Abstract
Implementation of nonlinear flows by explicit schemes can be very convenient, due to their simplicity and low-computational cost per time step. A well known drawback is the small time step bound, referred to as the CFL condition, which ensures a stable flow. For p-Laplacian flows, with \(1< p <2\), explicit schemes without gradient regularization require, in principle, a time step approaching zero. However, numerical implementations show explicit flows with small time-steps are well behaved. We can now explain and quantify this phenomenon.
In this paper we examine explicit p-Laplacian flows by analyzing the evolution of nonlinear eigenfunctions, with respect to the p-Laplacian operator. For these cases analytic solutions can be formulated, allowing for a comprehensive analysis. A generalized CFL condition is presented, relating the time step to the inverse of the nonlinear eigenvalue. Moreover, we show that the flow converges and formulate a bound on the error of the discrete scheme. Finally, we examine general initial conditions and propose a dynamic time-step bound, which is based on a nonlinear Rayleigh quotient.
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Andreu, F., Ballester, C., Caselles, V., Mazón, J.M., et al.: Minimizing total variation flow. Differ. Integr. Eqn. 14(3), 321–360 (2001)
Baravdish, G., Svensson, O., Åström, F.: On backward p (x)-parabolic equations for image enhancement. Numer. Funct. Anal. Optim. 36(2), 147–168 (2015)
Burger, M., Gilboa, G., Moeller, M., Eckardt, L., Cremers, D.: Spectral decompositions using one-homogeneous functionals. SIAM J. Imaging Sci. 9(3), 1374–1408 (2016)
Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1–2), 89–97 (2004)
Chen, J., Guo, J.: Image restoration based on adaptive p-Laplace diffusion. In: 3rd International Congress on Image and Signal Processing, vol. 1, pp. 143–146. IEEE (2010)
Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006)
Cohen, I., Gilboa, G.: Shape preserving flows and the p-Laplacian spectra, HAL preprint hal-01870019
Cohen, I., Gilboa, G.: Energy dissipating flows for solving nonlinear eigenpair problems. J. Comput. Phys. 375, 1138–1158 (2018)
Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen differenzengleichungen der mathematischen physik. Math. Ann. 100(1), 32–74 (1928)
Crank, J., Nicolson, P.: A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Adv. Comput. Math. 6(1), 207–226 (1996)
García Azorero, J., Peral Alonso, I.: Existence and nonuniqueness for the p-Laplacian. Commun. Partial. Differ. Equ. 12(12), 126–202 (1987)
Gawronska, E., Sczygiol, N.: Relationship between eigenvalues and size of time step in computer simulation of thermomechanics phenomena. In: Proceedings of the International MultiConference of Engineers and Computer Scientists, vol. 2 (2014)
Gilboa, G.: A total variation spectral framework for scale and texture analysis. SIAM J. Imaging Sci. 7(4), 1937–1961 (2014)
Grewenig, S., Weickert, J., Bruhn, A.: From box filtering to fast explicit diffusion. In: Goesele, M., Roth, S., Kuijper, A., Schiele, B., Schindler, K. (eds.) DAGM 2010. LNCS, vol. 6376, pp. 533–542. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15986-2_54
Hafiene, Y., Fadili, J., Elmoataz, A.: Nonlocal p-Laplacian evolution problems on graphs. SIAM J. Numer. Anal. 56(2), 1064–1090 (2018)
Huang, C., Zeng, L.: Level set evolution model for image segmentation based on variable exponent p-Laplace equation. Appl. Math. Model. 40(17–18), 7739–7750 (2016)
Iserles, A.: A First Course in the Numerical Analysis of Differential Equations. No. 44. Cambridge University Press, Cambridge (2009)
Kuijper, A.: Image processing by minimising \(l^p\) norms. Pattern Recogn. Image Anal. 23(2), 226–235 (2013)
Kuijper, A.: p-Laplacian driven image processing. In: IEEE International Conference on Image Processing, ICIP 2007, vol. 5, p. V-257. IEEE (2007)
Kuijper, A.: Geometrical PDEs based on second-order derivatives of gauge coordinates in image processing. Image Vis. Comput. 27(8), 1023–1034 (2009)
Lax, P.D., Richtmyer, R.D.: Survey of the stability of linear finite difference equations. Commun. Pure Appl. Math. 9(2), 267–293 (1956)
Liu, Q., Guo, Z., Wang, C.: Renormalized solutions to a reaction-diffusion system applied to image denoising. DCDS-B 21(6), 1839–1858 (2016)
Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. PAMI 12(7), 629–639 (1990)
Wei, W., Zhou, B.: A p-Laplace equation model for image denoising. Inform. Technol. J. 11, 632–636 (2012)
Weickert, J.: Anisotropic Diffusion in Image Processing, vol. 1. Teubner, Stuttgart (1998)
Widrow, B., McCool, J.M., Larimore, M.G., Johnson, C.R.: Stationary and nonstationary learning characteristics of the LMS adaptive filter. Proc. IEEE 64(8), 1151–1162 (1976)
Acknowledgements
This project is supported by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 777826. We acknowledge support by the Israel Science Foundation (grant No. 718/15). This work was supported by the Technion Ollendorff Minerva Center.
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Cohen, I., Falik, A., Gilboa, G. (2019). Stable Explicit p-Laplacian Flows Based on Nonlinear Eigenvalue Analysis. In: Lellmann, J., Burger, M., Modersitzki, J. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2019. Lecture Notes in Computer Science(), vol 11603. Springer, Cham. https://doi.org/10.1007/978-3-030-22368-7_25
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DOI: https://doi.org/10.1007/978-3-030-22368-7_25
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