Algorithms of the q2r×q2r-point 2D discrete Fourier transform

Two methods of calculation of the 2-D DFT are analyzed.  The q2r×q2r-point 2-D DFT can be calculated by the traditional columnrow method with 2(q2r) 1-D DFTs, and we also propose the fast algorithm which splits each 1-D DFT by the short transforms by means of the fast paired transforms.  The q2r×q2r-point 2-D DFT can be calculated by the tensor or paired representations of the image, when the image is represented as a set of 1-D signals which define the 2-D transform in the different subsets of frequency-points and they all together cover the complete set of frequencies. In this case, the splittings of the q2r×q2r-point 2-D DFT are performed by the 2-D discrete tensor or paired transforms, respectively, which lead to the calculation with a minimum number of 1-D DFTs.