In order to process the wrapped phase map with strong noise and under-sampled area simultaneously, four kinds of phase unwrapping algorithms based on fast Fourier transform are studied in depth and the speed, accuracy and the range of applications are expounded. The effective of denoising and dealing with the under-sampled situations of four kinds of algorithms are verified by computer simulations and experimental demonstrations. The results demonstrate that when the data contain strong noises, the quartic Fourier transform algorithm is the best, the combination algorithm based on lateral shearing interferometry and Fourier transform is the worst. While the data have under-sampled areas, the combination algorithm based on lateral shearing interferometry and Fourier transform is the best, followed by the quartic Fourier transform algorithm. When the experimental data contain both noises and under-sampled areas, the quartic Fourier transform algorithm can obtain the highest-precision results.