Abstract
The notions of maximum and minimum are the key to the powerful tools of greyscale morphology. Unfortunately these notions do not carry over directly to tensor-valued data. Based upon the Loewner ordering for symmetric matrices this chapter extends the maximum and minimum operation to the tensor-valued setting. This provides the ground to establish matrix-valued analogues of the basic morphological operations ranging from erosion/dilation to top hats. In contrast to former attempts to develop a morphological machinery for matrices, the novel definitions of maximal/minimal matrices depend continuously on the input data, a property crucial for the construction of morphological derivatives such as the Beucher gradient or a morphological Laplacian. These definitions are rotationally invariant and preserve positive semidefiniteness of matrix fields as they are encountered in DTMRI data. The morphological operations resulting from a component-wise maximum/minimum of the matrix channels disregarding their strong correlation fail to be rotational invariant. Experiments on DT-MRI images as well as on indefinite matrix data illustrate the properties and performance of our morphological operators.
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References
J. M. Borwein and A. S. Lewis. Convex Analysis and Nonlinear Optimization. Springer, New York, 1999.
T. Brox, J. Weickert, B. Burgeth, and P. Mrázek. Nonlinear structure tensors. Technical report, Dept. of Mathematics, Saarland University, Saarbrücken, Germany, July 2004.
B. Burgeth, M. Welk, C. Feddern, and J. Weickert. Morphological operations on matrix-valued images. In T. Pajdla and J. Matas, editors, Computer Vision — ECCV 2004, Part IV, volume 3024 of Lecture Notes in Computer Science, pp. 155–167. Springer, Berlin, 2004.
C. Feddern, J. Weickert, B. Burgeth, and M. Welk. Curvature-driven PDE methods for matrix-valued images. International Journal of Computer Vision, 2005. In press.
B. Gärtner. http://www2.inf.ethz.ch/personal/gaertner/. WebPage last visited: July 2nd, 2004.
J. Goutsias, L. Vincent, and D. S. Bloomberg, editors. Mathematical Morphology and its Applications to Image and Signal Processing, volume 18 of Computational Imaging and Vision. Kluwer, Dordrecht, 2000.
R. M. Haralick. Digital step edges from zero crossing of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(1):58–68, 1984.
H. J. A. M. Heijmans. Morphological Image Operators. Academic Press, Boston, 1994.
R. Kimmel and A. M. Bruckstein. Regularized Laplacian zero crossings as optimal edge integrators. International Journal of Computer Vision, 53(3):225–243, 2003.
H. P. Kramer and J. B. Bruckner. Iterations of a non-linear transformation for enhancement of digital images. Pattern Recognition, 7:53–58, 1975.
D. Marr and E. Hildreth. Theory of edge detection. Proceedings of the Royal Society of London, Series B, 207:187–217, 1980.
G. Matheron. Eléments pour une théorie des milieux poreux. Masson, Paris, 1967.
J. Serra. Echantillonnage et estimation des phénomènes de transition minier. PhD thesis, University of Nancy, France, 1967.
J. Serra. Image Analysis and Mathematical Morphology, volume 1. Academic Press, London, 1982.
J. Serra. Image Analysis and Mathematical Morphology, volume 2. Academic Press, London, 1988.
P. Soille. Morphological Image Analysis. Springer, Berlin, 1999.
H. Talbot and R. Beare, editors. Proc. Sixth International Symposium on Mathematical Morphology and its Applications. Sydney, Australia, April 2002. http://www.cmis.csiro.au/ismm2002/proceedings/.
D. Tschumperlé and R. Deriche. Diffusion tensor regularization with constraints preservation. In Proc. 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, volume 1, pp. 948–953, Kauai, HI, December 2001. IEEE Computer Society Press.
L. J. van Vliet, I. T. Young, and A. L. D. Beckers. A nonlinear Laplace operator as edge detector in noisy images. Computer Vision, Graphics and Image Processing, 45(2):167–195, 1989.
J. Weickert and T. Brox. Diffusion and regularization of vector-and matrix-valued images. In M. Z. Nashed and O. Scherzer, editors, Inverse Problems, Image Analysis, and Medical Imaging, volume 313 of Contemporary Mathematics, pp. 251–268. AMS, Providence, 2002.
M. Welk, C. Feddern, B. Burgeth, and J. Weickert. Median filtering of tensor-valued images. In B. Michaelis and G. Krell, editors, Pattern Recognition, volume 2781 of Lecture Notes in Computer Science, pp. 17–24, Berlin, 2003. Springer.
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Burgeth, B., Welk, M., Feddern, C., Weickert, J. (2006). Mathematical Morphology on Tensor Data Using the Loewner Ordering. In: Weickert, J., Hagen, H. (eds) Visualization and Processing of Tensor Fields. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31272-2_22
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DOI: https://doi.org/10.1007/3-540-31272-2_22
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