American Institute of Mathematical Sciences

September  2006, 5(3): 617-637. doi: 10.3934/cpaa.2006.5.617

Young measure solutions of the two-dimensional Perona-Malik equation in image processing

 1 College of Resource and Environment, China Agricultural University, Beijing 100094, China 2 School of Mathematical Sciences, University of Sussex, Falmer, Brighton, BN1 9QH

Received  May 2005 Revised  February 2006 Published  June 2006

For a given smooth initial value $u_0$, we construct sequences of approximate solutions $u_j$ in $W^{1,\infty}$ for the well-known Perona-Malik anisotropic diffusion model in image processing defined by $u_t-$ div $[\rho(|\nabla u|^2)\nabla u]=0$ under the homogeneous Neumann condition, where $\rho(|X|^2)X=X/(1+|X|^2)$ for $X\in\mathbb R^2$. The Perona-Malik diffusion equation is of non-coercive forward-backward type. Our constructed approximate solutions satisfy the equation in the sense that $(u_j)_t-$ div$_x [\rho(|\nabla u_j|^2)\nabla u_j]\to 0$ strongly in $W^{-1,p}(Q_T)$ for all $1\leq p<\infty$, where $Q_T=(0,T)\times \Omega$ with $\Omega\subset\mathbb R^2$ the unit square. We also show, for any non-constant initial value $u_0$ that the approximate solutions $u_j$ do not converge to a solution, rather, they converge weakly to Young measure-valued solutions which can be represented partially explicitly. Our main idea is to convert the equation into a differential inclusion problem.
Citation: Yan Chen, Kewei Zhang. Young measure solutions of the two-dimensional Perona-Malik equation in image processing. Communications on Pure & Applied Analysis, 2006, 5 (3) : 617-637. doi: 10.3934/cpaa.2006.5.617
 [1] 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 [2] Francesca Faraci, Antonio Iannizzotto. Three nonzero periodic solutions for a differential inclusion. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 779-788. doi: 10.3934/dcdss.2012.5.779 [3] Alice Fiaschi. Rate-independent phase transitions in elastic materials: A Young-measure approach. Networks & Heterogeneous Media, 2010, 5 (2) : 257-298. doi: 10.3934/nhm.2010.5.257 [4] Alice Fiaschi. Young-measure quasi-static damage evolution: The nonconvex and the brittle cases. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 17-42. doi: 10.3934/dcdss.2013.6.17 [5] Luca Calatroni, Bertram Düring, Carola-Bibiane Schönlieb. ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 931-957. doi: 10.3934/dcds.2014.34.931 [6] Kristian Bredies. Weak solutions of linear degenerate parabolic equations and an application in image processing. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1203-1229. doi: 10.3934/cpaa.2009.8.1203 [7] John B. Greer, Andrea L. Bertozzi. $H^1$ Solutions of a class of fourth order nonlinear equations for image processing. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 349-366. doi: 10.3934/dcds.2004.10.349 [8] Wei Wang, Na Sun, Michael K. Ng. A variational gamma correction model for image contrast enhancement. Inverse Problems & Imaging, 2019, 13 (3) : 461-478. doi: 10.3934/ipi.2019023 [9] Ziqing Yuana, Jianshe Yu. Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion. Communications on Pure & Applied Analysis, 2020, 19 (1) : 391-405. doi: 10.3934/cpaa.2020020 [10] Mieczysław Cichoń, Bianca Satco. On the properties of solutions set for measure driven differential inclusions. Conference Publications, 2015, 2015 (special) : 287-296. doi: 10.3934/proc.2015.0287 [11] Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053 [12] Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515 [13] Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167 [14] Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032 [15] Antonia Chinnì, Roberto Livrea. Multiple solutions for a Neumann-type differential inclusion problem involving the $p(\cdot)$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 753-764. doi: 10.3934/dcdss.2012.5.753 [16] Dina Kalinichenko, Volker Reitmann, Sergey Skopinov. Asymptotic behavior of solutions to a coupled system of Maxwell's equations and a controlled differential inclusion. Conference Publications, 2013, 2013 (special) : 407-414. doi: 10.3934/proc.2013.2013.407 [17] Piotr Bogusław Mucha, Milan Pokorný, Ewelina Zatorska. Approximate solutions to a model of two-component reactive flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1079-1099. doi: 10.3934/dcdss.2014.7.1079 [18] D. Criaco, M. Dolfin, L. Restuccia. Approximate smooth solutions of a mathematical model for the activation and clonal expansion of T cells. Mathematical Biosciences & Engineering, 2013, 10 (1) : 59-73. doi: 10.3934/mbe.2013.10.59 [19] T. Gallouët, J.-C. Latché. Compactness of discrete approximate solutions to parabolic PDEs - Application to a turbulence model. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2371-2391. doi: 10.3934/cpaa.2012.11.2371 [20] David Henry, Octavian G. Mustafa. Existence of solutions for a class of edge wave equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1113-1119. doi: 10.3934/dcdsb.2006.6.1113

2018 Impact Factor: 0.925