New algorithms for computing directional discrete Fourier transforms

New algorithms for computing the discrete Fourier transform (DFT) spectra along different directions are derived and implemented. For computing the DFT spectrum along any given direction (containing N DFT frequencies), a new algorithm is presented that requires N(N-1) additions and a single 1-D FFT. As expected, for a single direction, the directional FFT algorithm is significantly faster than standard 2-D FFT algorithms that compute the entire spectrum (all results are compared against FFTW and FFTPACK). A scalable extension of the unidirectional algorithm for computing the entire DFT spectrum is also derived and implemented. The three most promising features of the new algorithm are that: (i) computation scales nearly linearly with the number of DFT frequencies computed, (ii) the algorithm uses a reduced number of multiplications (yet uses more additions), and (iii) it is more accurate