COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, Vol.5(2005), No.2, pp.170–200 c  2005 Institute of Mathematics of the National Academy of Sciences of Belarus EXPLICIT FV SCHEME FOR THE PERONA—MALIK EQUATION ZUZANA KRIV ´ A Department of Mathematics, Slovak University of Technology Radlinsk´ eho 11, 813 68 Bratislava, Slovakia Abstract — The Perona—Malik equation for nonlinear diffusion is the well-known equation widely used in image processing. This equation can be applied with great success for selective smoothing of images, where the noise is represented by small gradients and the spurious edges are represented by the large ones. For a numerical solution of this equation we use the finite volume method. We analyze convergence of a corresponding explicit finite volume scheme to its weak solution and show some numerical experiments. We suggest also some modifications of this scheme to obtain better CPU time performance of the algorithm. 2000 Mathematics Subject Classification: 35K55; 65M12. Keywords: image processing, Perona—Malik equation, finite volume method, explicit scheme, weak solution, convergence analysis. 1. Formulation and assumptions of the studied problem In this section, we in detail formulate the problem, we are going to deal with ∂ t u −∇.(g(|∇G σ ∗ u|)∇u)=0 in Q T ≡ I × Ω, (1) ∂ ν u =0 on I × ∂ Ω, (2) u(0, ·)= u 0 in Ω, (3) where Ω ⊂ R d is the rectangular domain, I = [0,T ] is the scaling interval, and g(s) is the decreasing smooth function, g(0) = 1, 0 <g(s) → 0 for s →∞, (4) G σ ∈ C ∞ (R d ) is the smoothing kernel with  R d G σ (x)dx =1 (5) and G σ (x) → δ x for σ → 0,δ x is the Dirac function at the point x, u 0 ∈ L 2 (Ω). (6) 2. Introduction to the regularized Perona—Malik equation Equation (1) is based on the Gaussian smoothing. Witkin and Koenderink [24] noticed that the convolution of the image with Gaussians of a certain variance was equivalent to the solving of the heat equation with the image as the initial datum for a corresponding time. If we embed the original data denoted by u 0 into a family of gradually simplified versions of