A new technique for the acceleration of iterative image restoration algorithms is proposed. The method is based on the principles of vector extrapolation and does not require the minimization of a cost function. The algorithm is derived and its performance illustrated with Richardson-Lucy (R-L) and maximum entropy (ME) deconvolution algorithms and the Gerchberg-Saxton magnitude and phase retrieval algorithms. Considerable reduction in restoration times is achieved with little image distortion or computational overhead per iteration. The speedup achieved is shown to increase with the number of iterations performed and is easily adapted to suit different algorithms. An example R-L restoration achieves an average speedup of 40 times after 250 iterations and an ME method 20 times after only 50 iterations. An expression for estimating the acceleration factor is derived and confirmed experimentally. Comparisons with other acceleration techniques in the literature reveal significant improvements in speed and stability.