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Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 596348, 16 pages
Research Article

Multivariate Self-Dual Morphological Operators Based on Extremum Constraint

1School of Electronic and Information Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China
2School of Electronics and Information, Northwestern Polytechnical University, Xi’an 710072, China

Received 2 January 2015; Revised 15 June 2015; Accepted 21 June 2015

Academic Editor: Babak Shotorban

Copyright © 2015 Tao Lei 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.


Self-dual morphological operators (SDMO) do not rely on whether one starts the sequence with erosion or dilation; they treat the image foreground and background identically. However, it is difficult to extend SDMO to multichannel images. Based on the self-duality property of traditional morphological operators and the theory of extremum constraint, this paper gives a complete characterization for the construction of multivariate SDMO. We introduce a pair of symmetric vector orderings (SVO) to construct multivariate dual morphological operators. Furthermore, utilizing extremum constraint to optimize multivariate morphological operators, we construct multivariate SDMO. Finally, we illustrate the importance and effectiveness of the multivariate SDMO by applications of noise removal and segmentation performance. The experimental results show that the proposed multivariate SDMO achieves better results, and they suppress noises more efficiently without losing image details compared with other filtering methods. Moreover, the proposed multivariate SDMO is also shown to have the best segmentation performance after the filtered images via watershed transformation.