Analysis of a Semidiscrete Scheme for Solving Image Smoothing Equation of Mean Curvature Flow Type

in a domain Ω ⊂ IR accompanied with homogeneous Neumann boundary conditions and an initial condition. Equation (1.1) is useful in image processing for selective smoothing of images and shapes. Numerical experiments in processing of 2D and 3D images are presented in [10]. Here, we present analysis of a special semidiscrete scheme for solving (1.1). Equation (1.1) is a degenerate parabolic equation and is related to the so-called level set equation ((1.1) with g(s) ≡ 1) which has been proposed by Osher & Sethian [16],[21] for computation of moving fronts in interfacial dynamics. The level set equation moves each level line (surface) of 2D (3D) image with the velocity proportional to its normal mean curvature field. This causes intrinsic smoothing of level sets. By means of the Perona-Malik function g (for which a typical choice is, e.g., g(s) = 1/(1 + s)) we control the motion of level sets which are also edges. The smoothing of silhouettes on which the gradient of intensity is large can be slowed down by using g. In analysis and also in computations (see [10]) we use the following Evans-Spruck regularization,