Image Segmentation by Mathematical Morphology: An Approach through Linear, Bilinear and Conformal Transformation

Authors: Dibyendu Ghoshal, Pinaki Pratim Acharjya

Abstract:

Image segmentation process based on mathematical morphology has been studied in the paper. It has been established from the first principles of the morphological process, the entire segmentation is although a nonlinear signal processing task, the constituent wise, the intermediate steps are linear, bilinear and conformal transformation and they give rise to a non linear affect in a cumulative manner.

Keywords: Image segmentation, linear transform, bilinear transform, conformal transform, mathematical morphology.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1094741

References:

[1] Milan Sonka et. al. Image Processing, Analysis and Machine Vision. PWS Publishing, second edition, 1999.
[2] Pierre Soille, Morphological Image Analysis. Springer-Verlag, 2003.
[3] Rafael C. Gonzalez, Richard E Woods. Digital Image Processing. Prentice Hall, second edition, 2002.
[4] P. Suetnes, P.Fua and A. J. Hanson, "Computational strategies for object recognition,” ACM Computing Surveys, Vol. 24, pp. 05-61, 1992.
[5] R. Besl, R. Jain, "Three dimensional object recognition,” ACM Computing Surveys, Vol. 17, pp. 75-145, 1985.
[6] K. Hohne, H. Fuchs, S. Pizer, "3D imaging in medicine: Algorithms, systems, Applications”, Berlin, Germany, Springer –Verlag, 1990.
[7] M. Kunt, M. Bernard and R. Leonardi, "Recent results in high compression image coding,” IEEE Trans. on Circuits and Systems, Vol.34, pp.1306-1336, 1987.
[8] M. Bomans, K. Hohne, U. Tiede and M. Riemer, "3D segmentation of MR images of the head for 3D display,” IEEE Transactions on Medical imaging, Vol.9, pp. 253-277, 1990.
[9] Phani krishna kishore M and Venugopal IVS, "A Cryptographic Algorithm based on Bilinear Transformation”, International Journal of Computer Science Issues, Vol. 8, Issue 5, No 2, pp. 298-302, September 2011.
[10] R K jain and S R K lyengar, Advanced engineering mathematics, Narosa publication, pp. 14.1-14.28.
[11] E.P. Dolzhenko (2001), "Conformal mapping", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.
[12] Dai Okano, Hidenori Ogata, Kaname Amano, Masaaki Sugihara, "Numerical conformal mappings of bounded multiply connected domains by the charge simulation method”, Journal of Computational and Applied Mathematics, pp. 109117, 2003.
[13] M.E.Klonowska and W.J.Prosnak,” On an effective method forconfor malmapping of multiply connected domains”, ActaMechanica, pp. 3552, 1996.
[14] A.P. Ilayaraja, M. P. Gopinath, "An Analysis of Morphological Operation Using Arbitrary Structuring Element”, International Journal of Modern Engineering Research, Vol.2, Issue.2, pp-354-357, 2012.
[15] S. Thilagamani, N.Shanthi, "A Novel Recursive Clustering Algorithm for Image Oversegmentation”, European Journal of Scientific Research, Vol.52, No.3, pp.430-436, 2011.
[16] S. Beucher, "Watershed, hierarchical segmentation and water fall algorithm,” in Mathematical Morphology and Its Applications to Image Processing, Dordrecht, The Netherlands: Kluwer, 1994, pp. 69–76.
[17] Beucher, S., and Meyer, F. The morphological approach to segmentation: the watershed transformation. In Mathematical Morphology in Image Processing, E. R. Dougherty, Ed. Marcel Dekker, New York, ch. 12, pp. 433-481, 1993.
[18] F. Meyer, S. Beucher, "Morphological Segmentation,” Journal of Visual Communication and Image Representation,vol. 1, pp. 21-46, 1990.
[19] C. R. Giardina and E. R. Dougherty, Morphological Methods in Image and Signal Processing, Prentice-Hall, Upper Saddle River, NJ, USA, 1988.
[20] Santiago Velasco-Forero, Student Member, IEEE, and Jesus Angulo, "Random Projection Depth for Multivariate Mathematical Morphology”, IEEE Journal Of Selected Topics In Signal Processing, Vol. 6, No. 7, November 2012.
[21] S. Velasco-Forero and J. Angulo, "Mathematical morphology for vector images using statistical depth,” in Mathematical Morphology and Its Applications to Image and Signal Processing, ser. Lecture Notes in Computer Science, P. Soille, M. Pesaresi, and G. Ouzounis, Eds. Berlin/Heidelberg, Germany: Springer, 2011, vol. 6671, pp.355–366.
[22] Rafael C. Gonzalez, Richard E. Woods, Steven L. Eddins, "Digital Image Processing Using MATLAB,” Second Edition, Gatesmark Publishing, 2009.
[23] P. Jackway, "Gradient watersheds in morphological scalespace,” IEEE Trans. Image Processing vol. l5, pp. 913–921, June, 1996.
[24] P. Soille, Morphological Image Analysis. New York: Springer Verlag, 1999.
[25] Image Analysis and Mathematical Morphology by Jean Serra, ISBN 0-12-637240-3, 1982.
[26] Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances by Jean Serra, ISBN 0-12-637241-1, 1988.
[27] An Introduction to Morphological Image Processing by Edward R. Dougherty, ISBN 0-8194-0845-X, 1992.