168 (Pubitemid 35377414)
28. Z. Wang B. C. Vemuri Y. Chen T. H. Mareci 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 10.1109/TMI.2004.831218
29. Weickert, J. (1998). Anisotropic diffusion in image processing. Stuttgart: Teubner.
30. Weickert, J., and Hagen, H. (Eds.) (2005). Visualization and processing of tensor fields. Berlin: Springer.