Interpretation of uniform translational image motion: DCT versus FT

We study properties of the discrete cosine transform (DCT) when applied to an image sequence formed by uniformly translating a still image. The Fourier transform (FT) applied to such a sequence has non-zero content only on a spatio-temporal frequency plane orthogonal to the direction of motion. We derive an equivalent spectrum for the DCT case. The spectrum function is more complicated than in the FT case and cannot be easily interpreted analytically. However, its numerical evaluation demonstrates that spectral occupancy in the DCT domain is limited to a narrow band around a plane similar to one in the FT case with two important differences: the plane is subject to folding, and the DCT coefficient amplitude is strongly attenuated for larger temporal "frequencies". We verify the theoretical derivations experimentally on images. The obtained result opens an interesting possibility for the computation of constant-velocity motion in the DCT domain. We demonstrate some preliminary results of motion estimation in the 3D DCT domain by identifying directions of spectral occupancy with respect to transform coefficients.