Abstract
The median filter is one of the fundamental filters in image processing. Its standard realisation relies on a rank ordering of given data which is easy to perform if the given data are scalar values. However, the generalisation of the median filter to multivariate data is a delicate issue. One of the methods of potential interest for computing a multivariate median is the convex-hull-stripping median from the statistics literature. Its definition is of purely algorithmical nature, and it offers the advantageous property of affine equivariance.
While it is a classic result that the standard median filter approximates mean curvature motion, no corresponding assertion has been established up to now for the convex-hull-stripping median. The aim of our paper is to close this gap in the literature. In order to provide a theoretical foundation for the convex-hull-stripping median of multivariate images, we investigate its continuous-scale limit. It turns out that the resulting evolution is described by the well-known partial differential equation of affine curvature motion. Thus we have established in this paper a relation between two important models from image processing and statistics. We also present some experiments that support our theoretical findings.
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References
Alvarez, L., Guichard, F., Lions, P.L., Morel, J.M.: Axioms and fundamental equations in image processing. Arch. Ration. Mech. Anal. 123, 199–257 (1993)
Austin, T.L.: An approximation to the point of minimum aggregate distance. Metron 19, 10–21 (1959)
Barnett, V.: The ordering of multivariate data. J. Roy. Stat. Soc. A 139(3), 318–355 (1976)
Chakraborty, B., Chaudhuri, P.: On a transformation and re-transformation technique for constructing an affine equivariant multivariate median. Proc. AMS 124(6), 2539–2547 (1996)
Fletcher, P., Venkatasubramanian, S., Joshi, S.: The geometric median on Riemannian manifolds with applications to robust atlas estimation. NeuroImage 45, S143–S152 (2009)
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)
Guichard, F., Morel, J.M.: Geometric partial differential equations and iterative filtering. In: Heijmans, H., Roerdink, J. (eds.) Mathematical Morphology and its Applications to Image and Signal Processing, pp. 127–138. Kluwer, Dordrecht (1998)
Hayford, J.F.: What is the center of an area, or the center of a population? J. Am. Stat. Assoc. 8(58), 47–58 (1902)
Jackson, D.: Note on the median of a set of numbers. Bull. Am. Math. Soc. 27, 160–164 (1921)
Oja, H.: Descriptive statistics for multivariate distributions. Stat. Probab. Lett. 1, 327–332 (1983)
Rao, C.R.: Methodology based on the \(l_1\)-norm in statistical inference. Sankhyā A 50, 289–313 (1988)
Sapiro, G., Tannenbaum, A.: Affine invariant scale-space. Int. J. Comput. Vis. 11, 25–44 (1993)
Seheult, A.H., Diggle, P.J., Evans, D.A.: Discussion of Professor Barnett’s paper. J. Roy. Stat. Soc. A 139(3), 351–352 (1976)
Sethian, J.A.: Level Set and Fast Marching Methods. Cambridge University Press, Cambridge (1999)
Small, C.G.: A survey of multidimensional medians. Int. Stat. Rev. 58(3), 263–277 (1990)
Spence, C., Fancourt, C.: An iterative method for vector median filtering. In: Proceedings of 2007 IEEE International Conference on Image Processing, vol. 5, pp. 265–268 (2007)
Tukey, J.W.: Exploratory Data Analysis. Addison-Wesley, Menlo Park (1971)
Tukey, J.W.: Mathematics and the picturing of data. In: Proceedings of the International Congress of Mathematics, pp. 523–532. Vancouver, Canada (1974)
Vardi, Y., Zhang, C.H.: A modified Weiszfeld algorithm for the Fermat-Weber location problem. Math. Program. A 90, 559–566 (2001)
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 Math. J. 43, 355–386 (1937)
Welk, M.: Partial differential equations of bivariate median filters. In: Aujol, J.-F., Nikolova, M., Papadakis, N. (eds.) SSVM 2015. LNCS, vol. 9087, pp. 53–65. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-18461-6_5
Welk, M.: Multivariate median filters and partial differential equations. J. Math. Imaging Vis. 56, 320–351 (2016)
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, Amsterdam (2014)
Welk, M., Weickert, J., Becker, F., Schnörr, C., Feddern, C., Burgeth, B.: Median and related local filters for tensor-valued images. Sig. Process. 87, 291–308 (2007)
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Welk, M., Breuß, M. (2019). The Convex-Hull-Stripping Median Approximates Affine Curvature Motion. In: Lellmann, J., Burger, M., Modersitzki, J. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2019. Lecture Notes in Computer Science(), vol 11603. Springer, Cham. https://doi.org/10.1007/978-3-030-22368-7_16
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