ABSTRACT
The discrete cosine transform (DCT) is one of the widely used tools in digital signal and image processing, especially in image compression and transformbased coding in telecommunication [45]-[49]. This transform approaches the Karhuner-Loeve transform for first-order Markov stationary random data and can be used in linear filtration and pattern recognition [50, 51]. Since introducing the DCT [2], many fast algorithms have been proposed to reduce the computational complexity of the DCT. We note the fast 1-D DCT proposed in [56], radix-2 (including cases with even sequence of lengths) and vectorradix [61, 62], fast approximate computation of the DCT [63], as well as different recursive algorithms for the DCT with effective VLSI hardware implementation [52, 57, 58], algorithms based on Clenshaw’s recurrent formula [59], and recursive order reduction of the Tchebycheff polynomial [53, 55]. We consider effective algorithms for splitting the 2r-point DCT by short 2k-point DCTs, k = 1 : (r− 1). The splitting is defined by the paired representation of signals with respect to the cosine transform. Two recursive forms of splitting the N -point type-II DCT are described. We also consider a method of calculating the DCT which is based on using the Coxeter type matrices. The cases N = 4 and 8 are illustrated in detail. The integer approximations of the DCT are also considered, and methods of control bits, nonlinear equations, and lifting scheme are described.
