Wavefront Reconstruction using the Fast Fourier Transform and spatial filtering is shown to be computationally tractable and sufficiently accurate for use in large Shack Hartmann-based adaptive optics systems (up to at least 10,000 actuators). This method is significantly faster and can have lower noise propagation than traditional VMM reconstructors. The boundary problem which prevented the accurate reconstruction of phase in circular apertures using square-grid FTs is identified and solved. The methods are adapted for use on the Fried-geometry. Detailed performance analysis of mean squared error and noise propagation for FT methods is presented, using both theory and simulation.