New shift, scaling, and derivative properties for the DCT
Roger Reeves · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1998
The DCT is used in image and video compression standards JPEG, MPEG and H.261. A set of properties for shifting and scaling by fractional amounts, and taking linear operations such as differentiation and integration is described. The properties take as input the DCT coefficients of a sampled signal, subject them to a linear transform, and return the DCT coefficients of a sampled signal which has been subject to the corresponding operation. The properties are derived by considering the inverse discrete transform as a sum of continuous basis functions. Mathematically, the properties are equivalent to taking the inverse transform of the DCT coefficients; reconstructing the continuous signal using an infinite sum of sinc functions, performing the desired operation (shift, scale, differentiate, integrate) on the reconstructed signal, resampling the result, and then taking the DCT of the resulting samples. It is proved that such an approach is valid for the type 2 DCT, a 2D version of which is used in JPEG, MPEG and H.261. A consequence of this method is that the original signal is assumed to be symmetrically extended and periodically repeated with period 2N, where N is the size of the DCT. Operations which result in points outside the DCT window in the reconstructed signal will return a point on the symmetric extension of the signal. In most cases this will result in an error being introduced because no actual information exists on what the value of the signal should be except within the DCT window. This approach has an exact analog in the signal domain, and can also be incorporated into the reverse transform. The techniques may prove useful in compressed domain processing applications, and are interesting because they allow operations from the continuous domain such as integration and differentiation to be interpreted in the discrete domain, using the sampling theorem.