Median and related local filters for tensor-valued images

Signal Processing

Volumn 87, Issue 2, 2007, Pages 291-308

  Welk, Martin ^a   Weickert, Joachim ^a   Becker, Florian ^b   Schnörr, Christoph ^b   Feddern, Christian ^a   Burgeth, Bernhard ^a  

^a SAARLAND UNIVERSITY   (Germany)

^b UNIVERSITY OF MANNHEIM   (Germany)

Author keywords

Adaptive structure tensor; Convex optimisation; Local image filter; Median filtering; Tensor image processing

Indexed keywords

COMPUTATIONAL GEOMETRY; DATA ACQUISITION; MATRIX ALGEBRA; OPTIMIZATION; SIGNAL FILTERING AND PREDICTION; TENSORS;

ADAPTIVE STRUCTURE TENSOR; CONVEX OPTIMISATION; LOCAL IMAGE FILTER; MEDIAN FILTERING; TENSOR IMAGE PROCESSING;

IMAGE PROCESSING;

EID: 33750983926     PISSN: 01651684     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.sigpro.2005.12.013     Document Type: Article

References (38)

1. Pierpaoli C., Jezzard P., Basser P.J., Barnett A., and Di Chiro G. Diffusion tensor MR imaging of the human brain. Radiology 201 3 (1996) 637-648
2. W. Förstner, E. Gülch, a fast operator for detection and precise location of distinct points, corners and centres of circular features, in: Proceedings of the ISPRS Intercommission Conference on Fast Processing of Photogrammetric Data, Interlaken, Switzerland, June 1987, pp. 281-304.
3. M. Welk, C. Feddern, B. Burgeth, J. Weickert, Median filtering of tensor-valued images, in: B. Michaelis, G. Krell (Eds.), Pattern Recognition, Lecture Notes in Computer Science, vol. 2781, Springer, Berlin, 2003, pp. 17-24.
4. M. Welk, C. Feddern, B. Burgeth, J. Weickert, Tensor median filtering and M-smoothing, in: J. Weickert, H. Hagen (Eds.), Visualization and Processing of Tensor Fields, Springer, Berlin, 2006, pp. 345-356.
5. M. Welk, F. Becker, C. Schnörr, J. Weickert, Matrix-valued filters as convex programs, in: R. Kimmel, N. Sochen, J. Weickert (Eds.), Scale-Space and PDE Methods in Computer Vision, Lecture Notes in Computer Science, vol. 3459, Springer, Berlin, 2005, pp. 204-216.
6. F. Becker, Matrix-valued filters as convex programs, Diploma Thesis, CVGPR group, University of Mannheim, 2004.
7. B. Burgeth, M. Welk, C. Feddern, J. Weickert, Morphological operations on matrix-valued images, in: T. Pajdla, J. Matas (Eds.), Computer Vision-ECCV 2004, Lecture Notes in Computer Science, vol. 3024, Springer, Berlin, 2004, pp. 155-167.
8. B. Burgeth, M. Welk, C. Feddern, J. Weickert, Mathematical morphology on tensor data using the Loewner ordering, in: J. Weickert, H. Hagen (Eds.), Visualization and Processing of Tensor Fields, Springer, Berlin, 2006, pp. 357-368.
9. C. Westin, S.E. Maier, B. Khidhir, P. Everett, F.A. Jolesz, R. Kikinis, Image processing for diffusion tensor magnetic resonance imaging, in: C. Taylor, A. Colchester (Eds.), Medical Image Computing and Computer-Assisted Intervention-MICCAI 1999, Lecture Notes in Computer Science, vol. 1679, Springer, Berlin, 1999, pp. 441-452.
10. K. Hahn, S. Pigarin, B. Pütz, Edge preserving regularization and tracking for diffusion tensor imaging, in: W.J. Niessen, M.A. Viergever (Eds.), Medical Image Computing and Computer-Assisted Intervention-MICCAI 2001, Lecture Notes in Computer Science, vol. 2208, Springer, Berlin, 2001, pp. 195-203.
11. D. Tschumperlé, R. Deriche, Diffusion tensor regularization with constraints preservation, in: Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 1, Kauai, HI, December 2001, IEEE Computer Society Press, Silver Spring, MD, pp. 948-953.
12. C. Poupon, J. Mangin, V. Frouin, J. Régis, F. Poupon, M. Pachot-Clouard, D. Le Bihan, I. Bloch, Regularization of MR diffusion tensor maps for tracking brain white matter bundles, in: W.M. Wells, A. Colchester, S. Delp (Eds.), Medical Image Computing and Computer-Assisted Intervention-MICCAI 1998. Lecture Notes in Computer Science, vol. 1496, Springer, Berlin, 1998, pp. 489-498.
13. Parker G.J.M., Schnabel J.A., Symms M.R., Werring D.J., and Barker G.J. Nonlinear smoothing for reduction of systematic and random errors in diffusion tensor imaging. J. Magnetic Resonance Imaging 11 (2000) 702-710
14. Tukey J.W. Exploratory Data Analysis (1971), Addison-Wesley, Menlo Park
15. E.R. Dougherty, J. Astola (Eds.), Nonlinear Filters for Image Processing. SPIE Press, Bellingham, 1999.
16. Klette R., and Zamperoni R. Handbook of Image Processing Operators (1996), Wiley, New York
17. Barnett V. The ordering of multivariate data. J. Roy. Statist. Soc. A 139 3 (1976) 318-355
18. A. Koschan, M. Abidi, A comparison of median filter techniques for noise removal in color images, in: Proceedings of the Seventh German Workshop on Color Image Processing, Erlangen, Germany, 2001, pp. 69-79.
19. Astola J., Haavisto P., and Neuvo Y. Vector median filters. Proc. IEEE 78 4 (1990) 678-689
20. Caselles V., Sapiro G., and Chung D.H. Vector median filters, inf-sup operations, and coupled PDE's: theoretical connections. J. Math. Imaging and Vision 8 (2000) 109-119
21. Barni M., Buit F., Bartolini F., and Cappellini V. A quasi-Euclidean norm to speed up vector median filtering. IEEE Trans. Image Process. 9 10 (2000) 1704-1709
22. Austin T.L. An approximation to the point of minimum aggregate distance. Metron 19 (1959) 10-21
23. Seymour D.R. Note on Austin's "An approximation to the point of minimum aggregate distance". Metron 28 (1970) 412-421
24. Megiddo N. The weighted Euclidean 1-center problem. Math. Oper. Res. 8 4 (1983) 498-504
25. Fekete S.P., Mitchell J.S.B., and Beurer K. On the continuous Fermat-Weber problem. Oper. Res. 53 1 (2005) 61-76
26. Boyd S., and Vandenberghe L. Convex Optimization (2004), Cambridge University Press, Cambridge
27. Keuchel J., Schnörr C., Schellewald C., and Cremers D. Binary partitioning, perceptual grouping, and restoration with semidefinite programming. IEEE Trans. Pattern Anal. Mach. Intelligence 25 11 (2003) 1364-1379
28. J. Weickert, T. Brox, Diffusion and regularization of vector- and matrix-valued images, in: M.Z. Nashed, O. Scherzer (Eds.), Inverse Problems, Image Analysis, and Medical Imaging, Contemporary Mathematics, vol. 313, American Mathematical Society, Providence, RI, 2002, pp. 251-268.
29. T. Brox, M. Rousson, R. Deriche, J. Weickert, Unsupervised segmentation incorporating colour, texture, and motion, in: N. Petkov, M.A. Westenberg (Eds.), Computer Analysis of Images and Patterns, Lecture Notes in Computer Science, vol. 2756, Berlin, Springer, 2003, pp. 353-360.
30. M. Rousson, T. Brox, R. Deriche, Active unsupervised texture segmentation on a diffusion based feature space, Technical Report No. 4695, Odyssée, INRIA Sophia-Antipolis, France, 2003.
31. T. Brox, J. Weickert, Nonlinear matrix diffusion for optic flow estimation, in: L. Van Gool (Ed.), Pattern Recognition, Lecture Notes in Computer Science, vol. 2449, Berlin, Springer, 2002, pp. 446-453.
32. R. van den Boomgaard, J. van de Weijer, Least squares and robust estimation of local image structure, in: L.D. Griffin, M. Lillholm (Eds.), Scale Space Methods in Computer Vision, Lecture Notes in Computer Science, vol. 2695, Springer, Berlin, 2003, pp. 237-254.
33. T. Brox, R. van den Boomgaard, F. Lauze, J. van de Weijer, J. Weickert, P. Mrázek, P. Kornprobst, Adaptive structure tensors and their applications, in: J. Weickert, H. Hagen (Eds.), Visualization and Processing of Tensor Fields, Springer, Berlin, 2006, pp. 17-47.
34. F. Guichard, J.M. Morel, Partial differential equations and image iterative filtering, in: I.S. Duff, G.A. Watson (Eds.), The State of the Art in Numerical Analysis, IMA Conference Series (New Series), No. 63, Clarendon Press, Oxford, 1997, pp. 525-562.
35. J. Barral Souto, El modo y otras medias, casos particulares de una misma expresión matemática, Cuadernos de Trabajo, No. 3, Instituto de Biometria, Universidad Nacional de Buenos Aires, Argentina, 1938.
36. Torroba P.L., Cap N.L., Rabal H.J., and Furlan W.D. Fractional order mean in image processing. Opt. Eng. 33 2 (1994) 528-534
37. Winkler G., Aurich V., Hahn K., and Martin A. Noise reduction in images: some recent edge-preserving methods. Pattern Recognition and Image Anal. 9 4 (1999) 749-766
38. O. Coulon, D.C. Alexander, S.A. Arridge, A regularization scheme for diffusion tensor magnetic resonance images, in: M.F. Insana, R.M. Leahy (Eds.), Information Processing in Medical Imaging-IPMI 2001, Lecture Notes in Computer Science, vol. 2082, Springer, Berlin, 2001, pp. 92-105.