||In fractal image compression, a digital image is approximated by the unique fixed point of a contractive affine mapping. The decoding consists of iterating the affine mapping starting from an arbitrary image until convergence to the fixed point. We show that the decoding can be accelerated by using the new pixel intensities of an image iterate as soon as they become available. We provide a mathematical formulation of the proposed decoding and prove its convergence in a general setting. We show that under some mild restrictions the asymptotic rate of convergence of the proposed method is greater than or equal to that of the conventional method. We also discuss the use of standard iterative methods for the decoding. Finally, we show that the convergence of the proposed method can be enhanced by an ordering technique.