Fourth-order partial differential equations for image enhancement

Abstract

Second-order partial differential equations have been studied as a useful tool for noise removal. The Perona–Malik model [P. Perona, Malik, Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell. 12 (1990) 629–639] has an edge preserving property but have sometimes the undesirable blurred effect. In this paper, we propose improved models by combining Catté et al.’s model [F. Catte’, P.L. Lions, J.M. Morel, T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal. 129 (1992) 182–193] with fourth-order terms. We prove the existence and uniqueness of the proposed models. Then, we show numerical evidence of the power of resolution of these models with respect to other known models as the Perona–Malik model, the Catté et al.’s model, the modified total variation model by Chan et al. [T. Chan, A. Marquina, P. Mulet, Second order differential functionals in total variation-based image restoration. Available from: <http://www.math.ucla.edu/chan>], etc.