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 & 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.   Google Scholar

[2]

S. Kichenassamy, The Perona-Malik paradox,, SIAM J. Appl. Math., 57 (1997), 1328.  doi: 10.1137/S003613999529558X.  Google Scholar

[3]

B. Kawohl and N. Kutev, Maximum and comparison principle for one-dimensional anisotropic diffusion,, Math. Ann., 311 (1998), 107.  doi: 10.1007/s002080050179.  Google Scholar

[4]

P. Guidotti, A new nonlocal nonlinear diffusion of image processing,, Journal of Differential Equations, 246 (2009), 4731.  doi: 10.1016/j.jde.2009.03.017.  Google Scholar

[5]

P. Guidotti, A new well-posed nonlinear nonlocal diffusion,, Nonlinear Analysis, 72 (2010), 4625.  doi: 10.1016/j.na.2010.02.040.  Google Scholar

[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.  doi: 10.1080/03605302.2010.542672.  Google Scholar

[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.   Google Scholar

[8]

G.-H. Cottet and M. E. Ayyadi, A volterra type model for image processing,, IEEE Trans. Image Processing, 7 (1998), 292.  doi: 10.1109/83.661179.  Google Scholar

[9]

F. Weber, On products on non-commuting sectorial operators,, Annali della Scuola Normale Superiore di Pisa Cl. Sci. (4), 27 (1998), 499.   Google Scholar

[10]

A. Belahmidi, "Equations aux Dérivées Partielles Appliquées à la Restoration et à l'Agrandissement des Images,", Ph.D. Thesis, (2003).   Google Scholar

[11]

A. Belahmidi and A. Chambolle, Time-delay regularization of anisotropic diffusion and image processing,, M2AN Math. Model. Numer. Anal., 39 (2005), 231.   Google Scholar

[12]

H. Amann, Time-delayed Perona-Malik problems,, Acta Math. Univ. Comenianae (N.S.), 76 (2007), 15.   Google Scholar

[13]

P. Guidotti and J. Lambers, Two new nonlinear nonlocal diffusions for noise reduction,, Journal of Mathematical Imaging and Vision, 33 (2009), 25.  doi: 10.1007/s10851-008-0108-z.  Google Scholar

[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.   Google Scholar

[15]

L. Grafakos, "Classical and Modern Fourier Analysis,", Pearson Education, (2004).   Google Scholar

[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.  doi: 10.1016/j.jmaa.2004.11.034.  Google Scholar

[17]

K. Zhang, Existence of infinitely many solutions for the one-dimensional Perona-Malik model,, Calc. Var. Partial Differential Equations, 26 (2006), 171.   Google Scholar

[18]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000).   Google Scholar

show all references

References:
[1]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion,, IEEE Transactions Pattern Anal. Machine Intelligence, 12 (1990), 161.   Google Scholar

[2]

S. Kichenassamy, The Perona-Malik paradox,, SIAM J. Appl. Math., 57 (1997), 1328.  doi: 10.1137/S003613999529558X.  Google Scholar

[3]

B. Kawohl and N. Kutev, Maximum and comparison principle for one-dimensional anisotropic diffusion,, Math. Ann., 311 (1998), 107.  doi: 10.1007/s002080050179.  Google Scholar

[4]

P. Guidotti, A new nonlocal nonlinear diffusion of image processing,, Journal of Differential Equations, 246 (2009), 4731.  doi: 10.1016/j.jde.2009.03.017.  Google Scholar

[5]

P. Guidotti, A new well-posed nonlinear nonlocal diffusion,, Nonlinear Analysis, 72 (2010), 4625.  doi: 10.1016/j.na.2010.02.040.  Google Scholar

[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.  doi: 10.1080/03605302.2010.542672.  Google Scholar

[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.   Google Scholar

[8]

G.-H. Cottet and M. E. Ayyadi, A volterra type model for image processing,, IEEE Trans. Image Processing, 7 (1998), 292.  doi: 10.1109/83.661179.  Google Scholar

[9]

F. Weber, On products on non-commuting sectorial operators,, Annali della Scuola Normale Superiore di Pisa Cl. Sci. (4), 27 (1998), 499.   Google Scholar

[10]

A. Belahmidi, "Equations aux Dérivées Partielles Appliquées à la Restoration et à l'Agrandissement des Images,", Ph.D. Thesis, (2003).   Google Scholar

[11]

A. Belahmidi and A. Chambolle, Time-delay regularization of anisotropic diffusion and image processing,, M2AN Math. Model. Numer. Anal., 39 (2005), 231.   Google Scholar

[12]

H. Amann, Time-delayed Perona-Malik problems,, Acta Math. Univ. Comenianae (N.S.), 76 (2007), 15.   Google Scholar

[13]

P. Guidotti and J. Lambers, Two new nonlinear nonlocal diffusions for noise reduction,, Journal of Mathematical Imaging and Vision, 33 (2009), 25.  doi: 10.1007/s10851-008-0108-z.  Google Scholar

[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.   Google Scholar

[15]

L. Grafakos, "Classical and Modern Fourier Analysis,", Pearson Education, (2004).   Google Scholar

[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.  doi: 10.1016/j.jmaa.2004.11.034.  Google Scholar

[17]

K. Zhang, Existence of infinitely many solutions for the one-dimensional Perona-Malik model,, Calc. Var. Partial Differential Equations, 26 (2006), 171.   Google Scholar

[18]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000).   Google Scholar

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