Abstract
Multivariate median filters have been proposed as generalisations of the well-established median filter for grey-value images to multi-channel images. As multivariate median, most of the recent approaches use the \(L^1\) median, i.e. the minimiser of an objective function that is the sum of distances to all input points. Many properties of univariate median filters generalise to such a filter. However, the famous result by Guichard and Morel about approximation of the mean curvature motion PDE by median filtering does not have a comparably simple counterpart for \(L^1\) multivariate median filtering. We discuss the affine equivariant Oja median as an alternative to \(L^1\) median filtering. We derive the PDE approximated by Oja median filtering in the bivariate case, and demonstrate its validity by a numerical experiment.
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References
Alvarez, L., Lions, P.-L., Morel, J.-M.: Image selective smoothing and edge detection by nonlinear diffusion. II. SIAM Journal on Numerical Analysis 29, 845–866 (1992)
Astola, J., Haavisto, P., Neuvo, Y.: Vector median filters. Proceedings of the IEEE 78(4), 678–689 (1990)
Austin, T.L.: An approximation to the point of minimum aggregate distance. Metron 19, 10–21 (1959)
Chakraborty, B., Chaudhuri, P.: On a transformation and re-transformation technique for constructing an affine equivariant multivariate median. Proceedings of the AMS 124(6), 2539–2547 (1996)
Chakraborty, B., Chaudhuri, P.: A note on the robustness of multivariate medians. Statistics and Probability Letters 45, 269–276 (1999)
Chung, D.H., Sapiro, G.: On the level lines and geometry of vector-valued images. IEEE Signal Processing Letters 7(9), 241–243 (2000)
Eckhardt, U.: Root images of median filters. Journal of Mathematical Imaging and Vision 19, 63–70 (2003)
Gini, C., Galvani, L.: Di talune estensioni dei concetti di media ai caratteri qualitativi. Metron 8, 3–209 (1929)
Guichard, F., Morel, J.-M.: Partial differential equations and image iterative filtering. In: Duff, I.S., Watson, G.A. (eds.) The State of the Art in Numerical Analysis. IMA Conference Series (New Series), vol. 63, pp. 525–562. Clarendon Press, Oxford (1997)
Hayford, J.F.: What is the center of an area, or the center of a population? Journal of the American Statistical Association 8(58), 47–58 (1902)
Hettmansperger, T.P., Randles, R.H.: A practical affine equivariant multivariate median. Biometrika 89(4), 851–860 (2002)
Kleefeld, A., Breuß, M., Welk, M., Burgeth, B.: Adaptive filters for color images: median filtering and its extensions. In: Trémeau, A., Schettini, R., Tominaga, S. (eds.) CCIW 2015. LNCS, vol. 9016, pp. 149–158. Springer, Heidelberg (2015)
Oja, H.: Descriptive statistics for multivariate distributions. Statistics and Probability Letters 1, 327–332 (1983)
Rao, C.R.: Methodology based on the \(l_1\)-norm in statistical inference. Sankhyā A 50, 289–313 (1988)
Small, C.G.: A survey of multidimensional medians. International Statistical Review 58(3), 263–277 (1990)
Spence, C., Fancourt, C.: An iterative method for vector median filtering. In: Proc. 2007 IEEE International Conference on Image Processing, vol. 5, pp. 265–268 (2007)
Tukey, J.W.: Exploratory Data Analysis. Addison-Wesley, Menlo Park (1971)
Weber, A.: Über den Standort der Industrien. Mohr, Tübingen (1909)
Weiszfeld, E.: Sur le point pour lequel la somme des distances de \(n\) points donnés est minimum. Tôhoku Mathematics Journal 43, 355–386 (1937)
Welk, M., Breuß, M.: Morphological amoebas and partial differential equations. In: Hawkes, P.W. (ed.) Advances in Imaging and Electron Physics, vol. 185, pp. 139–212. Elsevier Academic Press (2014)
Welk, M., Breuß, M., Vogel, O.: Morphological amoebas are self-snakes. Journal of Mathematical Imaging and Vision 39, 87–99 (2011)
Welk, M., Feddern, C., Burgeth, B., Weickert, J.: Median filtering of tensor-valued images. In: Michaelis, B., Krell, G. (eds.) DAGM 2003. LNCS, vol. 2781, pp. 17–24. Springer, Heidelberg (2003)
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Welk, M. (2015). Partial Differential Equations of Bivariate Median Filters. In: Aujol, JF., Nikolova, M., Papadakis, N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2015. Lecture Notes in Computer Science(), vol 9087. Springer, Cham. https://doi.org/10.1007/978-3-319-18461-6_5
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DOI: https://doi.org/10.1007/978-3-319-18461-6_5
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