Signal Processing based on Stable radix-2 DCT I-IV Algorithms having Orthogonal Factors

Abstract

Abstract. This paper presents stable, radix-2, completely recursive discrete cosine transform algorithms DCT-I and DCT-III solely based on DCT-I, DCT-II, DCT-III, and DCT-IV having sparse and orthogonal factors. Error bounds for computing the completely recursive DCT-I, DCT-II, DCT-III, and DCT-IV algorithms having sparse and orthogonal factors are addressed. Signal flow graphs are demonstrated based on the completely recursive DCT-I, DCT-II, DCT-III, and DCT-IV algorithms having orthogonal factors. Finally image compression results are presented based on the recursive 2D DCT-II and DCT-IV algorithms for image size 512 × 512 pixels with transfer block sizes 8 × 8, 16 × 16, and 32 × 32 with 93.75% absence of coefficients in each transfer block.

Low-complexity Scaling Methods for DCT-II Approximations

“…The DCT-II computation can be performed by means of the Chen algorithm [45]. Such algorithm is based on a decomposition that expresses C II 2N in terms of C II N and C IV N [9,16,18,20,[46][47][48]. The matrix form of the Chen algorithm is given by [2, p. 96] 1…”

“…Second, a collection of scaling methods capable of extending current approaches for DCT approximations which render low-complexity hardware implementations is sought. Direct matrix factorizations [13][14][15][16][17][18][19][20][21] and a relation between the DCT-II and the discrete sine (DST) transform of type IV (DST-IV) are employed to derive the sought methods. As contributions, we also provide 16-and 32-point DCT-II approximations with the associated performance analysis compared to the scaled approximations obtained from the JAM method.…”

Sparse Matrix Based Low-Complexity, Recursive, and Radix-2 Algorithms for Discrete Sine Transforms

“…Though one can find orthogonal matrix factorizations for the DCT and DST in [7], the resulting algorithms in [7] are not completely recursive and hence do not lead to simple recursive algorithms. An alternative factorization for DCT I in [8] and DST-I in [9] can be seen in [10], [11], but the factorizations in the latter papers do not solely depend on DCT I-IV or DST I-IV. Moreover, [11] has used the same factorization for DST-II and IV as in [7].…”

“…We refer to many more historical remarks on DST factorizations and the corresponding signal flow graphs in [12]. We note here that the DCT algorithms in [8], [9], [13] and the relationship between cosine and sine transform matrices in [10] can be used to obtain the DST algorithms in [12]. On the other hand, one can use an elegant factorization for Chebyshev-like Vandermonde matrices to factor the DCT and DST matrices as described in [14].…”