This paper provides a fast algorithm to order in a strict way the integer gray values in digital images which can be applied for exact histogram specification. Our algorithm is based on the ordering procedure by the variational approach in [9]. This variational method was shown to be superior to 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 the functionals. The fast convergence of our algorithm results from known analytical properties of the model. In particular the original image is a good starting point for the iterations and the involved inverse functions admit a simple explicit form. Only a few iteration steps of this algorithm provide an image whose pixels can be ordered in a strict and faithful way. Numerical experiments confirm that our algorithm outperforms by far its main competitors in speed and quality. Moreover, in contrast to other ordering algorithms we can handle large images commonly taken by commercial cameras. Concerning applications the proposed ordering algorithm is the basis of the hue and range preserving color image enhancement method proposed in [11].
