Time-delay Regularization of Anisotropic Diffusion and Image Processing

We study a time-delay regularization of the anisotropic diffusion model for image denoising of Perona and Malik [IEEE Trans. Pattern Anal. Mach. Intell 12 (1990) 629–639], which has been proposed by Nitzberg and Shiota [IEEE Trans. Pattern Anal. Mach. Intell 14 (1998) 826–835]. In the two-dimensional case, we show the convergence of a numerical approximation and the existence of a weak solution. Finally, we show some experiments on images. Mathematics Subject Classification. 68U10, 35K55, 35M10. Received: March 2, 2004. Introduction In a well-known paper, Perona and Malik [18] have proposed a model for image restoration based on the following partial differential equation: ∂u ∂t = div ( g(|Du|2)Du u(·, 0) = u0. (1) Here u0 is the grey level intensity of the original image, u(·, t) is the restored version, that depends on the scale parameter t, and g is a smooth non-increasing positive function with g(0) = 1 and sg(s) → 0 at infinity. The main idea is that the restoration process obtained by the equation is conditional: if x is an edge point, where the gradient is large, then the diffusion will be stopped and therefore the edge will be kept. If x is in a homogeneous area, the gradient has to be small, and the diffusion will tend to smooth around x. By introducing an edge stopping function g(|Du|2) in the diffusion process, the model has been considered as an important improvement of the theory of edge detection [15]. The experiments of Perona and Malik were very impressive, edges remained stable over a very long time. In their paper [18], they claim that edge detection based on this process clearly outperforms the Canny edge detector [3]. Unfortunately, the Perona-Malik model is ill-posed. Indeed, among the functions which Perona and Malik advocate in their papers, we find g(s) = 1/(1+ s) or g(s) = e−s 2 for which no correct theory of equation (1)