Shifted Fourier transform-based tensor algorithms for the 2-D DCT

In this paper, tensor algorithms for calculating the two-dimensional (2-D) discrete cosine transform (DCT) are presented. The tensor approach is based on the concept of the covering revealing the transforms, which yields in particular the splitting of the shifted 2/sup r//spl times/2/sup r/-point Fourier and cosine transforms into 2/sup r-1/3 one-dimensional (1-D) incomplete 2/sup r/-point transforms. The multiplicative complexity of the 2-D 2/sup r//spl times/2/sup r/-point discrete cosine transforms in terms of the tensor representation is 4/sup r/3-2/sup r-2/(r/sup 2/+7r+14), which is reduced to 4/sup r/8/3-2/sup r/(r/sup 2/+7r+10)-20/3 when using the improved tensor algorithm. The multiplicative complexity in the general L/sup r//spl times/L/sup r/ case, with a prime L>2, as well as in the L/sub 1/L/sub 2//spl times/L/sub 1/L/sub 2/ case, with arbitrary co-prime L/sub 1/, L/sub 2/>1, is provided. The examples of the tensor algorithms for calculating the 8/spl times/8-point DCT through 104, 88, and 84 multiplications are given in detail. Based on the proposed concept, the fast algorithm for calculating the 1-D DCT-I is also developed. The multiplicative complexity of the 2/sup r/-point DCT-I is 2/sup r+1/-(r-2)(r+5)/2-8. The comparative estimates with the known algorithms are given.