Abstract
Among the major developments in Mathematical Morphology in the last two decades are the interrelated subjects of connectivity classes and connected operators. Braga-Neto and Goutsias have proposed an extension of the theory of connectivity classes to a multiscale setting, whereby one can assign connectivity to an object observed at different scales. In this paper, we study connected operators in the context of multiscale connectivity. We propose the notion of a σ-connected operator, that is, an operator connected at scale σ. We devote some attention to the study of binary σ-grain operators. In particular, we show that families of σ-grain openings and σ-grain closings, indexed by the connectivity scale parameter, are granulometries and anti-granulometries, respectively. We demonstrate the use of multiscale connected operators with image analysis applications. The first is the scale-space representation of grayscale images using multiscale levelings, where the role of scale is played by the connectivity scale. Then we discuss the application of multiscale connected openings in granulometric analysis, where both size and connectivity information are summarized. Finally, we describe an application of multiscale connected operators to an automatic target recognition problem.
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S. Batman and E. Dougherty, “Size distributions for multivariate morphological granulometries: Texture classification and statistical properties”Optical Engineering, Vol. 36, pp. 1518–1529, 1997.
G. Birkhoff,Lattice Theory, 3rd edition, Vol. 25, American Mathematical Society, Providence, Rhode Island,1967.
U.M. Braga-Neto and J. Goutsias, “Constructing multiscale connectivities” to appear inComputer Vision and Image Understanding, 2005.
U.M. Braga-Neto, M. Choudhary, and J. Goutsias, “Automatic target detection and tracking on forward-looking infrared image sequences using morphological connected operators”Journal of Electronic Imaging, Vol. 13, No. 4, pp. 802–813, 2004.
U.M. Braga-Neto and J. Goutsias, “Multiresolution connectivity: An axiomatic approach” in Mathematical Morphology and its Applications to Image and Signal Processing, J. Goutsias, L. Vincent, and D.S. Bloomberg (Eds.), Kluwer: Boston, Massachusetts, 2000, pp. 159–168.
U.M. Braga-Neto and J. Goutsias, “Connectivity on complete lattices: New results”Computer Vision and Image Understanding, Vol. 85, No. 1, pp. 22–53, 2002.
U.M. Braga-Neto and J. Goutsias, “A multiscale approach to connectivity”Computer Vision and Image Understanding, Vol. 89, No. 1, pp. 70–107,2003.
U.M. Braga-Neto and J. Goutsias, “A theoretical tour of connectivity in image processing and analysis”Journal of Mathematical Imaging and Vision, Vol. 19, No. 1, pp. 5–31, 2003.
J. Crespo and R.W. Schafer, “Locality and adjacency stability constraints for morphological connected operators”Journal of Mathematical Imaging and Vision, Vol. 7, pp. 85–102, 1997.
J. Crespo, J. Serra, and R.W. Schafer, “Theoretical aspects of morphological filters by reconstruction”Signal Processing, Vol. 47, No. 2, pp. 201–225, 1995.
H.J.A.M. Heijmans, “Connected morphological operators for binary images”Computer Vision and Image Understanding, Vol. 73, pp. 99–120, 1999.
H.J.A.M. Heijmans,Morphological Image Operators, Academic Press: Boston, MA, 1994.
R. Jones, “Connected filtering and segmentation using component trees”Computer Vision and Image Understanding, Vol. 75, pp. 215–228, 1999.
J.J. Koenderink, “The structure of images”Biological Cybernetics, Vol. 50, pp. 363–370, 1984.
M. Kunt, A. Ikonomopoulos, and M. Kocher, “Second generation image coding techniques”Proceedings of the IEEE, Vol. 73, pp. 549–574, 1985.
P. Maragos, “Pattern spectrum and multiscale shape representation”IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 11, pp. 701–716, 1989.
F. Meyer, “From connected operators to levelings” in Mathematical Morphology and its Applications to Image and Signal Processing, H.J.A.M. Heijmans and J.B.T.M. Roerdink (Eds.),Kluwer: Dordrecht, 1998, pp. 191–198.
F. Meyer, “The levelings.” InMathematical Morphology and its Applications to Image and Signal Processing, H.J.A.M. Heijmans and J.B.T.M. Roerdink (Eds.), Kluwer: Dordrecht, 1998, pp. 199–206.
F. Meyer and P. Maragos, “Morphological scale-space representation with levelings” inScale-Space’99 Symposium, M. Nielsen et al. (Eds.), Springer-Verlag: Berlin Heidelberg, 1999, pp. 187–198.
C. Ronse, “Set-theoretic algebraic approaches to connectivity in continuous or digital spaces”Journal of Mathematical Imaging and Vision, Vol. 8, pp. 41–58, 1998.
C. Ronse and J. Serra, “Geodesy and connectivity in lattices”Fundamenta Informaticae, Vol. 46, pp. 349–395, 2001.
A. Rosenfeld and A.C. Kak,Digital Picture Processing, 2nd edition. Academic Press: Orlando, Florida, 1982.
P. Salembier, A. Oliveras, and L. Garrido, “Antiextensive connected operators for image and sequence processing”IEEE Transactions on Image Processing, Vol. 7, pp. 555–570, 1998.
P. Salembier and M. Pardàs, “Hierarchical morphological segmentation for image sequence coding”IEEE Transactions on Image Processing, Vol. 3, pp. 639–651, 1994.
P. Salembier and J. Serra, “Flat zones filtering, connected operators, and filters by reconstruction”IEEE Transactions on Image Processing, Vol. 4, pp. 1153–1160, 1995.
P. Salembier and H. Sanson, “Robust motion estimation using connected operators” inProceedings of the IEEE International Conference on Image Processing, Vol. 1. Santa Barbara, California, 1997, pp. 77–80.
J. Serra (Ed.).Image Analysis and Mathematical Morphology. Vol. 2 Theoretical Advances, Academic Press: London, England, 1988.
J. Serra, “Connectivity on complete lattices”Journal of Mathematical Imaging and Vision, Vol. 9, pp. 231–251, 1998.
J. Serra, “Connections for sets and functions”Fundamenta Informaticae, Vol. 41, pp. 147–186, 2000.
J. Serra and P. Salembier, “Connected operators and pyramids” inProceedings of the SPIE Conference on Image Algebra and Morphological Image Processing IV, Vol. 2030, San Diego, California, 1993, pp. 65–76.
V. Vilaplana and F. Marques, “Face segmentation using connected operators” in Mathematical Morphology and its Applications to Image and Signal Processing, H.J.A.M. Heijmans and J.B.T.M. Roerdink (Eds.), Kluwer: Boston, Massachusetts, 1998, pp. 207–214.
L. Vincent, “Morphological area openings and closings for grayscale images” inProceedings of NATO Shape in Picture Workshop, Driebergen: The Netherlands, 1993, pp. 22–27.
L. Vincent, “Morphological grayscale reconstruction in image analysis: Applications and efficient algorithms”IEEE Transactions on Image Processing, Vol. 2, pp. 176–201, 1993.
A.P. Witkin, “Scale-space filtering” inProceedings of 7th International Joint Conference on Artificial Intelligence, Karlsruhe, West Germany, 1983, pp. 1019–1022.
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Ulisses Braga-Neto received the Baccalaureate degree in Electrical Engineering from the Universidade Federal de Pernambuco (UFPE), Brazil, in 1992, the Master’s degree in Electrical Engineering from the Universidade Estadual de Campinas, Brazil, in 1994, the M.S.E. degree in Electrical and Computer Engineering and the M.S.E. degree in Mathematical Sciences, both from The Johns Hopkins University, in 1998, and the Ph.D. degree in Electrical and Computer Engineering, from The Johns Hopkins University, in 2001. He was a Post-Doctoral Fellow at the University of Texas MD Anderson Cancer Center and a Visiting Scholar at Texas A&M University, from 2002 to 2004. He is currently an Associate Researcher at the Aggeu Magalhães Research Center of the Osvaldo Cruz Foundation, Brazilian Ministry of Health. His research interests include Bioinformatics, Pattern Recognition, Image Analysis, and Mathematical Morphology.
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Braga-Neto, U. Multiscale Connected Operators. J Math Imaging Vis 22, 199–216 (2005). https://doi.org/10.1007/s10851-005-4890-6
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DOI: https://doi.org/10.1007/s10851-005-4890-6