A nonlinear primal-dual method for total variation-based image restoration. (English) Zbl 0929.68118

Chan, Tony F.; Golub, Gene H.; Mulet, Pep

SIAM J. Sci. Comput. 20, No. 6, 1964-1977 (1999).

Summary: We present a new method for solving Total Variation (TV) minimization problems in image restoration. The main idea is to remove some of the singularity caused by the nondifferentiability of the quantity \(| \nabla u| \) in the definition of the TV-norm before we apply a linearization technique such as Newton’s method. This is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function, which can be interpreted as the normal vector to the level sets of the image u. Our method can be viewed as a primal-dual method as proposed by A. R. Conn and M. L. Overton [A Primal-Dual Interior Point Method for Minimizing a Sum of Euclidean Norms, preprint (1994)] and K. D. Andersen [Ph.D. thesis, Odense University, Denmark (1995)] for the minimization of a sum of Euclidean norms. In addition to possessing local quadratic convergence, experimental results show that the new method seems to be globally convergent.

MSC:

68U10	Computing methodologies for image processing
65K10	Numerical optimization and variational techniques
65F10	Iterative numerical methods for linear systems

Keywords:

image restoration; total variation; primal-dual; conjugate gradient

Software:

KELLEY