Methods for extrapolating gradient data outside a circular aperture from measurements obtained within a circular aperture are presented. The proposed methods are required to be computationally efficient and to avoid the excitation of additional waffle modes in Fried alignment. It is shown that, using an octagon as an intermediate step from the circle to the square in the extrapolation process, the computations or residual reconstruction error can be reduced. The resulting computational cost is as low as $O(N^{1/2})$ , where N is the number of measurement points. The performances of the extrapolation methods are studied in connection with a recently developed $O(N)$ wavefront reconstruction algorithm based on wavelet filter banks [IEEE J. Sel. Top. Signal Process. 2, 781 (2008)] Experiments indicate that, as expected, there is a significant reconstruction error if no extrapolation is used. Further, the proposed extrapolation techniques lead to a reconstruction with data that are marginally different from a pupil masked reconstruction using data from a square aperture.