Abstract
Median filters for scalar-valued data are well-known tools for image denoising and analysis. They preserve discontinuities and are robust under noise. We generalise median filtering to matrix-valued data using a minimisation approach. Experiments on DT-MRI and fluid dynamics tensor data demonstrate that tensor-valued median filtering shares important properties of its scalar-valued counterpart, including the robustness as well as the existence of non-trivial steady states (root signals).
A straightforward extension of the definition allows the introduction of matrix-valued mid-range filters and, more general, M-smoothers. Mid-range filters can also serve as a building block in constructing further (e.g. supremum-based) tensor image filters.
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References
Astola, J., Haavisto, P., Neuvo, Y. (1990) Vector median filters. Proceedings of the IEEE 78(4), 678–689
Austin, T. L., Jr. (1959) An approximation to the point of minimum aggregate distance. Metron 19, 10–21
Barnett, V. (1976) The ordering of multivariate data. Journal of the Royal Statistical Society A, 139(3), 318–355
Barral Souto, J. (1938) El modo y otras medias, casos particulares de una misma expresión matemática. Cuadernos de Trabajo No. 3, Instituto de Biometria, Universidad Nacional de Buenos Aires, Argentina
Brox, T., Weickert, J. (2002) Nonlinear matrix diffusion for optic flow estimation. In: Van Gool, L. (Ed.) Pattern Recognition. Lecture Notes in Computer Science, Vol. 2449, Springer, Berlin, 446–453
Coulon, O., Alexander, D. C., Arridge, S. A. (2001) A regularization scheme for diffusion tensor magnetic resonance images. In: Insana, M. F., Leahy, R. M. (Eds.) Information Processing in Medical Imaging — IPMI 2001. Lecture Notes in Computer Science, Vol. 2082, Springer, Berlin, 92–105
Dougherty, E. R., Astola, J. (Eds.) (1999) Nonlinear Filters for Image Processing. SPIE Press, Bellingham
Guichard, F., Morel, J.-M. (1997) Partial differential equations and image iterative filtering. In: Duff, I. S., Watson, G. A. (Eds.) The State of the Art in Numerical Analysis, no. 63 IMA Conference Series (New Series), Clarendon Press, Oxford, 525–562
Hahn, K., Pigarin, S., Pütz, B. (2001) Edge preserving regularization and tracking for diffusion tensor imaging. In: Niessen, W. J., Viergever, M. A. (Eds.) Medical Image Computing and Computer-Assisted Intervention — MICCAI 2001. Lecture Notes in Computer Science, Vol. 2208, Springer, Berlin, 195–203
Klette, R., Zamperoni, R. (1996) Handbook of Image Processing Operators. Wiley, New York
Koschan, A., Abidi, M. (2001) A comparison of median filter techniques for noise removal in color images. In: Proc. Seventh German Workshop on Color Image Processing, Erlangen, Germany, 69–79
Parker, G. J. M., Schnabel, J. A., Symms, M. R., Werring, D. J., Barker, G. J. (2000) Nonlinear smoothing for reduction of systematic and random errors in diffusion tensor imaging. Journal of Magnetic Resonance Imaging 11, 702–710
Poupon, C., Mangin, J., Frouin, V., Régis, J., Poupon, F., Pachot-Clouard, M., Le Bihan, D., Bloch, I. (1998) Regularization of MR diffusion tensor maps for tracking brain white matter bundles. In: Wells, W. M., Colchester, A., Delp, S. (Eds.) Medical Image Computing and Computer-Assisted Intervention — MICCAI 1998. Lecture Notes in Computer Science, Vol. 1496, Springer, Berlin, 489–498
Torroba, P. L., Cap, N. L., Rabal, H. J., Furlan, W. D. (1994) Fractional order mean in image processing. Optical Engineering 33(2), 528–534
Tschumperlé, D., Deriche, R. (2001) Diffusion tensor regularization with constraints preservation. In: Proc. 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Vol. 1, Kauai, HI, Dec. 2001. IEEE Computer Society Press, 948–953
Tukey, J. W. (1971) Exploratory Data Analysis. Addison-Wesley, Menlo Park
Weickert, J., Brox, T. (2002) Diffusion and regularization of vector-and matrix-valued images. In: Nashed, M. Z., Scherzer, O. (Eds.) Inverse Problems, Image Analysis, and Medical Imaging. Contemporary Mathematics, Vol. 313, AMS, Providence, 251–268
Welk, M., Becker, F., Schnörr, C., Weickert, J. (2005) Matrix-valued filters as convex programs. In Kimmel, R., Sochen, N., Weickert, J. (Eds.) Scale-Space and PDE Methods in Computer Vision. Lecture Notes in Computer Science, Springer, Berlin, to appear.
Welk, M., Feddern, C., Burgeth, B., Weickert, J. (2003) Median filtering of tensor-valued images. In: Michaelis, B., Krell, G. (Eds.) Pattern Recognition. Lecture Notes in Computer Science, Vol. 2781, Springer, Berlin, 17–24
Westin, C., Maier, S. E., Khidhir, B., Everett, P., Jolesz, F. A., Kikinis, R. (1999) Image processing for diffusion tensor magnetic resonance imaging. In: Taylor, C., Colchester, A. (Eds.) Medical Image Computing and Computer-Assisted Intervention — MICCAI 1999. Lecture Notes in Computer Science, Vol. 1679, Springer, Berlin, 441–452
Winkler, G., Aurich, V., Hahn, K., Martin, A. (1999) Noise reduction in images: some recent edge-preserving methods. Pattern Recognition and Image Analysis 9(4), 749–766
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Welk, M., Feddern, C., Burgeth, B., Weickert, J. (2006). Tensor Median Filtering and M-Smoothing. 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_21
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DOI: https://doi.org/10.1007/3-540-31272-2_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25032-6
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