Why use the PEF?

PEF-based texture synthesis can only achieve the results of the Fourier transform method (Figures 2 and 3) in the limit $N_a \rightarrow N_d$, which is unrealistic in practical situations, where N_d is very large. Least squares estimation of the filter in this case is certainly costlier than three Fast Fourier transforms. On the other hand, if the filter size can be limited without compromising quality, which is the case for stationary, simply correlated images, then the PEF-based method is more flexible. Unlike the Fourier transform a PEF can be estimated easily when data are missing. Figure 8 shows that the PEF estimated from the incomplete data captures enough features of the data's spectrum to make a fairly convincing texture synthesis result. The output of the PEF-based method can be of any size, while the output of a Fourier transform is generally constrained to be the same size as the input.