# Nonlinear diffusion filtering influenced by mean curvature

## Main Article Content

## Abstract

The paper introduces a new nonlinear diffusion filtering method on closed surfaces such as a sphere, ellipsoid or the Earth's surface. Our new model extends the regularized surface Perona-Malik model by including a local extrema detector based on a mean curvature of processed data. The model is thus represented by a nonlinear diffusion equation which filters noise while preserves main edges, local extrema and details important for a correct interpretation of data. We define a surface finite-volume method to approximate numerically the nonlinear parabolic partial differential equation on a closed surface. The closed surface is approximated by a polyhedral surface created by planar triangles representing subdivision of an initial icosahedron grid and we use a piece-wise linear approximation of a solution in space and the backward Euler time discretization. Numerical experiments present nonlinear diffusion filtering of artificial data and real measurements, namely the GOCE satellite observations. They aim to point out a main advantage of the new nonlinear model which, on the contrary of Perona-Malik model, preserves local extremal values of filtered data.

## Article Details

How to Cite

KOLLÁR, Michal; MIKULA, Karol; ČUNDERLÍK, Róbert.
Nonlinear diffusion filtering influenced by mean curvature.

**Proceedings of the Conference Algoritmy**, [S.l.], p. 183-193, feb. 2016. Available at: <http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/407>. Date accessed: 22 sep. 2017.
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Articles

## References

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[2] Catté F, Lions PL, Morel JM, Coll T, Image selective smoothing and edge detection by nonlinear diffusion., SIAM J Numer Anal 29. 1992, p. 182193

[3] Čunderlík R, Mikula K, Tunega M, Nonlinear diffusion filtering of data on the Earth’s surface, Journal of Geodesy. 2012, p. 143-160, ISSN 0949-7714

[4] 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, DOI 10.1007/1345 2015 211, Springer (in press)

[5] 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, 1999, ESTEC, Noordwijk, The Netherlands.

[6] Eymard R, Gallouet T, Herbin R, Finite Volume Methods, Handbook of Numerical Analysis P.G. Ciarlet, 1997, J.L. Lions eds, vol 7, pp 713-1020

[7] Perona P, Malik J, Scale space and edge detection using anisotropic diffusion., Proceedings of the IEEE society workshop on computer vision. 1987