Abstract
In order to describe anisotropy in image processing models or physical measurements, matrix fields are a suitable choice. In diffusion tensor magnetic resonance imaging (DT-MRI), for example, information about the diffusive properties of water molecules is captured in symmetric positive definite matrices. The corresponding matrix field reflects the structure of the tissue under examination. Recently, morphological partial differential equations (PDEs) for dilation and erosion known for grey scale images have been extended to matrix-valued data.
In this article we consider an adaptive, PDE-driven dilation process for matrix fields. The anisotropic morphological evolution is steered with a matrix constructed from a structure tensor for matrix valued data. An important novel ingredient is a directional variant of the matrix-valued Rouy-Tourin scheme that enables our method to complete or enhance anisotropic structures effectively. Experiments with synthetic and real-world data substantiate the gap-closing and line-completing properties of the proposed method.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Matheron, G.: Eléments pour une théorie des milieux poreux. Masson, Paris (1967)
Serra, J.: Echantillonnage et estimation des phénomènes de transition minier. PhD thesis, University of Nancy, France (1967)
Matheron, G.: Random Sets and Integral Geometry. Wiley, New York (1975)
Serra, J.: Image Analysis and Mathematical Morphology, vol. 1. Academic Press, London (1982)
Serra, J.: Image Analysis and Mathematical Morphology, vol. 2. Academic Press, London (1988)
Soille, P.: Morphological Image Analysis, 2nd edn. Springer, Berlin (2003)
Alvarez, L., Guichard, F., Lions, P.L., Morel, J.M.: Axioms and fundamental equations in image processing. Archive for Rational Mechanics and Analysis 123, 199–257 (1993)
Arehart, A.B., Vincent, L., Kimia, B.B.: Mathematical morphology: The Hamilton–Jacobi connection. In: Proc. Fourth International Conference on Computer Vision, Berlin, pp. 215–219. IEEE Computer Society Press, Los Alamitos (1993)
Brockett, R.W., Maragos, P.: Evolution equations for continuous-scale morphological filtering. IEEE Transactions on Signal Processing 42, 3377–3386 (1994)
Sapiro, G., Kimmel, R., Shaked, D., Kimia, B.B., Bruckstein, A.M.: Implementing continuous-scale morphology via curve evolution. Pattern Recognition 26, 1363–1372 (1993)
van den Boomgaard, R.: Mathematical Morphology: Extensions Towards Computer Vision. PhD thesis, University of Amsterdam, The Netherlands (1992)
Breuß, M., Weickert, J.: A shock-capturing algorithm for the differential equations of dilation and erosion. Journal of Mathematical Imaging and Vision 25(2), 187–201 (2006)
Verdú-Monedero, R., Angulo, J.: Spatially-variant directional mathematical morphology operators based on a diffused average squared gradient field. In: Blanc-Talon, J., Bourennane, S., Philips, W., Popescu, D., Scheunders, P. (eds.) ACIVS 2008. LNCS, vol. 5259, pp. 542–553. Springer, Heidelberg (2008)
Breuß, M., Burgeth, B., Weickert, J.: Anisotropic continuous-scale morphology. In: Martí, J., Benedí, J.M., Mendonça, A., Serrat, J. (eds.) IbPRIA 2007. LNCS, vol. 4478, pp. 515–522. Springer, Heidelberg (2007)
Burgeth, B., Breuß, M., Pizarro, L., Weickert, J.: PDE-driven adaptive morphology for matrix fields. In: Tai, X.-C., et al. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 247–258. Springer, Heidelberg (2009)
Burgeth, B., Didas, S., Florack, L., Weickert, J.: A generic approach to diffusion filtering of matrix-fields. Computing 81, 179–197 (2007)
Rouy, E., Tourin, A.: A viscosity solutions approach to shape-from-shading. SIAM Journal on Numerical Analysis 29, 867–884 (1992)
Förstner, W., Gülch, E.: A fast operator for detection and precise location of distinct points, corners and centres of circular features. In: Proc. ISPRS Intercommission Conference on Fast Processing of Photogrammetric Data, Interlaken, Switzerland, June 1987, pp. 281–305 (1987)
Bigün, J., Granlund, G.H., Wiklund, J.: Multidimensional orientation estimation with applications to texture analysis and optical flow. IEEE Transactions on Pattern Analysis and Machine Intelligence 13(8), 775–790 (1991)
Bigün, J.: Vision with Direction. Springer, Berlin (2006)
Brox, T., Weickert, J., Burgeth, B., Mrázek, P.: Nonlinear structure tensors. Image and Vision Computing 24(1), 41–55 (2006)
Feddern, C., Weickert, J., Burgeth, B., Welk, M.: Curvature-driven PDE methods for matrix-valued images. International Journal of Computer Vision 69(1), 91–103 (2006)
Di Zenzo, S.: A note on the gradient of a multi-image. Computer Vision, Graphics and Image Processing 33, 116–125 (1986)
Burgeth, B., Didas, S., Weickert, J.: A general structure tensor concept and coherence-enhancing diffusion filtering for matrix fields. In: Laidlaw, D., Weickert, J. (eds.): Visualization and Processing of Tensor Fields. Mathematics and Visualization, pp. 305–323. Springer, Heidelberg (2009)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)
Burgeth, B., Bruhn, A., Didas, S., Weickert, J., Welk, M.: Morphology for matrix-data: Ordering versus PDE-based approach. Image and Vision Computing 25(4), 496–511 (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pizarro, L., Burgeth, B., Breuß, M., Weickert, J. (2009). A Directional Rouy-Tourin Scheme for Adaptive Matrix-Valued Morphology. In: Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds) Mathematical Morphology and Its Application to Signal and Image Processing. ISMM 2009. Lecture Notes in Computer Science, vol 5720. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03613-2_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-03613-2_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03612-5
Online ISBN: 978-3-642-03613-2
eBook Packages: Computer ScienceComputer Science (R0)