On this paper, we focus on a numerical method for image processing. After recalling image processing modeling by the heat equation, we prove that a better model can be given by a nonlinear problem, namely the p-laplacian. We prove the existence and uniqueness of the solution of the nonlinear problem by variational method. Using Matlab, we build a numerical algorithm which permits to reconstruct digital images altered by noise.

image processing, p-laplacian, variational method, numerical images.

Received: November 24, 2020; Accepted: December 13, 2020; Published: January 19, 2021

How to cite this article: Serigne Diouf, Mamadou M. Diop and Alassane Sy, Numerical Imaging Using Nonlinear Optimisation, International Journal of Numerical Methods and Applications 20(1) (2021), 1-16. DOI: 10.17654/NM020010001

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References: 

[1] R. P. Agarwal, M. E. Philippakis, D. O’Regan and N. S. Papageorgiou, Constant Sign and Nodal Solutions for Problems with the p-laplacian and a nonsmooth potential using variational techniques, Bound Value Probl. 2009, 820237 (2009). https://doi.org/10.1155/2009/820237
[2] G. Aubert and J. F. Aujol, Optimal partitions, regularized solutions, and application to image classification, Applicable Analysis 84(1) (2005), 15-35.
[3] J.-F. Aujol, G. Aubert and L. Blanc-Féraud, Wavelet-based level set evolution for classification of textured images, IEEE Trans. on Image Processing 12(12) (2003), 1634-1641.
[4] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Springer Verlag, Applied Mathematical Sciences, Vol. 147, 2001.
[5] G. Aubert and L. Vese, A variational method in image recovery, SIAM Journal of Numerical Analysis 34(5) (1997), 1948-1979.
[6] H. Bréziz, Analyse Fonctionnelle: Théorie et application, Masson, 1987.
[7] J. J. Koenderink, The structure of images, Biol. Cybrem. 50 (1984), 363-370.
[8] J. Malik and P. Perona, A sacle space and edge detection using anisotropic diffusion, Proc IEEE Computer Soc. Workshop on Computer Vision, 1987.
[9] S. Masnou and J. M. Morel, Level lines based discocclusion, Proc. of Intl. Conf. Imag. Proc., 1998.
[10] J. M. Morel and S. Solimini, Variational Methods in Image Segmentation, Birkhäuser, 1994.
[11] D. Mumford and J. Shah, Optimal Approximations by Piecewise Smooth Functions and Associated Variation Problems Communications on Pures and Applied Mathematics, Vol. XLII N, 1989.
[12] N. K. Nordström, Variational Edge Detection, PH.D. Disertation, Department of Electrical Engineering, University of California at Berkeley, 1990.
[13] N. Paragios and R. Deriche, Geodesic active regions and level set methods for supervised texture segmentation, Int. Jour. Computer Vision 46(3) (2002), 223-247.
[14] S. Oruganti and R. Shivanji, Existence Result for Class of P-Laplacian Semipositone Equations Boundary Value Problems, Vol. 2006, 1-7.
[15] A. Sy and D. Seck, Topological optimization with the p-laplacian operator and an application in image processing, Bound Value Probl 2009, 896813 (2009). https://doi.org/10.1155/2009/896813
[16] J. Weickert, Anisotropic diffusion in image processing, Ph.D. Thesis, University of Kaiserslautern, Germany, 1996.
[17] J. Weickert, Efficient image segmentation using partial differential equations and morphology, Pattern Recognition 34(9) (2001), 1813-1824.
[18] Eric Goncalvs da Silva, Methodes et Analyse Numriques, Engineering school, Institut Polytechnique de Grenoble, 2007, pp. 99. cel-00556967.
[19] J.-A., Dsidri, Analyse des érreurs d’approximation dans les problèmes classiques d’EDP pour la Mcanique des milieux continus, 2012.
[20] G. Faccanoni, EDP linaires, Aide-mmoire, 2016.