Convergence of Newton's method for a minimization problem in impulse noise removal

Recently, two-phase schemes for removing salt-and-pepper and random-valued impulse noise are proposed in [6, 7]. The first phase uses decision-based median filters to locate those pixels which are likely to be corrupted by noise (noise candidates). In the second phase, these noise candidates are restored using a detail-preserving regularization method which allows edges and noise-free pixels to be preserved. As shown in [18], this phase is equivalent to solving a one-dimensional nonlinear equation for each noise candidate. One can solve these equations by using Newton's method. However, because of the edge-preserving term, the domain of convergence of Newton's method will be very narrow. In this paper, we determine the initial guesses for these equations such that Newton's method will always converge.