# American Institute of Mathematical Sciences

June  2012, 5(3): 581-590. doi: 10.3934/dcdss.2012.5.581

## A family of nonlinear diffusions connecting Perona-Malik to standard diffusion

 1 Department of Mathematics, University of California, 340 Rowland Hall, Irvine, CA 92697-3975, United States

Received  September 2010 Published  October 2011

A one parameter family of equations is considered which connects the well-known Perona-Malik equation to standard diffusion. The parameter acts as a regularization parameter which gradually modifies the ill-posed Perona-Malik equation, through a strongly locally well-posed equation, to a strongly globally well-posed one which exhibits a behavior akin to that of standard diffusion, which is itself obtained in the limit. In the locally well-posed regime, the equation is degenerate parabolic and the onset of singularities can not be ruled out. Using a classical regularization approach, a-priori estimates can be derived which allow to go to limit with the regularization parameter globally in time. It is, however, not clear how to obtain a proper definition of weak solution for the limiting equation of interest.
Citation: Patrick Guidotti. A family of nonlinear diffusions connecting Perona-Malik to standard diffusion. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 581-590. doi: 10.3934/dcdss.2012.5.581
##### References:
 [1] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions Pattern Anal. Machine Intelligence, 12 (1990), 161-192. [2] S. Kichenassamy, The Perona-Malik paradox, SIAM J. Appl. Math., 57 (1997), 1328-1342. doi: 10.1137/S003613999529558X. [3] B. Kawohl and N. Kutev, Maximum and comparison principle for one-dimensional anisotropic diffusion, Math. Ann., 311 (1998), 107-123. doi: 10.1007/s002080050179. [4] P. Guidotti, A new nonlocal nonlinear diffusion of image processing, Journal of Differential Equations, 246 (2009), 4731-4742. doi: 10.1016/j.jde.2009.03.017. [5] P. Guidotti, A new well-posed nonlinear nonlocal diffusion, Nonlinear Analysis, 72 (2010), 4625-4637. doi: 10.1016/j.na.2010.02.040. [6] M. Ghisi and M. Gobbino, An example of global transcritical solution for the Perona-Malik equation, Communications in Partial Differential Equations, 36 (2011), 1318-1352. doi: 10.1080/03605302.2010.542672. [7] F. Catté, P.-L. Lions, J.-M. Morel and T. Coll, Image selective smoothing and edge-detection by non-linear diffusion, SIAM J. Numer. Anal., 29 (1992), 182-193. [8] G.-H. Cottet and M. E. Ayyadi, A volterra type model for image processing, IEEE Trans. Image Processing, 7 (1998), 292-303. doi: 10.1109/83.661179. [9] F. Weber, On products on non-commuting sectorial operators, Annali della Scuola Normale Superiore di Pisa Cl. Sci. (4), 27 (1998), 499-531. [10] A. Belahmidi, "Equations aux Dérivées Partielles Appliquées à la Restoration et à l'Agrandissement des Images," Ph.D. Thesis, Université Paris-Dauphine, Paris, 2003. [11] A. Belahmidi and A. Chambolle, Time-delay regularization of anisotropic diffusion and image processing, M2AN Math. Model. Numer. Anal., 39 (2005), 231-251. [12] H. Amann, Time-delayed Perona-Malik problems, Acta Math. Univ. Comenianae (N.S.), 76 (2007), 15-38. [13] P. Guidotti and J. Lambers, Two new nonlinear nonlocal diffusions for noise reduction, Journal of Mathematical Imaging and Vision, 33 (2009), 25-37. doi: 10.1007/s10851-008-0108-z. [14] Y. You, W. Xu, A. Tannenbaum and M. Kaveh, Behavioral analysis of anisotropic diffusion in image processing, IEEE Transaction on Image Processing, 5 (1996), 1539-1553. [15] L. Grafakos, "Classical and Modern Fourier Analysis," Pearson Education, Inc., Upper Saddle River, NJ, 2004. [16] S. Taheri, Q. Tang and K. Zhang, Young measure solutions and instability of the one-dimensional Perona-Malik equation, J. Math. Anal. Appl., 308 (2005), 467-490. doi: 10.1016/j.jmaa.2004.11.034. [17] K. Zhang, Existence of infinitely many solutions for the one-dimensional Perona-Malik model, Calc. Var. Partial Differential Equations, 26 (2006), 171-199. [18] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.

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##### References:
 [1] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions Pattern Anal. Machine Intelligence, 12 (1990), 161-192. [2] S. Kichenassamy, The Perona-Malik paradox, SIAM J. Appl. Math., 57 (1997), 1328-1342. doi: 10.1137/S003613999529558X. [3] B. Kawohl and N. Kutev, Maximum and comparison principle for one-dimensional anisotropic diffusion, Math. Ann., 311 (1998), 107-123. doi: 10.1007/s002080050179. [4] P. Guidotti, A new nonlocal nonlinear diffusion of image processing, Journal of Differential Equations, 246 (2009), 4731-4742. doi: 10.1016/j.jde.2009.03.017. [5] P. Guidotti, A new well-posed nonlinear nonlocal diffusion, Nonlinear Analysis, 72 (2010), 4625-4637. doi: 10.1016/j.na.2010.02.040. [6] M. Ghisi and M. Gobbino, An example of global transcritical solution for the Perona-Malik equation, Communications in Partial Differential Equations, 36 (2011), 1318-1352. doi: 10.1080/03605302.2010.542672. [7] F. Catté, P.-L. Lions, J.-M. Morel and T. Coll, Image selective smoothing and edge-detection by non-linear diffusion, SIAM J. Numer. Anal., 29 (1992), 182-193. [8] G.-H. Cottet and M. E. Ayyadi, A volterra type model for image processing, IEEE Trans. Image Processing, 7 (1998), 292-303. doi: 10.1109/83.661179. [9] F. Weber, On products on non-commuting sectorial operators, Annali della Scuola Normale Superiore di Pisa Cl. Sci. (4), 27 (1998), 499-531. [10] A. Belahmidi, "Equations aux Dérivées Partielles Appliquées à la Restoration et à l'Agrandissement des Images," Ph.D. Thesis, Université Paris-Dauphine, Paris, 2003. [11] A. Belahmidi and A. Chambolle, Time-delay regularization of anisotropic diffusion and image processing, M2AN Math. Model. Numer. Anal., 39 (2005), 231-251. [12] H. Amann, Time-delayed Perona-Malik problems, Acta Math. Univ. Comenianae (N.S.), 76 (2007), 15-38. [13] P. Guidotti and J. Lambers, Two new nonlinear nonlocal diffusions for noise reduction, Journal of Mathematical Imaging and Vision, 33 (2009), 25-37. doi: 10.1007/s10851-008-0108-z. [14] Y. You, W. Xu, A. Tannenbaum and M. Kaveh, Behavioral analysis of anisotropic diffusion in image processing, IEEE Transaction on Image Processing, 5 (1996), 1539-1553. [15] L. Grafakos, "Classical and Modern Fourier Analysis," Pearson Education, Inc., Upper Saddle River, NJ, 2004. [16] S. Taheri, Q. Tang and K. Zhang, Young measure solutions and instability of the one-dimensional Perona-Malik equation, J. Math. Anal. Appl., 308 (2005), 467-490. doi: 10.1016/j.jmaa.2004.11.034. [17] K. Zhang, Existence of infinitely many solutions for the one-dimensional Perona-Malik model, Calc. Var. Partial Differential Equations, 26 (2006), 171-199. [18] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.
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