International Journal of Biomedical Imaging

International Journal of Biomedical Imaging / 2007 / Article
Special Issue

Mathematics in Biomedical Imaging

View this Special Issue

Research Article | Open Access

Volume 2007 |Article ID 027432 | https://doi.org/10.1155/2007/27432

Oddvar Christiansen, Tin-Man Lee, Johan Lie, Usha Sinha, Tony F. Chan, "Total Variation Regularization of Matrix-Valued Images", International Journal of Biomedical Imaging, vol. 2007, Article ID 027432, 11 pages, 2007. https://doi.org/10.1155/2007/27432

Total Variation Regularization of Matrix-Valued Images

Academic Editor: Hongkai Zhao
Received30 Oct 2006
Accepted13 Mar 2007
Published06 Jun 2007

Abstract

We generalize the total variation restoration model, introduced by Rudin, Osher, and Fatemi in 1992, to matrix-valued data, in particular, to diffusion tensor images (DTIs). Our model is a natural extension of the color total variation model proposed by Blomgren and Chan in 1998. We treat the diffusion matrix D implicitly as the product D=LLT, and work with the elements of L as variables, instead of working directly on the elements of D. This ensures positive definiteness of the tensor during the regularization flow, which is essential when regularizing DTI. We perform numerical experiments on both synthetical data and 3D human brain DTI, and measure the quantitative behavior of the proposed model.

References

  1. P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629–639, 1990. View at: Publisher Site | Google Scholar
  2. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D, vol. 60, no. 1–4, pp. 259–268, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. M. Lysaker, S. Osher, and X.-C. Tai, “Noise removal using smoothed normals and surface fitting,” IEEE Transactions on Image Processing, vol. 13, no. 10, pp. 1345–1357, 2004. View at: Publisher Site | Google Scholar | MathSciNet
  4. T. F. Chan and S. Esedoglu, “Aspects of total variation regularized L1 function approximation,” SIAM Journal on Applied Mathematics, vol. 65, no. 5, pp. 1817–1837, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. T. F. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, Philadelphia, Pa, USA, 2005. View at: Google Scholar
  6. J. Weickert, “A review of nonlinear diffusion filtering,” in Proceedings of the 1st International Conference on Scale-Space Theory in Computer Vision, vol. 1252 of Lecture Notes in Computer Science, pp. 3–28, Utrecht, The Netherlands, July 1997. View at: Google Scholar
  7. J. Weickert and T. Brox, “Diffusion and regularization of vector- and matrix-valued images,” Tech. Rep. preprint no. 58, Fachrichtung 6.1 Mathematik, Universitat des Saarlandes, Saarbrücken, Germany, 2002. View at: Google Scholar
  8. P. J. Basser, J. Mattiello, and D. LeBihan, “MR diffusion tensor spectroscopy and imaging,” Biophysical Journal, vol. 66, no. 1, pp. 259–267, 1994. View at: Google Scholar
  9. D. Le Bihan, J.-F. Mangin, and C. Poupon et al., “Diffusion tensor imaging: concepts and applications,” Journal of Magnetic Resonance Imaging, vol. 13, no. 4, pp. 534–546, 2001. View at: Publisher Site | Google Scholar
  10. C.-F. Westin, S. E. Maier, H. Mamata, A. Nabavi, F. A. Jolesz, and R. Kikinis, “Processing and visualization for diffusion tensor MRI,” Medical Image Analysis, vol. 6, no. 2, pp. 93–108, 2002. View at: Publisher Site | Google Scholar
  11. S. Mori and P. B. Barker, “Diffusion magnetic resonance imaging: its principle and applications,” The Anatomical Record, vol. 257, no. 3, pp. 102–109, 1999. View at: Publisher Site | Google Scholar
  12. S. Mori and P. C. M. van Zijl, “Fiber tracking: principles and strategies—a technical review,” NMR in Biomedicine, vol. 15, no. 7-8, pp. 468–480, 2002. View at: Publisher Site | Google Scholar
  13. R. Bammer, “Basic principles of diffusion-weighted imaging,” European Journal of Radiology, vol. 45, no. 3, pp. 169–184, 2003. View at: Publisher Site | Google Scholar
  14. E. O. Stejskal and J. E. Tanner, “Spin diffusion measurements: spin echoes in the presence of a time-dependent field gradient,” The Journal of Chemical Physics, vol. 42, no. 1, pp. 288–292, 1965. View at: Publisher Site | Google Scholar
  15. E. O. Stejskal, “Use of spin echoes in a pulsed magnetic-field gradient to study anisotropic, restricted diffusion and flow,” The Journal of Chemical Physics, vol. 43, no. 10, pp. 3597–3603, 1965. View at: Publisher Site | Google Scholar
  16. P. Tofts, Ed., Quantitative MRI of the Brain, John Wiley & Sons, New York, NY, USA, 2005. View at: Google Scholar
  17. C.-F. Westin, S. E. Maier, H. Mamata, A. Nabavi, F. A. Jolesz, and R. Kikinis, “Processing and visualization for diffusion tensor MRI,” Medical Image Analysis, vol. 6, no. 2, pp. 93–108, 2002. View at: Publisher Site | Google Scholar
  18. Z. Wang, B. C. Vemuri, Y. Chen, and T. H. Mareci, “A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI,” IEEE Transactions on Medical Imaging, vol. 23, no. 8, pp. 930–939, 2004. View at: Publisher Site | Google Scholar
  19. D. Tschumperlé and R. Deriche, “Variational frameworks for DT-MRI estimation, regularization and visualization,” in Proceedings of the 9th IEEE International Conference on Computer Vision (ICCV '03), vol. 1, pp. 116–121, Nice, France, October 2003. View at: Publisher Site | Google Scholar
  20. C.-F. Westin and H. Knutsson, “Tensor field regularization using normalized convolution,” in Proceedings of the 9th International Conference on Computer Aided Systems Theory (EUROCAST '03), R. Moreno-Diaz and F. Pichler, Eds., vol. 2809 of Lecture Notes in Computer Science, pp. 564–572, Las Palmas de Gran Canaria, Canary Islands, Spain, February 2003. View at: Google Scholar
  21. B. Chen and E. W. Hsu, “Noise removal in magnetic resonance diffusion tensor imaging,” Magnetic Resonance in Medicine, vol. 54, no. 2, pp. 393–401, 2005. View at: Publisher Site | Google Scholar
  22. C. Chefd'hotel, D. Tschumperlé, R. Deriche, and O. Faugeras, “Regularizing flows for constrained matrix-valued images,” Journal of Mathematical Imaging and Vision, vol. 20, no. 1-2, pp. 147–162, 2004. View at: Google Scholar
  23. Y. Gur and N. Sochen, “Denoising tensors via Lie group flows,” in Proceedings of the 3rd International Workshop on Variational, Geometric, and Level Set Methods in Computer Vision (VLSM '05), vol. 3752 of Lecture Notes in Computer Science, pp. 13–24, Springer, Beijing, China, October 2005. View at: Google Scholar
  24. D. Groisser, “Some differential-geometric remarks on a method for minimizing constrained functionals of matrix-valued functions,” Journal of Mathematical Imaging and Vision, vol. 24, no. 3, pp. 349–358, 2006. View at: Publisher Site | Google Scholar | MathSciNet
  25. P. Blomgren and T. F. Chan, “Color TV: total variation methods for restoration of vector-valued images,” IEEE Transactions on Image Processing, vol. 7, no. 3, pp. 304–309, 1998. View at: Publisher Site | Google Scholar
  26. T. F. Chan, G. H. Golub, and P. Mulet, “A nonlinear primal-dual method for total variation-based image restoration,” SIAM Journal of Scientific Computing, vol. 20, no. 6, pp. 1964–1977, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  27. G. Sapiro, “Color snakes,” Tech. Rep. HPL-95-113, Hewlett Packard Computer Peripherals Laboratory, Palo Alto, Calif, USA, 1995. View at: Google Scholar
  28. The Mathworks, “MatLab, The Language of Technical Computing,” http://www.mathworks.com/matlab/. View at: Google Scholar
  29. S. Pajevic and C. Pierpaoli, “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, vol. 42, no. 3, pp. 526–540, 1999. View at: Publisher Site | Google Scholar
  30. Mori and coworkers, “DTI-studio”, http://cmrm.med.jhmi.edu/.

Copyright © 2007 Oddvar Christiansen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Order printed copiesOrder
Views364
Downloads3476
Citations

Related articles

Article of the Year Award: Outstanding research contributions of 2020, as selected by our Chief Editors. Read the winning articles.