Abstract
Mathematical morphology is a successful branch of image processing with a history of more than four decades. Its fundamental operations are dilation and erosion, which are based on the notion of supremum and infimum with respect to an order. From dilation and erosion one can build readily other useful elementary morphological operators and filters, such as opening, closing, morphological top-hats, derivatives, and shock filters. Such operators are available for grey value images, and recently useful analogs of these processes for matrix-valued images have been introduced by taking advantage of the so-called Loewner order. There is a number of approaches to morphology for vector-valued images, that is, color images based on various orders, however, each with its merits and shortcomings. In this chapter we propose an approach to (elementary) morphology for color images that relies on the existing order based morphology for matrix fields of symmetric 2 × 2-matrices. An RGB-image is embedded into a field of those 2 × 2-matrices by exploiting the geometrical properties of the order cone associated with the Loewner order. To this end a modification of the HSL-color model and a relativistic addition of matrices is introduced. The experiments performed with various morphological elementary operators on synthetic and real images provide results promising enough to serve as a proof-of-concept.
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References
Agoston, M.K.: Computer Graphics and Geometric Modeling: Implementation and Algorithms. Springer, London (2005)
Aptoula, E., Lefèvre, S.: A comparative study on multivariate mathematical morphology. Pattern Recognit. 40(11), 2914–2929 (2007)
Barnett, V.: The ordering of multivariate data. J. Stat. Soc. A 139(3), 318–355 (1976)
Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization. Springer, New York (2000)
Burgeth, B., Welk, W.,Feddern, Ch., Weickert, J.: Morphological operations on matrix-valued images. In: Pajdla, T., Matas, J. (eds.) Computer Vision – ECCV 2004, Part IV, Prague. Lecture Notes in Computer Science, vol. 3024, pp. 155–167. Springer, Berlin (2004)
Burgeth, B., Bruhn, A., Papenberg, N., Welk, M., Weickert, J.: Mathematical morphology for tensor data induced by the Loewner ordering in higher dimensions. Signal Process. 87(2), 277–290 (2007)
Burgeth, B., Papenberg, N., Bruhn, A., Welk, M., Feddern, C., Weickert, J.: Mathematical morphology based on the loewner ordering for tensor data. In: Ronse, C., Najman, L., Decencière, E. (eds.) Mathematical Morphology: 40 Years On Computational Imaging and Vision, vol. 30, pp. 407–418. Springer, Dordrecht (2005)
Burgeth, B., Welk, M., Feddern, C., Weickert, J.: Mathematical morphology on tensor data using the Loewner ordering. In: Weickert, J., Hagen, H. (eds.) Visualization and Processing of Tensor Fields, pp. 357–367. Springer, Berlin (2006)
Comer, M.L., Delp, E.J.: Morphological operations for color image processing. J. Electron. Imaging 8(3), 279–289 (1999)
Serra, J.: Anamorphoses and function lattices. In: Dougherty, E.R. (ed.) Mathematical Morphology in Image Processing, pp. 483–523. Marcel Dekker, New York (1993)
Gaertner, B.: Smallest enclosing balls of points – fast and robust in c++. http://www.inf.ethz.ch/personal/gaertner/miniball.html. Last visited: 03 July 2012
Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 2nd edn. Addison–Wesley, Reading (2002)
Goutsias, J., Heijmans, H.J.A.M., Sivakumar, K.: Morphological operators for image sequences. Comput. Vis. Image Underst. 62, 326–346 (1995)
Haralick, R.M.: Digital step edges from zero crossing of second directional derivatives. IEEE Trans. Pattern Anal. Mach. Intell. 6(1), 58–68 (1984)
Heijmans, H.J.A.M.: Morphological Image Operators. Academic, Boston (1994)
Huppert, B.: Angewandte Lineare Algebra. de Gruyter, New York (1990)
Kimmel, R., Bruckstein, A.M.: Regularized Laplacian zero crossings as optimal edge integrators. Int. J. Comput. Vis. 53(3), 225–243 (2003)
Koenderink, J.J.: Color for the Sciences. MIT, Cambridge (2010)
Kramer, H.P., Bruckner, J.B.: Iterations of a non-linear transformation for enhancement of digital images. Pattern Recognit. 7, 53–58 (1975)
Marr, D., Hildreth, E.: Theory of edge detection. Proc. R. Soc. Lond. Ser. B 207, 187–217 (1980)
Matheron, G.: Eléments pour une théorie des milieux poreux. Masson, Paris (1967)
Serra, J.: Echantillonnage et estimation des phénomènes de transition minier. Ph.D. thesis, University of Nancy (1967)
Serra, J.: Image Analysis and Mathematical Morphology, vol. 1. Academic, London (1982)
Serra, J.: Image Analysis and Mathematical Morphology, vol. 2. Academic, London (1988)
Soille, P.: Morphological Image Analysis, 2nd edn. Springer, Berlin (2003)
van Vliet, L.J., Young, I.T., Beckers, A.L.D.: A nonlinear Laplace operator as edge detector in noisy images. Comput. Vis. Graph. Image Process. 45(2), 167–195 (1989)
Xu, S., Freund, R.M., Sun, J.: Solution methodologies for the smallest enclosing circle problem. High Perform. Comput. Eng. Syst. (HPCES). http://hdl.handle.net/1721.1/4015 (2003)
Acknowledgements
The authors would like to thank Sheng Xu, Robert M. Freund, and Jie Sun for the MATLAB source code of the subgradient method.
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Burgeth, B., Kleefeld, A. (2014). Order Based Morphology for Color Images via Matrix Fields. In: Westin, CF., Vilanova, A., Burgeth, B. (eds) Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54301-2_4
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DOI: https://doi.org/10.1007/978-3-642-54301-2_4
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