Mathematical morphology for matrix fields induced by the Loewner ordering in higher dimensions

Signal Processing

Volumn 87, Issue 2, 2007, Pages 277-290

  Burgeth, Bernhard ^a   Bruhn, Andres ^a   Papenberg, Nils ^a   Welk, Martin ^a   Weickert, Joachim ^a  

^a SAARLAND UNIVERSITY   (Germany)

Author keywords

Closing; Dilation; DT MRI; Erosion; Loewner ordering; Mathematical morphology; Matrix valued imaging; Mid range filter; Morphological derivatives; Opening; Shock filter; Top hats

Indexed keywords

EROSION; IMAGE PROCESSING; MAGNETIC RESONANCE IMAGING; MATHEMATICAL MORPHOLOGY; TENSORS;

CLOSING; DILATION; LOEWNER ORDERING; MATRIX-VALUED IMAGING; MID-RANGE FILTER; MORPHOLOGICAL DERIVATIVES; OPENING; SHOCK FILTER; TOP HATS;

MATRIX ALGEBRA;

EID: 33750148738     PISSN: 01651684     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.sigpro.2005.12.012     Document Type: Article

References (64)

1. Matheron G. Eléments pour une théorie des milieux poreux (1967), Masson, Paris
2. J. Serra, Echantillonnage et estimation des phénomènes de transition minier, Ph.D. Thesis, University of Nancy, France, 1967.
3. Haralick R.M., Sternberg S.R., and Zhuang X. Image analysis using mathematical morphology. IEEE Trans. Pattern Anal. Machine Intell. 9 4 (July 1987) 532-550
4. Sternberg S.R. Grayscale morphology. Comput. Vision Graph. Image Process. 35 (1986) 333-355
5. Comer M.L., and Delp E.J. Morphological operations for color image processing. J. Electron. Imaging 8 3 (1999) 279-289
6. Louverdis G., Vardavoulia M.I., Andreadis I., and Tsalides P. A new approach to morphological color image processing. Pattern Recognition 35 (2002) 1733-1741
7. Talbot H., Evans C., and Jones R. Complete ordering and multivariate mathematical morphology. In: Heijmans H.J.A.M., and Roerdink J.B.T.M. (Eds). Mathematical Morphology and its Applications to Image and Signal Processing, Computational Imaging and Vision vol. 12 (1998), Kluwer, Dordrecht
8. Goutsias J., Heijmans H.J.A.M., and Sivakumar K. Morphological operators for image sequences. Comput. Vision Image Understanding 62 (1995) 326-346
9. Heijmans H.J.A.M. Morphological Image Operators (1994), Academic Press, Boston
10. Matheron G. Random Sets and Integral Geometry (1975), Wiley, New York
11. Serra J. Image Analysis and Mathematical Morphology vol. 1 (1982), Academic Press, London
12. Serra J. Image Analysis and Mathematical Morphology vol. 2 (1988), Academic Press, London
13. Soille P. Morphological Image Analysis (1999), Springer, Berlin
14. J. Goutsias, L. Vincent, D.S. Bloomberg (Eds.), Mathematical morphology and its applications to image and signal processing, Computational Imaging and Vision, vol. 18, Kluwer, Dordrecht, 2000.
15. H.J.A.M. Heijmans, J.B.T.M. Roerdink (Eds.), Mathematical Morphology and its Applications to Image and Signal Processing, Computational Imaging and Vision, vol. 12, Kluwer, Dordrecht, 1998.
16. H. Talbot, R. Beare (Eds.), Proceedings of the Sixth International Symposium on Mathematical Morphology and its Applications, Sydney, Australia, April 2002, 〈http://www.cmis.csiro.au/ismm2002/proceedings/〉.
17. Basser P.J., Mattiello J., and LeBihan D. MR diffusion tensor spectroscopy and imaging. Biophysical J. 66 (1994) 259-267
18. Pierpaoli C., Jezzard P., Basser P.J., Barnett A., and Di Chiro G. Diffusion tensor MR imaging of the human brain. Radiology 201 3 (December 1996) 637-648
19. Granlund G.H., and Knutsson H. Signal Processing for Computer Vision (1995), Kluwer, Dordrecht
20. Förstner W., and Gülch E. A fast operator for detection and precise location of distinct points, corners and centres of circular features. Proceedings of the ISPRS Intercommission Conference on Fast Processing of Photogrammetric Data. Interlaken, Switzerland (June 1987) 281-305
21. Bigün J., Granlund G.H., and Wiklund J. Multidimensional orientation estimation with applications to texture analysis and optical flow. IEEE Trans. Pattern Anal. Machine Intell. 13 8 (August 1991) 775-790
22. Rao A.R., and Schunck B.G. Computing oriented texture fields. CVGIP: Graph. Models Image Process. 53 (1991) 157-185
23. Harris C.G., and Stephens M. A combined corner and edge detector. Proceedings of the Fourth Alvey Vision Conference. Manchester, England (August 1988) 147-152
24. Medioni G., Lee M.-S., and Tang C.-K. A Computational Framework for Segmentation and Grouping (2000), Elsevier, Amsterdam
25. C.-F. 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.
26. 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.
27. 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.
28. C. Poupon, J.-F. 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.
29. 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.
30. 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
31. Weickert J., and Brox T. Diffusion and regularization of vector- and matrix-valued images. In: Nashed M.Z., and Scherzer O. (Eds). Inverse Problems, Image Analysis, and Medical Imaging, Contemporary Mathematics vol. 313 (2002), AMS, Providence, RI 251-268
32. 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, Springer, Berlin, 2002, pp. 446-453.
33. M. Rousson, T. Brox, R. Deriche, Active unsupervised texture segmentation on a diffusion based feature space, in: Proceedings of the 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 2, Madison, WI, June 2003, IEEE Computer Society Press, Silver Spring, MD, pp. 699-704.
34. 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, Springer, Berlin, 2003, pp. 353-360.
35. C. Feddern, J. Weickert, B. Burgeth, Level-set methods for tensor-valued images, in: O. Faugeras N. Paragios (Eds.), Proceedings of the Second IEEE Workshop on Geometric and Level Set Methods in Computer Vision, Nice, France, October 2003, INRIA, pp. 65-72.
36. 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.
37. Castaño Moraga C.A., Westin C.-F., and Ruiz-Alzola J. Homomorphic filtering of DT-MRI fields. In: Ellis R.E., and Peters T.M. (Eds). Medical Image Computing and Computer-Assisted Intervention-MICCAI 2003, Lecture Notes in Computer Science (2003), Springer, Berlin
38. Astola J., Haavisto P., and Neuvo Y. Vector median filters. Proc. IEEE 78 4 (1990) 678-689
39. Barnett V. The ordering of multivariate data. J. Roy. Statist. Soc. Ser. A 139 3 (1976) 318-355
40. 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), vol. 63, Clarendon Press, Oxford, 1997, pp. 525-562.
41. Caselles V., Sapiro G., and Chung D.H. Vector median filters, inf-sup convolutions, and coupled PDE's: theoretical connections. J. Math. Imaging Vision 12 2 (2000) 109-119
42. p" and its use in multivariate image estimation. IEEE Trans. Circuits Systems Video Technol. 1 2 (1991) 197-209
43. 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.
44. Horn R.A., and Johnson C.R. Topics in Matrix Analysis (1994), Cambridge University Press, Cambridge, UK
45. B. Burgeth, M. Welk, C. Feddern, J. Weickert, Mathematical morphology on tensor data using the Loewner ordering, in: H. Hagen, J. Weickert (Eds.), Visualization and Processing of Tensor Fields, Springer, Berlin, 2006, pp. 357-368.
46. S. Beucher, C. Lantuéjoul, Use of watersheds in contour detection, in: Proceedings of the International Workshop on Image Processing, Real-Time Edge and Motion Detection/Estimation, Rennes, France, September 1979, pp. 2.1-2.12, IRISA Report No. 131.
47. van Vliet L.J., Young I.T., and Beckers A.L.D. A nonlinear Laplace operator as edge detector in noisy images. Comput. Vision Graph. Image Process. 45 2 (1989) 167-195
48. Marr D., and Hildreth E. Theory of edge detection. Proc. Roy. Soc. London Ser. B 207 (1980) 187-217
49. Haralick R.M. Digital step edges from zero crossing of second directional derivatives. IEEE Trans. Pattern Anal. Machine Intell. 6 1 (1984) 58-68
50. Kimmel R., and Bruckstein A.M. Regularized Laplacian zero crossings as optimal edge integrators. Internat. J. Comput. Vision 53 3 (2003) 225-243
51. Kramer H.P., and Bruckner J.B. Iterations of a non-linear transformation for enhancement of digital images. Pattern Recognition 7 (1975) 53-58
52. Osher S., and Rudin L.I. Feature-oriented image enhancement using shock filters. SIAM J. Numer. Anal. 27 (1990) 919-940
53. Guichard F., and Morel J.-M. A note on two classical enhancement filters and their associated PDE's. Internat. J. Comput. Vision 52 2/3 (2003) 153-160
54. Alvarez L., and Mazorra L. Signal and image restoration using shock filters and anisotropic diffusion. SIAM J. Numer. Anal. 31 (1994) 590-605
55. G. Gilboa, N.A. Sochen, Y.Y. Zeevi, Regularized shock filters and complex diffusion, in: A. Heyden, G. Sparr, M. Nielsen, P. Johansen (Eds.), Computer Vision-ECCV 2002, Lecture Notes in Computer Science, vol. 2350, Springer, Berlin, 2002, pp. 399-413.
56. Osher S., and Rudin L. Shocks and other nonlinear filtering applied to image processing. In: Tescher A.G. (Ed). Applications of Digital Image Processing XIV, Proceedings of SPIE vol. 1567 (1991), SPIE Press, Bellingham 414-431
57. Remaki L., and Cheriet M. Numerical schemes of shock filter models for image enhancement and restoration. J. Math. Imaging Vision 18 2 (March 2003) 153-160
58. J.G.M. Schavemaker, M.J.T. Reinders, R. van den Boomgaard, Image sharpening by morphological filtering, in: Proceedings of the 1997 IEEE Workshop on Nonlinear Signal and Image Processing, Mackinac Island, MI, September 1997, 〈www.ecn.purdue.edu/NSIP/〉.
59. J. Weickert, Coherence-enhancing shock filters, in: B. Michaelis, G. Krell (Eds.), Pattern Recognition, Lecture Notes in Computer Science, vol. 2781, Springer, Berlin, 2003, pp. 1-8.
60. A. Barvinok, A course in convexity, Graduate Studies in Mathematics, vol. 54, American Mathematical Society, Providence, RI, 2002.
61. Hiriart-Urruty J.-B., and Lemarechal C. Fundamentals of Convex Analysis (2001), Springer, Heidelberg
62. Borwein J.M., and Lewis A.S. Convex Analysis and Nonlinear Optimization (1999), Springer, New York
63. B. Gärtner, 〈http://www.inf.ethz.ch/personal/gaertner/〉, WebPage last visited: December 2nd, 2005.
64. Basser P.J. Inferring microstructural features and the physical state of tissues from diffusion-weighted images. Nuclear Magnetic Resonance Biomedicine 8 (1995) 333-334