General Factorization of Conjugate-Symmetric Hadamard Transforms

Complex-valued conjugate-symmetric Hadamard transforms (C-CSHT) are variants of complex Hadamard transforms and found applications in signal processing. In addition, their real-valued transform counterparts (R-CSHTs) perform comparably with Hadamard transforms (HTs) despite their lower computational complexity. Closed-form factorizations of C-CSHTs and R-CSHTs have recently been proposed to make calculations more efficient. However, there is still room to find effective and general factorizations. This paper presents a simple closed-form complete factorization of C-CSHTs based on that of R-CSHTs. The proposed factorization can be applied to both C- and R-CSHTs with one factorization and it provides several benefits: 1) It can save total implementation costs for both C-CSHTs and R-CSHTs; 2) the generalized R-CSHT factorization significantly reduces its computational cost; 3) memory-saved local orientation detection of images can be achieved; 4) a fast direction-aware transform can be attained; 5) it clarifies that C- and R-CSHTs are closely related to common block transforms, such as the discrete Fourier transform (DFT), binDCT, and HT; and 6) it achieves a new integer complex-valued transform, which can approximate the DFT better than the original C-CSHT. The image orientation estimation and performance in image coding of our R-CSHTs were evaluated through examples of practical applications based on the proposed factorization.