FFT interpolation for arbitrary factors: a comparison to cubic spline interpolation and linear interpolation

The authors applied FFT interpolation (FTI) for arbitrary factors to a common image task: 1D zoom, and compared it to cubic splines interpolation and linear interpolation. The authors also determined the relative error of the arbitrary factors FFT and the conventional FFT (integral powers of 2) for computing the Fourier transforms of trigonometric eigenfunctions. In the interpolation study, FTI proved to be satisfactory. If the data was band-limited, or nearly so, FTI was as effective as conventional interpolation. If the data was not band-limited, an anti-oscillation filter provided better results. The test data was a curve consisting of a rectangle, gaussian function and a 4th degree polynomial. The authors used interpolation to zoom the image by a factor of 3.1:1, and determined the rms (root mean square) errors by comparing the result to the known image values. To determine the relative error for computing eigenfunctions, the authors FFT'd trigonometric eigenfunctions. For an FFT of dimension 120, the relative error was less than 10^-10 over a wide range of eigenvalues and compared favorably with an FFT of dimension 128, which had a relative error of less than 10^-13 over the same range. The advantage of this approach is that the arbitrary factors algorithm is quite simple in design, and replaces an entire suite of FFT's based upon different prime factorizations. The authors also show that FTI is exact if the Fourier transform of the function is absolutely band-limited