Fast algorithms for multidimensional DCT-to-DCT computation between a block and its associated subblocks

In this paper, we first propose an efficient algorithm for computing one-dimensional (1-D) discrete cosine transform (DCT) for a signal block, given its two adjacent subblocks in the DCT domain and then introduce several algorithms for the fast computation of multidimensional (m-D) DCT with size N₁×N₂×...×N_m given 2^m subblocks of DCT coefficients with size N₁/2×N₂/2×...×N_m/2, where N_i(i=1,2,...,m) are powers of 2. Obviously, the row-column method, which employs the most efficient algorithms along each dimension, reduces the computational complexity considerably, compared with the traditional method, which employs only the one-dimensional (1-D) fast DCT and inverse DCT (IDCT) algorithms. However, when m≥2, the traditional method, which employs the most efficient multidimensional DCT/IDCT algorithms, has lower computational complexity than the row-column method. Besides, we propose a direct method by dividing the data into 2^m parts for independent fast computation, in which only two steps of r-dimensional (r=1,2,...,m) IDCT and additional multiplications and additions are required. If all the dimensional sizes are the same, the number of multiplications required for the direct method is only (2^m-1)/m2^m-1 times of that required for the row-column method, and if N≥2^2m-1, the computational efficiency of the direct method is surely superior to that of the traditional method, which employs the most efficient multidimensional DCT/IDCT algorithms.