General Factorization of Conjugate-Symmetric Hadamard Transforms

Abstract

Complex-valued conjugate-symmetric Hadamard transforms ( -CSHT) are variants of complex Hadamard transforms and found applications in signal processing. In addition, their real-valued transform counterparts ( -CSHTs) perform comparably with Hadamard transforms (HTs) despite their lower computational complexity. Closed-form factorizations of -CSHTs and -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 -CSHTs based on that of -CSHTs. The proposed factorization can be applied to both -and -CSHTs with one factorization and it provides several benefits: 1) It can save total implementation costs for both -CSHTs and -CSHTs; 2) the generalized -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 thatand -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 -CSHT. The image orientation estimation and performance in image coding of our -CSHTs were evaluated through examples of practical applications based on the proposed factorization.

Fast Jacket-Haar Transform with Any Size

“…x [7] x [8] x [9] x [10] x [11] x [12] x [1] x [5] x [2] x [4] x [ problem of the traditional constraint that the number of the points is the power of 2. With the proposed fast generation algorithms, the arbitrary length Jacket-Haar transform can be derived in a successive fashion.…”

“…. , Ψ 2, in the form described in (1) or (2), the (2 + 1)-point Jacket-Haar transform matrix derived from (11) is a sparse matrix since each of the rows possesses a maximum of two nonzero elements. For the other case, if there exists a sparse -point Jacket-Haar transform matrix Ψ with the given matrices Ψ 2,1 , Ψ 2,2 , .…”

“…Hadamard transform, Haar transform, discrete Fourier transform, and their derivatives are discrete orthogonal transforms with extensive applications in signal and image processing [1][2][3][4]. Jacket transform, motivated by the center weighted Hadamard transform [5], is a special transform with its inverse transform matrix being determined by the element or block-wise inverse of an original matrix [6].…”

“…Jacket transform, motivated by the center weighted Hadamard transform [5], is a special transform with its inverse transform matrix being determined by the element or block-wise inverse of an original matrix [6]. Jacket transform has been extensively applied in many fields, such as signal and data processing [1], digital wireless communications [7], cryptography [8], and encoding designs [9]. Meanwhile, several interesting matrices, such as Hadamard matrices and DFT matrices, belong to the Jacket transform matrix family.…”

Sequency Domain Signal Processing Using Complex Hadamard Transform

“…We are particularly interested in CS-SCHT, the elements of whose row vectors are arranged in increasing order of sequencies and the sequency spectrum of which is conjugate-symmetric like that of DFT. Development of fast algorithms for CS-SCHT and exploration of its applications has remained an active research domain in the last few years [7,11,12,14].…”