Abstract
Relationships between linear and morphological scale-spaces have been considered by various previous works. The aim of this paper is to study how to generalize the diffusion-based approaches in order to introduce nonlinear filters which effects mimic morphological dilation and erosion. A methodology based on the counter-harmonic mean is adopted here. Details of numerical implementation are discussed and results are provided to illustrate the behaviour of various studied cases: isotropic, nonlinear and coherence-enhanced diffusion. We also rediscover the classical link between Gaussian scale-space and dilation/erosion scale-spaces based on quadratic structuring functions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alvarez, L., Guichard, F., Lions, P.-L., Morel, J.-M.: Axioms and fundamental equations of image processing. Arch. for Rational Mechanics 123(3), 199–257 (1993)
Arehart, A.B., Vincent, L., Kimia, B.B.: Mathematical morphology: The Hamilton-Jacobi connection. In: Proc. of IEEE 4th Inter. Conf. on Computer Vision (ICCV 1993), pp. 215–219 (1993)
van den Boomgaard, R., Dorst, L.: The morphological equivalent of Gaussian scale-space. In: Proc. of Gaussian Scale-Space Theory, pp. 203–220. Kluwer, Dordrecht (1997)
van den Boomgaard, R.: Numerical solution schemes for continuous-scale morphology. In: Nielsen, M., Johansen, P., Fogh Olsen, O., Weickert, J. (eds.) Scale-Space 1999. LNCS, vol. 1682, pp. 199–210. Springer, Heidelberg (1999)
van den Boomgaard, R.: Decomposition of the Kuwahara-Nagao operator in terms of a linear smoothing and a morphological sharpening. In: Proc. of ISMM 2002, pp. 283–292. CSIRO Publishing (2002)
Breuß, M., Weickert, J.: Highly accurate PDE-based morphology for general structuring elements. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) Scale Space and Variational Methods in Computer Vision. LNCS, vol. 5567, pp. 758–769. Springer, Heidelberg (2009)
Brockett, R.W., Maragos, P.: Evolution equations for continuous-scale morphology. IEEE Trans. on Signal Processing 42(12), 3377–3386 (1994)
Bullen, P.S.: Handbook of Means and Their Inequalities, 2nd edn. Springer, Heidelberg (1987)
Burgeth, B., Weickert, J.: An Explanation for the Logarithmic Connection between Linear and Morphological System Theory. International Journal of Computer Vision 64(2-3), 157–169 (2005)
Catte, F., Lions, P.-L., Morel, J.-M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM Journal on Numerical Analysis 29(1), 182–193 (1992)
Dorst, L., van den Boomgaard, R.: Morphological Signal Processing and the Slope Transform. Signal Processing 38, 79–98 (1994)
Florack, L.: Image Structure. Kluwer Academic Publishers, Dordrecht (1997)
Florack, L., Maas, R., Niessen, W.: Pseudo-Linear Scale-Space Theory. International Journal of Computer Vision 31(2-3), 1–13 (1999)
Gonzalez, R.C., Woods, R.E.: Digital Image Processing. Addison-Wesley, Boston (1992)
Lindeberg, T.: Scale-Space Theory in Computer Vision. Kluwer Academic Publishers, Dordrecht (1994)
Maragos, P.: Slope Transforms: Theory and Application to Nonlinear Signal Processing. IEEE Trans. on Signal Processing 43(4), 864–877 (1995)
Maragos, P.: Differential morphology and image processing. IEEE Transactions on Image Processing 5(1), 922–937 (1996)
Perona, P., Malik, J.: Scale-Space and Edge Detection Using Anisotropic Diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)
Sapiro, G., Kimmel, R., Shaked, D., Kimia, B.B., Bruckstein, A.M.: Implementing continuous-scale morphology via curve evolution. Pattern recognition 26(9), 1363–1372 (1993)
Serra, J.: Image Analysis and Mathematical Morphology. Vol I, and Image Analysis and Mathematical Morphology. Vol II: Theoretical Advances. Academic Press, London (1982/1988)
Soille, P.: Morphological Image Analysis. Springer, Berlin (1999)
van Vliet, L.J.: Robust Local Max-Min Filters by Normalized Power-Weighted Filtering. In: Proc. of IEEE 17th International Conference of the Pattern Recognition (ICPR 2004), vol. 1, pp. 696–699 (2004)
Weickert, J.: Anisotropic Diffusion in Image Processing. ECMI Series. Teubner-Verlag, Stuttgart (1998)
Weickert, J.: Coherence-Enhancing Diffusion Filtering. Int. J. Comput. Vision 31(2-3), 111–127 (1999)
Weickert, J.: Efficient image segmentation using partial differential equations and morphology. Pattern Recognition 31(9), 1813–1824 (2001)
Welk, M.: Families of generalised morphological scale spaces. In: Griffin, L.D., Lillholm, M. (eds.) Scale-Space 2003. LNCS, vol. 2695, pp. 770–784. Springer, Heidelberg (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Angulo, J. (2010). Pseudo-morphological Image Diffusion Using the Counter-Harmonic Paradigm. In: Blanc-Talon, J., Bone, D., Philips, W., Popescu, D., Scheunders, P. (eds) Advanced Concepts for Intelligent Vision Systems. ACIVS 2010. Lecture Notes in Computer Science, vol 6474. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17688-3_40
Download citation
DOI: https://doi.org/10.1007/978-3-642-17688-3_40
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17687-6
Online ISBN: 978-3-642-17688-3
eBook Packages: Computer ScienceComputer Science (R0)