Numerical Analysis of the Topological Gradient Method for Fourth Order Models and Applications to the Detection of Fine Structures in Imaging

Abstract

In this paper we present the numerical analysis of the topological gradient method developed in [G. Aubert and A. Drogoul, Control Optim. Calc. Var., to appear] for the detection of fine structures (filaments and points in two dimensions). First used in mechanics of structures [S. Amstutz, I. Horchani, and M. Masmoudi, Control Cybernet., 34 (2005), pp. 81--101], this method has subsequently been applied in imaging for edge detection and image restoration [L. Jaafar Belaid, M. Jaoua, M. Masmoudi, and L. Siala, Engrg. Anal. Boundary Elements, 32 (2008), pp. 891--899; A. Drogoul and G. Aubert, J. Math. Imaging Vision, submitted; also available online from https://hal.archives-ouvertes.fr/hal-01018200/, 2014]. The model involves second order derivatives and leads to fourth order PDEs. We first develop the case of Gaussian noisy images, and then we extend the method to the more general case of blurred and Gaussian noisy images. We show that as for edge detection [A. Drogoul and G. Aubert, J. Math. Imaging Visi...