Abstract
The Laplacian of an image is one of the simplest and useful image processing tools which highlights regions of rapid intensity change and therefore it is applied for edge detection and contrast enhancement. This paper deals with the definition of the Laplacian operator on ultrametric spaces as well as its spectral representation in terms of the corresponding eigenfunctions and eigenvalues. The theory reviewed here provides the computational framework to process images or signals defined on a hierarchical representation associated to an ultrametric space. In particular, image regularization by ultrametric heat kernel filtering and image enhancement by hierarchical Laplacian are illustrated.
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References
Angulo, J., Velasco-Forero, S.: Morphological semigroups and scale-spaces on ultrametric spaces. In: Angulo, J., Velasco-Forero, S., Meyer, F. (eds.) ISMM 2017. LNCS, vol. 10225, pp. 28–39. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-57240-6_3
Bendikov, A., Grigor’yan, A., Pittet, C.: On a class of Markov semigroups on discrete ultra-metric spaces. Potential Anal. 37(2), 125–169 (2012)
Bendikov, A., Grigor’yan, A., Pittet, C., Woess, W.: Isotropic Markov semigroups on ultrametric spaces. Uspekhi Mat. Nauk 69(4), 3–102 (2014)
Bendikov, A., Krupski, P.: On the spectrum of the hierarchical Laplacian. Potential Anal. 41(4), 1247–1266 (2014)
Bendikov, A., Woess, W., Cygan, W.: Oscillating heat kernels on ultrametric spaces. J. Spectr. Theory, arXiv:1610.03292 (2017)
Cousty, J., Najman, L., Kenmochi, Y., Guimarães, S.: Hierarchical segmentations with graphs: quasi-flat zones, minimum spanning trees, and saliency maps. J. Math. Imaging Vis. 60(4), 479–502 (2018)
Khrennikov, A., Kozyrev, S., Zúñiga-Galindo, W.: Ultrametric Pseudo Differential Equations and Applications. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2018)
Lindqvist, P.: Notes on the Infinity Laplace Equation. Springer, Cham (2016)
Marr, D., Hildreth, E.C.: Theory of edge detection. Proc. Roy. Soc. Lond. Seri. B 207, 187–217 (1980)
van Vliet, L.J., Young, I.T., Beckers, G.L.: A nonlinear operator as edge detector in noisy images. Comput. Vis. Graph. Image Process. 45, 167–195 (1989)
Wetzler, A., Aflalo, Y., Dubrovina, A., Kimmel, R.: The Laplace-Beltrami operator: a ubiquitous tool for image and shape processing. In: Hendriks, C.L.L., Borgefors, G., Strand, R. (eds.) ISMM 2013. LNCS, vol. 7883, pp. 302–316. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38294-9_26
Meyer, F.: Hierarchies of partitions and morphological segmentation. In: Kerckhove, M. (ed.) Scale-Space 2001. LNCS, vol. 2106, pp. 161–182. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-47778-0_14
Meyer, F.: Watersheds on weighted graphs. Pattern Recog. Lett. 47, 72–79 (2014)
Peres, Y., Schramm, O., Sheffield, S., Wilson, D.B.: Tug-of-war and the Infinity Laplacian. J. Am. Math. Soc. 22(1), 167–210 (2009)
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Angulo, J. (2019). Hierarchical Laplacian and Its Spectrum in Ultrametric Image Processing. In: Burgeth, B., Kleefeld, A., Naegel, B., Passat, N., Perret, B. (eds) Mathematical Morphology and Its Applications to Signal and Image Processing. ISMM 2019. Lecture Notes in Computer Science(), vol 11564. Springer, Cham. https://doi.org/10.1007/978-3-030-20867-7_3
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DOI: https://doi.org/10.1007/978-3-030-20867-7_3
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