On the convergence of the lagged diffusivity fixed point method in total variation image restoration

In this paper we show that the lagged diffusivity fixed point algorithm introduced by Vogel and Oman in [SIAM J. Sci. Comput., 17 (1996), pp. 227-238] to solve the problem of total variation denoising, proposed by Rudin, Osher, and Fatemi in [Phys. D, 60 (1992), pp. 259-268], is a particular instance of a class of algorithms introduced by Voss and Eckhardt in [Computing, 25 (1980), pp. 243-251], whose origins can be traced back to Weiszfeld's original work for minimizing a sum of Euclidean lengths [Tohoku Math. J., 43 (1937), pp. 355-386]. There have recently appeared several proofs for the convergence of this algorithm [G. Aubert et al., Technical report 94-01, Informatique, Signaux et Systemes de Sophia Antipolis, 1994], [A. Chambolle and P.-L. Lions, Technical report 9509, CEREMADE, 1995], and [D. C. Dobson and C. R. Vogel, SIAM J. Numer. Anal., 34 (1997), pp. 1779-1791]. Here we present a proof of the global and linear convergence using the framework introduced in [H. Voss and U. Eckhart, Computing, 25 (1980), pp. 243-251] and give a bound for the convergence rate of the fixed point iteration that agrees with our experimental results. These results are also valid for suitable generalizations of the fixed point algorithm.