Abstract
The article deals with filtering of data on closed surfaces by using the linear and nonlinear diffusion equations. The linear diffusion filtering is given by the Laplace–Beltrami operator representing linear diffusion along the surface. For the nonlinear diffusion filtering, we introduce nonlinear diffusion equations with diffusion coefficient depending on surface gradient and/or surface Laplacian of solution. This allows adaptive filtering respecting edges and local extrema in the data. For numerical discretization we develop a surface finite-volume method to approximate the partial differential equations on surfaces like sphere, ellipsoid or the Earth surface. The surfaces are approximated by a polyhedral mesh created by planar triangles representing subdivision of an initial icosahedron or octahedron grids. Numerical experiments illustrate behaviour of the linear and nonlinear diffusion filters on testing data and on real measurements, namely the GOCE satellite observations and the satellite-only mean dynamic topography. They show advantages of the nonlinear filters which, on the contrary to the linear one, preserve important structures in processed geodesy data.
Similar content being viewed by others
References
Alvarez, L., Morel, J.M.: Formalization and computational aspects of image analysis. Acta Numer. 3, 1–59 (1994)
Alvarez, L., Guichard, F., Lions, P.L., Morel, J.M.: Axioms and fundamental equations of image processing. Arch. Ration. Mech. Anal. 123, 200–257 (1993)
Andersen, O., Knudsen, P., Stenseng, L: The DTU13 global mean sea surface from 20 years of satellite altimetry. Presented at the IAG Scientific Assembly in Potsdam (2013)
Bruinsma, S.L., Forste, C., Abrikosov, O., Marty, J.C., Rio, M.H., Mulet, S., Bonvalot, S.: The new ESA satellite-only gravity field model via the direct approach. Geophys. Res. Lett. 40, 3607–3612 (2003)
Caselles, V., Morel J.M., Sapiro, G., Tannenbaum, A.: Introduction to the special issue on partial differential equations and geometry-driven diffusion in image processing and analysis, IEEE Trans. Image Process. 7(3), 269–373 (1998)
Catté, F., Lions, P.L., Morel, J.M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29, 182–193 (1992)
Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (NS) 27, 1–67 (1992)
Čunderlík, R., Mikula, K., Tunega, M.: Nonlinear diffusion filtering of data on the Earth’s surface. J. Geodesy 87(2), 143–160 (2012)
Čunderlík, R.: Precise modelling of the static gravity field from GOCE second radial derivatives of the disturbing potential using the method of fundamental solutions. In: International Association of Geodesy Symposia. Springer (2015). doi:10.1007/1345_2015_211
DTU, DTU13MDT_5MIN. https://ftp.space.dtu.dk/pub/DTU13/5_MIN/
Dziuk, G., Elliott, C.M.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27(2), 262–292 (2007)
Dziuk, G., Elliott, C.M.: Finite element methods for surface PDEs. Acta Numer. 22, 289396 (2013)
ESA: GOCE User Toolbox GUT. https://earth.esa.int/web/guest/software-tools/gut/
ESA: Gravity field and steady-state ocean circulation mission. Report for mission selection of the four candidate earth explorer missions, ESA SP-1233(1). ESA Publications Division: ESTEC. Noordwijk, The Netherlands (1999)
Eymard, R., Gallouet, T., Herbin, R.: Finite Volume Methods, Handbook of Numerical Analysis, vol. 7, pp. 713–1020 (2003)
Faure, E. et al.: A workflow to process 3D+time microscopy images of developing organisms and reconstruct their cell lineage. Nat. Commun. 7. doi:10.1038/ncomms9674 (2016) (Article number: 8674)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer, Berlin (1988)
Kichenassamy, S.: The Perona–Malik paradox. SIAM J. Appl. Math. 57(5), 1328–1342 (1997)
Koenderink, J.: The structure of images. Biol. Cybern. 50, 363–370 (1984)
Kriva, Z., Mikula, K., Peyrieras, N., Rizzi, B., Sarti, A., Stasova, O.: 3D early embryogenesis image filtering by nonlinear partial differential equations. Med. Image Anal. 14(4), 510–526 (2010)
Lions, P.L.: Axiomatic derivation of image processing models. Math. Models Methods Appl. Sci. 4, 467–475 (1994)
Mayer-Grr, T., The GOCO Consortium: The New Combined Satellite Only Model GOCO03s. Presented at the GGHS-2012 in Venice, Italy, October 9–12 (2012)
Mikula, K., Ramarosy, N.: Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing. Numer. Math. 89(3), 561–590 (2001)
Nielsen, M., Johansen, P., Olsen, O.F., Weickert, J. (eds.): Scale-Space Theories in Computer Vision. Lecture Notes in Computer Science, vol. 1682. Springer, Berlin (1999)
Nitzberg, M., Shiota, T.: Nonlinear image filtering with edge and corner enhancement. IEEE Trans. Pattern Anal. Mach. Intell. 14(8), 826–833 (1992)
Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, Berlin (2003)
Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Computer Society Workshop on Computer Vision, Proc (1987)
Romeny, B.M.T.H. (ed.): Geometry Driven Diffusion in Computer Vision. Kluwer, Dodrecht (1994)
Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, Cambridge (2001)
Sethian, J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Material Science. Cambridge University Press, New York (1999)
Weickert, J.: Anisotropic Diffusion in Computer Vision. Teubner-Stuttgart, Stuttgart (1998)
Witkin, A.P.: Scale-space filtering. In: Proceedings of Eighth International Conference on Artificial Intelligence, vol. 2, pp. 1019–1022 (1983)
Young, D.M.: Iterative Solution of Large Linear Systems. Academic Press, London (1971)
Acknowledgments
This work was supported by Grants APVV-15-0522, VEGA 1/0608/15 and VEGA 1/0714/15.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Čunderlík, R., Kollár, M. & Mikula, K. Filters for geodesy data based on linear and nonlinear diffusion. Int J Geomath 7, 239–274 (2016). https://doi.org/10.1007/s13137-016-0087-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13137-016-0087-y
Keywords
- Data filtering
- Nonlinear diffusion equation
- Surface finite volume method
- GOCE data
- Satellite-only mean dynamic topography