The performance of the "weighted Fourier phase slope" centroiding algorithm at the subpupil image of a Shack-Hartmann wavefront sensor for point-like astronomical guiding sources is explored. This algorithm estimates the image's displacement in the Fourier domain by directly computing the phase slope at several spatial frequencies, without the intermediate step of computing the phase; it then applies optimized weights to the phase slopes at each spatial frequency obtained by a Bayesian estimation method. The idea was inspired by cepstrum deconvolution techniques, and this relationship is illustrated. The algorithm's tilt estimation performance is characterized and contrasted with other known centroiding algorithms, such as thresholded centre of gravity (TCoG) and cross correlation (CC), first through numerical simulations at the subpupil level, then at the pupil level, and finally at the laboratory test bench. Results show a similar sensitivity to that of the CC algorithm, which is superior to that of the TCoG algorithm when large fields of view are necessary, i.e., in an open-loop configured adaptive optics system, thereby increasing the guide star limiting magnitude by 0.6 to 0.7 mag. On the other side, its advantage over the CC algorithm is its lower computational cost by approximately an order of magnitude.