This paper provides a fast algorithm to order in a meaningful, strict way the integer gray values in digital (quantized) images. It can be used in any exact histogram specification-based application. Our algorithm relies on the ordering procedure based on the specialized variational approach. This variational method was shown to be superior to all other state-of-the art ordering algorithms in terms of faithful total strict ordering but not in speed. Indeed, the relevant functionals are in general difficult to minimize because their gradient is nearly flat over vast regions. In this paper, we propose a simple and fast fixed point algorithm to minimize these functionals. The fast convergence of our algorithm results from known analytical properties of the model. Our algorithm is equivalent to an iterative nonlinear filtering. Furthermore, we show that a particular form of the variational model gives rise to much faster convergence than other alternative forms. We demonstrate that only a few iterations of this filter yield almost the same pixel ordering as the minimizer. Thus, we apply only few iteration steps to obtain images, whose pixels can be ordered in a strict and faithful way. Numerical experiments confirm that our algorithm outperforms by far its main competitors.