A fast hybrid Jacket–Hadamard matrix based diagonal block-wise transform

doi:10.1016/j.image.2013.11.002

Signal Processing: Image Communication

Volume 29, Issue 1, January 2014, Pages 49-65

https://doi.org/10.1016/j.image.2013.11.002 Get rights and content

Highlights

•
A unified fast hybrid diagonal block-wise transform (FHDBT) algorithm is proposed.
•
Fast diagonal block matrix decomposition is made by the matrix product of successively lower order diagonal Jacket and Hadamard matrix.
•
The DCT-II, DST-II, DFT, and HWT matrices can be unified by using sparse matrix decomposition algorithm.
•
The proposed FHDBT exhibits less the computational complexity as its matrix size gets larger.
•
The proposed algorithm is also well matched to circulant channel matrix.

Abstract

In this paper, based on the block (element)-wise inverse Jacket matrix, a unified fast hybrid diagonal block-wise transform (FHDBT) algorithm is proposed. A new fast diagonal block matrix decomposition is made by the matrix product of successively lower order diagonal Jacket matrix and Hadamard matrix. Using a common lower order matrix in the form of $[\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}]$ , a fast recursive structure can be developed in the FHDBT, which is able to convert a newly developed discrete cosine transform (DCT)-II, discrete sine transform (DST)-II, discrete Fourier transform (DFT), and Haar-based wavelet transform (HWT). Since these DCT-II, DST-II, DFT, and HWT are widely used in different areas of applications, the proposed FHDBT can be applied to the heterogeneous system requiring several transforms simultaneously. Comparing with pre-existing DCT-II, DST-II, DFT, and HWT, it is shown that the proposed FHDBT exhibits less the complexity as its matrix size gets larger. The proposed algorithm is also well matched to circulant channel matrix. From the numerical experiments, it is shown that a better performance can be achieved by the use of DCT/DST-II compression scheme compared with the DCT-II only compression method.

Introduction

The last decade based on orthogonal transform has been seen a quiet revolution in digital video technology such as Moving Picture Experts Group (MPEG)-4, H.264, and high efficiency video coding (HEVC) [1], [2], [3], [4], [5], [6], [7]. Digital video is everywhere such as DVD, gaming players, computers and mobile handsets. Nowadays, many of the coexisting heterogeneous systems [7], [8] are likely to catch the latest news on the web as on the smart TV and iPhone. Video compression is essential to all these applications. The discrete cosine transform (DCT)-II is popular compression structures for MPEG-4, H.264, and HEVC, and is accepted as the best suboptimal transformation since its performance is very close to that of the statistically optimal Karhunen–Loeve transform (KLT) [1], [2], [3], [4], [5]. For practical consideration, the underlying H.264-advanced video coding (AVC) intra mode dictates the transform coding implementation within a block coder with a typical block of size up to 16×16. However, since a DCT-based block coder suffers from blocking effect, i.e., a disturbing discontinuity at the block boundaries, much research efforts have been leveraged to reduce the blocking effect. In [4], [7], a first-order Gauss–Markov model was assumed for the images, and then it was shown that the image can be decomposed into a boundary response and a residual process given the closed bound boundary information. The boundary response is an interpolation of the block content from its boundary data, whereas the residual process is the interpolation error. An approach in [4] showed that the KLT of the residual process became discrete sine transform (DST) and DCT when the boundary conditions are available in vertical and horizontal directions [4], [6], [7].

The discrete signal processing based on the discrete Fourier transform (DFT) is popular in orthogonal frequency division multiplexing (OFDM) wireless mobile communication systems [3] such as 3rd generation partnership project long-term evolution (3GPP-LTE), mobile worldwide interoperability for microwave access (WiMAX), international mobile telecommunications-advanced (IMT-Advanced) as well as wireless local area network (WLAN). In addition, wireless personal area network (WPAN), and broadcasting related applications (digital audio broadcasting (DAB), digital video broadcasting (DVB), digital multimedia broadcasting (DMB)) are based on DFT. Furthermore, the Haar-based wavelet transform (HWT) is also very useful in the joint photographic experts group committee in 2000 (JPEG-2000) standard [2], [9]. Thus, different applications require different types of unitary matrices and their decompositions. From this reason, in this paper we will propose a unified hybrid algorithm which can be used in the mentioned several applications in different purposes.

Compared with the conventional individual matrix decompositions, our main contributions are summarized as follows:

•
We propose the diagonal sparse matrix factorization for a unified hybrid algorithm based on the properties of the Jacket matrix [10], [11] and the decomposition of the sparse matrix. It has been shown that this matrix decomposition is useful in developing the fast algorithms and characters [20]. Individual DCT-II [1], [2], [3], [6], [7], [12], DST-II [4], [6], [7], [13], DFT [3], [5], [14], and HWT [9] matrices can be decomposed to one orthogonal character matrix and a corresponding special sparse matrix. The inverse of the sparse matrix can be easily obtained from the property of the block (element)-wise inverse Jacket matrix. However, there have been no previous works in the development of the common matrix decomposition supporting these transforms.
•
We propose a new unified hybrid algorithm which can be used in the multimedia applications, wireless communication systems, and broadcasting systems at almost the same computational complexity as those of the conventional unitary matrix decompositions as summarized in Table 1, Table 2. Compared with the existing unitary matrix decompositions, the proposed hybrid algorithm can be even used to the heterogeneous systems with hybrid multimedia terminals being serviced with different applications. The block (element)-wise diagonal decomposition of DCT-II, DST-II, DFT and HWT as shown in Fig. 1, Fig. 2, Fig. 3, Fig. 4 has a similar pattern as Cooley–Tukey's regular butterfly structures. Moreover, this unified hybrid algorithm can be also applied to the wireless communication terminals requiring a multiuser multiple input–multiple output (MIMO) singular value decomposition (SVD) block diagonalization systems [15], [19], [20], [24] and diagonal channels interference alignment management in macro and femto cell coexisting networks [16]. In [15], [16], [19], [24], [25], a block-diagonalized matrix can be applied to the wireless communications MIMO downlink channel [25].

The rest of this paper is organized as follows: in Sections 2 and 3, we present the diagonal block-wise inverse sparse matrix decomposition for the DCT-II and DST-II matrices, respectively. In 4 Diagonal element-wise inverse sparse matrix decomposition of DFT transform, 5 Diagonal element-wise inverse sparse matrix decomposition for HWT transform, we introduce the diagonal element-wise inverse sparse matrix decomposition for the DFT and HWT matrices, respectively. In Section 6, we propose hybrid diagonal block-wise Jacket matrices. The conclusion is given in Section 7.

Notation: The superscript ${(\cdot)}^{T}$ denotes transposition; $I_{N}$ denotes the N×N identity matrix; 0 denotes an all-zero matrix of appropriate dimensions; $C_{l}^{i} ≜ \cos (i π / l)$ ; $S_{l}^{i} ≜ \sin (i π / l)$ ; $W ≜ e^{(j 2 π / N)}$ ; ⊗ and ⊕, respectively, denote the Kronecker product and the direct sum.

Section snippets

Diagonal block-wise inverse sparse matrix decomposition the DCT-II transform

Definition 1

Let $J_{N} ≜ {a_{i, j}}$ be a matrix, then it is called the Jacket matrix when $J_{N}^{- 1} = (1 / N) {{(a_{i, j})}^{- 1}}^{T}$ . That is, the inverse of the Jacket matrix can be determined by its element-wise inverse [10], [11], [20]. An N×N row permutation matrix, denoted $P_{N}$ , is defined by $P_{N} ≜ [\begin{matrix} 1 & 0 & 0 & \dots & 0 & 0 & \dots & 0 \\ 0 & 0 & 0 & \dots & 1 & 0 & \dots & 0 \\ 0 & 1 & 0 & \dots & 0 & 0 & \dots & 0 \\ 0 & 0 & 0 & \dots & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 0 & 0 & \dots & 1 \end{matrix}] with P_{2} = I_{2}$ where P_N elements are determined by the following relationship: ${\begin{array}{l} p_{i, j} = 1 & if & i = 2 j, & 0 \leq j \leq \frac{N}{2} - 1, \\ p_{i, j} = 1 & if & i = (2 j + 1) \mod N, & \frac{N}{2} \leq j \leq N - 1, \\ p_{i, j} = 0 & others . \end{array}$ The block column permutation matrix, denoted by $Q_{N}$ , is

Diagonal block-wise inverse sparse matrix decomposition the DST-II transform

The DST-II matrix [1], [2], [3], [4], [7] can be expressed as follows: $S_{N} = \sqrt{\frac{2}{N}} [\begin{matrix} S_{4 N}^{2 k_{0} Φ_{0}} & S_{4 N}^{2 k_{0} Φ_{1}} & \dots & S_{4 N}^{2 k_{0} Φ_{N - 1}} \\ S_{4 N}^{2 k_{1} Φ_{0}} & S_{4 N}^{2 k_{1} Φ_{1}} & \dots & S_{4 N}^{2 k_{1} Φ_{N - 1}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ S_{4 N}^{2 k_{N - 2} Φ_{0}} & S_{4 N}^{2 k_{N - 2} Φ_{1}} & \dots & S_{4 N}^{2 k_{N - 2} Φ_{N - 1}} \\ \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} & \dots & - \frac{1}{\sqrt{2}} \end{matrix}] = \sqrt{\frac{2}{N}} Y_{N} .$ Similar to the procedure we have used in the DCT-II matrix, we first define the permuted DST-II matrix, ${\tilde{S}}_{N}$ , as follows: ${\tilde{S}}_{N} ≜ P_{N}^{- 1} S_{N} Q_{N}^{- 1} = \sqrt{\frac{2}{N}} P_{N}^{- 1} Y_{N} Q_{N}^{- 1} .$ From (16), we can have a recursive form for $Y_{N}$ as $Y_{N} = P_{N} [\begin{matrix} A_{N / 2} & 0 \\ 0 & Y_{N / 2} \end{matrix}] [\begin{matrix} I_{N / 2} & I_{N / 2} \\ I_{N / 2} & - I_{N / 2} \end{matrix}] Q_{N}$ where the submatrix $A_{N}$ can be calculated by $A_{N} = U_{N} Y_{N} D_{N}$ where $U_{N}$ and $D_{N}$ are,

Diagonal element-wise inverse sparse matrix decomposition of DFT transform

The DFT is a Fourier representation of a given sequence ${x (n)}$ , that is, $X (n) = \sum_{m = 0}^{N - 1} x (m) W^{n m}, 0 \leq n \leq N - 1$ where $W = e^{- j 2 π / N}$ . The $N$ -point DFT matrix can be denoted by $F_{N} ≜ {W^{n m}}$ . The $N \times N$ Sylvester Hadamard matrix is denoted by $H_{N}$ . The Sylvester Hadamard matrix is generated by the successive Kronecker products: $H_{N} = H_{2} \otimes H_{N / 2}$ for $N = 4, 8, \dots$ In (25), we have defined $H_{2} ≜ [\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}]$ . We decompose a sparse matrix $E_{N} ≜ P_{N} {\tilde{F}}_{N} W_{N}$ in the following way: $F_{N} = {[P_{N}]}^{T} {\tilde{F}}_{N}$ and ${\tilde{F}}_{N} = [\begin{matrix} {\tilde{F}}_{N / 2} & {\tilde{F}}_{N / 2} \\ E_{N / 2} & - E_{N / 2} \end{matrix}] = [\begin{matrix} {\tilde{F}}_{N / 2} & 0 \\ 0 & E_{N / 2} \end{matrix}] [\begin{matrix} I_{N / 2} & I_{N / 2} \\ I_{N / 2} & - I_{N / 2} \end{matrix}]$ where $E_{N /}$

Diagonal element-wise inverse sparse matrix decomposition for HWT transform

The discrete Haar wavelet transform (HWT) is expressed by an $N \times N$ matrix $H_{N}$ . We can show that $H_{N} = {H_{N}}^{- 1} = {H_{N}}^{T}$ and $H_{2} = [\begin{matrix} r & r \\ r & - r \end{matrix}]$ with $r = 1 \sqrt{2}$ . Let us define following two matrices: $Π_{N} ≜ [\begin{matrix} 0 & I_{N / 2} \\ I_{N / 2} & 0 \end{matrix}] and Ψ_{N} ≜ P_{N} [\begin{matrix} I_{N / 2} & 0 \\ 0 & - I_{N / 2} \end{matrix}] for N \geq 4 .$ Inverses of these defined matrices in (30) are also defined by $Π_{N}^{- 1} = Π_{N} = {\tilde{Π}}_{N} and {\tilde{Ψ}}_{N} ≜ r Ψ_{N}^{- 1} = r [\begin{matrix} I_{N / 2} & 0 \\ 0 & - I_{N / 2} \end{matrix}] P_{N}^{T} .$ Notice that $Π_{N} = {\bar{I}}_{N}$ and $Π_{2} = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$ , $Ψ_{2} = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$

We can also develop a recursive form for a permuted HWT matrix in the following way, so that the permuted HWT matrix is defined by ${\tilde{H}}_{N} = Π_{N} H_{N} Ψ_{N} =$

Proposed hybrid architecture for fast computations of DCT-II, DST-II, DFT and HWT matrices

We have derived the recursive formulas for DCT-II, DST-II, DFT, and HWT. The derived results show that DCT-II, DST-II, DFT, and HWT matrices can be unified by using a similar sparse matrix decomposition algorithm, which is based on the block-wise Jacket matrix and diagonal recursive architecture with different characters. The conventional method is only converted from DFT to DCT-II, DST-II, and DWT. However, the proposed method can be universally switching from DCT-II to DST-II, DFT and HWT.

Conclusion

In this paper, we have derived a unified fast hybrid diagonal block-wise transform based on Jacket–Hadamard matrix. The proposed FHDBT have shown that DCT-II, DST-II, DFT, and HWT can be unified by using the diagonal sparse matrix based on the Jacket matrix and recursive structure with some characters changed from DCT-II to DST-II, DFT and HWT. The FHDBT has used the matrix product of recursively lower order diagonal sparse matrix and Hadamard matrix. The resulting signal flow graphs of DCT-II,

Acknowledgment

This work was supported by the MEST 2012-002521, NRF, Korea. The first author would like to thank to Professor Gilbert Strang, Department of Mathematics, MIT, for technical discussion.

References (26)

K.R. Rao et al.
Discrete Cosine Transform: Algorithms, Advantages, Applications
(1990)
I.E. Richardson, The H.264 Advanced Video Compression Standard, 2nd ed., John Wiley and Sons, Hoboken, NJ, April...
K.R. Rao et al.
Fast Fourier Transform: Algorithm and Applications
(2010)
A.K. Jain
Fundamentals of Digital Image Processing
(1987)
R. Wang
Introduction to Orthogonal Transforms: With Applications in Data Processing and Analysis
(2012)
ITU-T SG16 WP3/JCT-VC, CE 7.5, Performance Analysis of Adaptive DCT/DST Selection, July...
J. Hai et al.
Jointly optimized spatial prediction and block transform for video and image coding
IEEE Trans. Image Process.
(2012)
A. Mellouk et al.
Quality of service based routing algorithms for heterogeneous networks
IEEE Commun. Mag.
(2007)
G. Strang et al.
Wavelets and Filer Banks
(1996)
M.H. Lee
A new reverse Jacket transform and its fast algorithm
IEEE Trans. Circuits Syst. II
(2000)

Z. Chen et al.

Fast cocyclic Jacket transform

IEEE Trans. Signal Process.

(2008)

C.L. Wang et al.

High-throughput VLSI architectures for the 1-D and 2-D discrete cosine transform

IEEE Trans. Circuits Syst. Video Technol.

(1995)

Z. Wang

Fast Algorithm for the discrete W transform and for the discrete Fourier transform

IEEE Trans. Acoust. Speech Signal Process.

(1984)

Cited by (13)

Complexity reduction, self/completely recursive, radix-2 DCT I/IV algorithms
2020, Journal of Computational and Applied Mathematics
This paper proposes fast, self/completely recursive, radix-2 Discrete Cosine Transform (DCT) of types I/IV algorithms. Implementations of these new algorithms are stated through signal-flow graphs. These algorithms are derived mainly using a matrix factorization technique to factor the DCT I/IV matrices into sparse and scaled orthogonal matrices. This paper fills the gap of addressing the remaining DCTs in connection to our former work in the lowest multiplication complexity, self recursive, radix-2 DCT II/III algorithms. There has been no radix-2 DCT-I algorithm which is self recursive and has sparse and scaled orthogonal factors as proposed in this paper. The paper also establishes a novel relationship between DCT-I and DCT-III matrices having sparse factors for any $n = 2^{t}$ where $t \geq 1$ . This enables one to observe all traditional completely recursive DCT-I algorithms as the self recursive DCT-I algorithm proposed in this paper. Apart from the novel DCT-I algorithm, we also establish a novel completely recursive DCT-IV algorithm which executes recursively with the lowest multiplication complexity, self recursive, radix-2 DCT-II algorithm which was derived in our former paper. The proposed DCT-IV algorithm attains the lowest possible multiplication counts.
Signal flow graph approach to efficient and forward stable DST algorithms
2018, Linear Algebra and Its Applications
Citation Excerpt :
DST can be used to analyze image reconstruction via signal transition through a square-optical fiber lens [51], and spectral interference and additive wideband noise on the accuracy of the normalized frequency estimator can be investigated with a discrete-time sine-wave [1], to mention a few. Along with these applications, the engagement of DCT and DST in image processing, signal processing, fingerprint enhancement, quick response code (QR code), and multimode interface can also be seen in e.g., [2,7,12,13,18,20–23,25–28,42,43,46–48]. This section presents fast, efficient, and completely recursive DST algorithms defined solely via corresponding DST I–IV, having sparse, scaled orthogonal, rotational, and rotational-reflection factors by modifying the radix-2 stable DST I–IV algorithms introduced in [31].
In this paper, signal flow graphs are mainly addressed for the efficient and forward stable Discrete Sine Transform (DST) algorithms having sparse, scaled orthogonal, rotational, rotational-reflection, and butterfly matrices. In electrical engineering, theoretical computer science, control theory, system engineering, etc., we often use signal flow graphs as a modeling tool which interconnects the system components, or represents the realization of a system as electronic devices. The objective in this paper is to establish the connection between algebraic operations used in sparse and scaled orthogonal factorizations of DST I–IV matrices, with the signal flow graph building blocks. This paper elaborates signal flow graphs for the foundational frameworks with $n = 8$ and $n = 16$ , and use these together with the digital structures to describe the DST algorithms having a $(n - 1)$ -point signal flow graph for DST I and n-point signal flow graphs for DST II–IV. The presented DST algorithms are completely recursive, and solely based on corresponding matrices DST I–IV. These DST algorithms have low arithmetic complexity, especially the low number of multiplications, and significant speed improvement factor as opposed to most existing algorithms. Finally, this paper establishes that the presented algorithms are forward stable DST algorithms.
Medical data sheet in safe havens - A tri-layer cryptic solution
2015, Computers in Biology and Medicine
Citation Excerpt :
The fusion of traditional cryptic and modern chaotic features, namely confusion, diffusion, ergodicity, and randomness have evolved as a stronger tool in the form of image encryption [6–8], which can be adopted for establishing secured digital communication. Recently, many researchers have proposed and implemented image encryption algorithms using different transform techniques, chaotic systems [9–14], multiplexing [15–17], compression [18] and fusion [19–22] techniques. Authors have implemented different transform domain based image encryption algorithms, especially in gyrator domains.
Secured sharing of the diagnostic reports and scan images of patients among doctors with complementary expertise for collaborative treatment will help to provide maximum care through faster and decisive decisions. In this context, a tri-layer cryptic solution has been proposed and implemented on Digital Imaging and Communications in Medicine (DICOM) images to establish a secured communication for effective referrals among peers without compromising the privacy of patients. In this approach, a blend of three cryptic schemes, namely Latin square image cipher (LSIC), discrete Gould transform (DGT) and Rubik׳s encryption, has been adopted. Among them, LSIC provides better substitution, confusion and shuffling of the image blocks; DGT incorporates tamper proofing with authentication; and Rubik renders a permutation of DICOM image pixels. The developed algorithm has been successfully implemented and tested in both the software (MATLAB 7) and hardware Universal Software Radio Peripheral (USRP) environments. Specifically, the encrypted data were tested by transmitting them through an additive white Gaussian noise (AWGN) channel model. Furthermore, the sternness of the implemented algorithm was validated by employing standard metrics such as the unified average changing intensity (UACI), number of pixels change rate (NPCR), correlation values and histograms. The estimated metrics have also been compared with the existing methods and dominate in terms of large key space to defy brute force attack, cropping attack, strong key sensitivity and uniform pixel value distribution on encryption.
Implementation of Omar Pigeon Space-Time (OPST) Algorithm to Mitigate the Interference and Peak-to-Average Power Ratio (PAPR) Using RPR Mobile and HST-HM in the 5G
2022, Traitement du Signal
Hybrid Architecture for Sinusoidal and Non-sinusoidal Transforms
2022, Circuits, Systems, and Signal Processing
Block-Circulant Inverse Orthogonal Jacket Matrices and Its Applications to the Kronecker MIMO Channel
2019, Circuits, Systems, and Signal Processing

View all citing articles on Scopus

^☆: This work was supported by the MEST 2012-002521, NRF, Korea.

¹: This work was done when he was with Inha University, Incheon, Korea.

View full text

A fast hybrid Jacket–Hadamard matrix based diagonal block-wise transform☆

Highlights

Abstract

Introduction

Section snippets

Diagonal block-wise inverse sparse matrix decomposition the DCT-II transform

Diagonal block-wise inverse sparse matrix decomposition the DST-II transform

Diagonal element-wise inverse sparse matrix decomposition of DFT transform

Diagonal element-wise inverse sparse matrix decomposition for HWT transform

Proposed hybrid architecture for fast computations of DCT-II, DST-II, DFT and HWT matrices

Conclusion

Acknowledgment

Discrete Cosine Transform: Algorithms, Advantages, Applications

Fast Fourier Transform: Algorithm and Applications

Fundamentals of Digital Image Processing

Introduction to Orthogonal Transforms: With Applications in Data Processing and Analysis

Jointly optimized spatial prediction and block transform for video and image coding

IEEE Trans. Image Process.

Quality of service based routing algorithms for heterogeneous networks

IEEE Commun. Mag.

Wavelets and Filer Banks

A new reverse Jacket transform and its fast algorithm