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
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