The approximation method proposed in this paper converts each constant in a DCT algorithm into a minimum number of signed digits. Then any multiplication in the transform can be turned into a number of add-and-shift operations on an integer value. With this technique, any fast DCT algorithm can be made multiplierless. To reduce the algorithm complexity and the word length requirement, we developed an effective algorithm for converting any constant into a signed digits string with minimum number of non-zero signed digits and a reduced word length. The approximation errors of different constants could affect differently the MSE of the approximated algorithm. The simple strategy is to assign more digits to those constants whose errors are more sensitive to the MSE of the algorithm whose complexity will however increase when the total number of digits assigned to the constants becomes larger. This paper devised an efficient algorithm to find an optimized signed digits configuration for minimizing the MSE of the algorithm with a specified complexity. Experiment results show that Lee?s fast DCT algorithm approximated by the proposed method can be used to reconstruct images with high visual quality in terms of PSNR.