Lowest Complexity Self-Recursive Radix-2 DCT II/III Algorithms

Abstract

This paper presents the lowest multiplication complexity, self-recursive, radix-2 DCT II/III algorithms for any $n=2^t; (t \geq 1)$ with implementations in signal-flow graphs and image compression. These algorithms are derived mainly using a matrix factorization technique to factor DCT II/III matrices into sparse and scaled orthogonal matrices. Although there are small length DCT II algorithms available only for $n=8$ and $n=16$, there is no existing self-recursive fast radix-2 DCT II/III algorithms for any $n$. To fill this gap, this paper presents the lowest multiplication complexity radix-2 self-recursive DCT II/III algorithms. We also attain the lowest theoretical multiplication bound for $n=8$ with the new DCT II algorithm and establish new lowest bounds for DCT III for any $n$ and for DCT II for any $n \geq 32$. The paper also establishes a novel relationship between DCT II an DCT IV having sparse factors. This enables one to see the connection of most traditional factorization of DCT II/III matrices with the proposed DCT II/III matrices.