A General structure tensor concept and coherence-enhancing diffusion filtering for matrix fields

Mathematics and Visualization

Volumn 0, Issue 9783540883777, 2009, Pages 305-323

  Burgeth, Bernhard ^a   Didas, Stephan ^a   Weickert, Joachim ^a  

^a SAARLAND UNIVERSITY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

DIFFUSION; IMAGE ENHANCEMENT; MAGNETIC RESONANCE IMAGING; MATRIX ALGEBRA; TENSORS; THREE DIMENSIONAL COMPUTER GRAPHICS; TISSUE;

ALGEBRAIC PROPERTIES; COHERENCE-ENHANCING DIFFUSION; DIFFUSION TENSOR MAGNETIC RESONANCE IMAGING; GENERAL STRUCTURES; NUMERICAL EXPERIMENTS; PHYSICAL MEASUREMENT; STRUCTURE TENSORS; SYMMETRIC MATRICES;

DIFFUSION TENSOR IMAGING;

EID: 70349344554     PISSN: 16123786     EISSN: 2197666X     Source Type: Book Series    
DOI: 10.1007/978-3-540-88378-4_15     Document Type: Chapter

References (28)

1. V. Arsigny, P. Fillard, X. Pennec, and N. Ayache. Fast and simple calculus on tensors in the log-Euclidean framework. In J. Duncan and G. Gerig, editors, Medical Image Computing and Computer-Assisted Intervention - MICCAI 2005, Part I, volume 3749 of LNCS, Springer, Berlin, pp. 115–122, 2005.
2. V. Arsigny, P. Fillard, X. Pennec, and N. Ayache. Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magnetic Resonance in Medicine, 56(2):411–421, 2006.
3. J. Bigun. Vision with Direction. Springer, Berlin, 2006.
4. T. Brox and J. Weickert. A TV flow based local scale estimate and its application to texture discrimination. Journal of Visual Communication and Image Representation, 17(5):1053–1073, 2006.
5. T. Brox, J. Weickert, B. Burgeth, and P. Mrázek. Nonlinear structure tensors. Image and Vision Computing, 24(1):41–55, 2006.
6. B. Burgeth, A. Bruhn, S. Didas, J. Weickert, and M. Welk. Morphology for matrix-data: Ordering versus PDE-based approach. Image and Vision Computing, 25(4):496–511, 2007.
7. B. Burgeth, S. Didas, L. Florack, and J. Weickert. A generic approach for singular PDEs for the processing of matrix fields. In F. Sgallari, F. Murli, and N. Paragios, editors, Scale Space and Variational Methods in Computer Vision, volume 4485 of Lecture Notes in Computer Science, Springer, Berlin, pp. 556–567, 2007.
8. B. Burgeth, S. Didas, L. Florack, and J. Weickert. A generic approach to diffusion filtering of matrix-fields. Computing, 81:179–197, 2007.
9. B. Burgeth, N. Papenberg, A. Bruhn, M. Welk, C. Feddern, and J. Weickert. Morphology for higher-dimensional tensor data via Loewner ordering. In C. Ronse, L. Najman, and E. Decencière, editors, Mathematical Morphology: 40 Years On, volume 30 of Computational Imaging and Vision, Springer, Dordrecht, pp. 407–418, 2005.
10. C. Chefd’Hotel, D. Tschumperlé, R. Deriche, and O. Faugeras. Constrained flows of matrix-valued functions: Application to diffusion tensor regularization. In A. Heyden, G. Sparr, M. Nielsen, and P. Johansen, editors, Computer Vision – ECCV 2002, volume 2350 of Lecture Notes in Computer Science, Springer, Berlin, pp. 251–265, 2002.
11. O. Coulon, D. C. Alexander, and S. A. Arridge. A regularization scheme for diffusion tensor magnetic resonance images. In M. F. Insana and R. M. Leahy, editors, Information Processing in Medical Imaging – IPMI 2001, volume 2082 of Lecture Notes in Computer Science, Springer, Berlin, pp. 92–105, 2001.
12. S. Di Zenzo. A note on the gradient of a multi-image. Computer Vision, Graphics and Image Processing, 33:116–125, 1986.
13. C. Feddern, J. Weickert, B. Burgeth, and M. Welk. Curvature-driven PDE methods for matrix-valued images. International Journal of Computer Vision, 69(1):91–103, 2006.
14. W. Förstner and E. Gülch. A fast operator for detection and precise location of distinct points, corners and centres of circular features. In Proc. ISPRS Intercommission Conference on Fast Processing of Photogrammetric Data, Interlaken, Switzerland, pp. 281–305, 1987.
15. Y. Gur and N. Sochen. Diffusion over tensor fields via lie group pde flows. In M. Farber, R. Ghrist, M. Burger, and D. Koditschek, editors, Topology and Robotics, Contemporary Mathematics, Springer, Berlin, pp. 59–74, 2007.
16. R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, UK, 1990.
17. H. Knutsson. Representing local structure using tensors. In The 6th Scandinavian Conference on Image Analysis, pp. 244–251, Oulu, Finland, June 1989. Report LiTH–ISY–I–1019, Computer Vision Laboratory, Linköping University, Sweden, 1989.
18. C. Lenglet, M. Rousson, R. Deriche, and O. Faugeras. Statistics on the manifold of multivariate normal distributions: theory and application to diffusion tensor mri processing. Journal of Mathematical Imaging and Vision, 25(3):423–444, 2006.
19. T. McGraw, B. C. Vemuri, Y. Chen, M. Rao, and T. H. Mareci. DT-MRI denoising and neuronal fiber tracking. Medical Image Analysis, 8:95–111, 2004.
20. X. Pennec, P. Fillard, and N. Ayache. A riemannian framework for tensor computing. International Journal of Computer Vision, 66(1):41–66, 2006.
21. T. Schultz, B. Burgeth, and J. Weickert. Flexible segmentation and smoothing of DT-MRI fields through a customizable structure tensor. In G. Bebis, R. Boyle, B. Parvin, D. Koracin, P. Remagnino, A. Nefian, G. Meenakshisundaram, V. Pascucci, J. Zara, J. Molineros, H. Theisel, and T. Malzbender, editors, Advances in Visual Computing, volume 4291 of Lecture Notes in Computer Science, Springer, Berlin, pp. 454–464, 2006.
22. D. Tschumperlé and R. Deriche. Diffusion tensor regularization with constraints preservation. In Proc. 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, volume 1, Kauai, HI, pp. 948–953, December 2001. IEEE Computer Society Press.
23. D. Tschumperlé and R. Deriche. Orthonormal vector sets regularization with PDE’s and applications. International Journal of Computer Vision, 50(3):237–252, 2002.
24. J. Weickert. Anisotropic Diffusion in Image Processing. Teubner, Stuttgart, 1998.
25. J. Weickert. Coherence-enhancing diffusion filtering. International Journal of Computer Vision, 31(2/3):111–127, 1999.
26. J. Weickert. Coherence-enhancing diffusion of colour images. Image and Vision Computing, 17(3/4):199–210, 1999.
27. J. Weickert and T. Brox. Diffusion and regularization of vector- and matrix-valued images. In M. Z. Nashed and O. Scherzer, editors, Inverse Problems, Image Analysis, and Medical Imaging, volume 313 of Contemporary Mathematics, AMS, Providence, pp. 251–268, 2002.
28. J. Weickert and H. Hagen, editors. Visualization and Processing of Tensor Fields. Springer, Berlin, 2006.