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Jacket Matrix Based Recursive Fourier Analysis and Its Applications

Written By

Daechul Park and Moon Ho Lee

Submitted: 24 May 2014 Published: 03 June 2015

DOI: 10.5772/59353

From the Edited Volume

Fourier Transform - Signal Processing and Physical Sciences

Edited by Salih Mohammed Salih

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1. Introduction

The last decade based on orthogonal transform has been seen a quiet revolution in digital video technology as in Moving Picture Experts Group (MPEG)-4, H.264, and high efficiency video coding (HEVC) [1–7]. 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) [1-5].

The discrete signal processing based on the discrete Fourier transform (DFT) is popular in wide range of applications depending on specific targets: orthogonal frequency division multiplexing (OFDM) wireless mobile communication systems in 3GPP-LTE [3], mobile worldwide interoperability for microwave access (WiMAX), international mobile telecommunications-advanced (IMT-Advanced), broadcasting related applications such as digital audio broadcasting (DAB), digital video broadcasting (DVB), digital multimedia broadcasting (DMB)) 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], [8]. Thus, different applications require different types of unitary matrices and their decompositions. From this reason, in this book chapter 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 [9], [10] and the recursive decomposition of the sparse matrix. It has been shown that this matrix decomposition is useful in developing the fast algorithms [11]. Individual DCT-II [1–3], [6], [7], [12], DST-II [4], [6], [7], [13], DFT [3], [5], [14], and HWT [8] 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 and 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 decompositions of DCT-II, DST-II, DFT and DWT have 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) SVD block diagonalization systems [15], [11,19], [22] and diagonal channels interference alignment management in macro/femto cell coexisting networks [16]. In [15-16, 19, 22- 23], a block-diagonalized matrix can be applied to wireless communications MIMO downlink channel.

In Section 2, we present recursive factorization algorithms of DCT-II, DST-II, and DFT matrix for fast computation. In Section 3, hybrid architecture is proposed for fast computations of DCT-II, DST-II, and DFT matrices. Also numerical simulations follow. The conclusion is given in Section 4.

Notation: The superscript ()T denotes transposition; IN denotes the N×N identity matrix; 0 denotes an all-zero matrix of appropriate dimensions; Cli=cos(iπ/l) ; Sli=sin(iπ/l) ; W=ej2πN ; and , respectively, denote the Kronecker product and the direct sum.


2. Jacket matrix based recursive decompositions of Fourier matrix

2.1. Recursive decomposition of DCT-II

Definition 1: Let JN={ai,j} be a matrix, then it is called the Jacket matrix when JN1=1N{(ai,j)1}T.

That is, the inverse of the Jacket matrix can be determined by its element-wise inverse [9-11]. The row permutation matrix, PN is defined by


where PN elements are determined by the following relation:


The block column permutation matrix, QN is defined by


where I¯N/2 denotes reversed identity matrix. Note that QN1=QN and PN1PN, whereas QN1=QNT and PN1=PNT.

Proposition 1: With the use of the Kronecker product and Hadamard matrices, a higher order block-wise inverse Jacket matrix (BIJM) can be recursively obtained by




where the lowest order Hadamard matrix is defined by H2=[1111]

Proof: A proof of this proposition is given in Appendix 6.A.

Note that since the BIJM requires a matrix transposition and then normalization by its size, a class of transforms can be easily inverted as follows:


Due to a simple operation of the BIJM, we can reduce the complexity order as the matrix size increases. In the following, we shall use this property of the BIJM in developing a hybrid diagonal block-wise transform.

According to [1-4] and [7], the DCT-II matrix is defined as follows:


where Φi=2i+1 and ki=i+1. We first define a permuted DCT-II matrix C˜N=PN1CNQN1=2NPN1XNQN1. We can readily show that the matrix XN can be constructed recursively as follows:


Here, the matrix BN in (7) is given as:


where f(m,1)=2m1 and f(m,n+1)=f(m,n)+2f(m,1) for m,n{1,2,,N/2}. For example, the matrix B4 is given by


Since XN/21=4NXN/2T and BN/21=4NBN/2T, the matrix decomposition in (7) is the form of the matrix product of diagonal block-wise inverse Jacket and Hadamard matrices. The matrix BN/2 is recursively factorized using Lemma 1.

Lemma 1: The matrix BN can be decomposed as:


where a lower triangular matrix LN is defined by LN={LN(m,n)} with elements


and a diagonal matrix DN is defined by DN=diag{C4NΦ0,C4NΦ1,,C4NΦN1}.

Proof: A proof of this Lemma is provided in Appendix 6.B.

Using (10), we first rewrite (7) as


which can be evaluated recursively as follows:


Note that in (13) a 2×2 Hadamard matrix is defined by X2=[1111]. Also, applying the Kronecker product of I2 and X4,X8 can be obtained. Keep applying the Kronecker product of I2 and XN/2, the final equivalent form of XN is obtained. Thus, the proposed systematic decomposition is based on the Jacket and Hadamard matrices.

In [17], the author proposed a recursive decimation-in-frequency algorithm, where the same decomposition specified in (10) was used. However, due to using a different permutation matrix, a different recursive form was obtained. Different recursive decomposition was proposed in [18]. Four different matrices, such as the first matrix, the last matrix, the odd numbered matrix, and the even number matrix, were proposed. Compared to the decomposition in [18], the proposed decomposition is seen to be more systematic and requires less numbers of additions and multiplications. We show a complexity comparison among the proposed decomposition and other methods in Table 1-2.

Reference number Conventional methods Proposed
Addition Multiplication Addition Multiplication
W. H. Chen at el [18]
3N/2(log2N1)+2 Nlog2N(3N/2)+4 Nlog2N N/2(log2N+1)
Z. Wang[13] DST-II N(74log2(N)2)+3 N(34log2(N)1)+3 Nlog2N N/2(log2N+1)
Cooley and Tukey [21] DFT Nlog2N (N/2)log2N Nlog2N (N/2)log2N

Table 1.

The comparison of computation complexity of conventional independent the DCT-II, DST-II, DFT, and hybrid DCT-II/DST-II/DFT

MatrixSize, N Conventional Proposed
Addition Multiplication Addition Multiplication
DCT-II 4 8 6 8 6
8 26 16 24 16
16 74 44 64 40
32 194 116 160 96
64 482 292 384 224
128 1154 708 896 512
256 2690 1668 2048 1152
DST-II 4 9 5 8 6
8 29 13 24 16
16 83 35 64 40
32 219 91 160 96
64 547 227 384 224
128 1315 547 896 512
256 3075 1283 2048 1152
DFT 4 8 4 8 4
8 24 12 24 12
16 64 32 64 32
32 160 80 160 80
64 384 192 384 192
128 896 448 896 448
256 2048 1024 2048 1024

Table 2.

Computational Complexity: DCT-II/DST-II/DFT

Applying (13), we can readily compute CN=2NXN. The inverse of CN can be obtained from the properties of the sparse Jacket matrix inverse:


The corresponding butterfly data flow diagram of CN is given in Fig. 1.

Figure 1.

Regular systematic butterfly data flow of DCT-II.

2.2. Recursive decomposition of the DST-II

The DST-II matrix [1-4] and [7] can be expressed as follows:


Similar to the procedure we have used in the DCT-II matrix, we first define the permuted DST-II matrix, S˜N as follows:


From (16), we can have a recursive form for YN as


where the submatrix AN can be calculated by


where UN and DN are, respectively, upper triangular and diagonal matrices. The upper triangular matrix UN={UN(m,n)} is defined as follows:


whereas the matrix DN is defined as before in (10). The derivation of (18) is given in Appendix C. Recursively applying (18) in (17), Recursively applying (18) in (17), we can find that


Further applying (17) to the Kronecker product [I2YN/2], the following general recursive form for DST-II matrix can be obtained as:


Note that if we compare (21) and (13), a similarity can be found in the proposed matrix decompositions. That is, starting from the common lowest order Y2=[1111], the discrete sine kernel matrix is recursively constructed. Especially, applying the relationship of UN=I˜NL˜NI˜N, where I˜N=[001010100] denotes the opposite diagonal identity matrix, the butterfly data flow of the DST-II matrix can be obtained from the corresponding that of the proposed DCT-II decomposition. The butterfly data flow graph of the DST-II matrix is shown in Fig. 2.

Now utilizing the properties of the BIJM, we can first obtain


such that the inverse of the matrix SN is given by


Note that applying again the properties of the BIJM and (18), a recursive form of the inverse DST-II can be easily obtained.

Figure 2.

Regular systematic butterfly data flow of DST-II.

2.3. Recursive decomposition of DFT

The DFT is a Fourier representation of a given sequence {x(n)},


where W=ej2π/N. The N -point DFT matrix can be denoted by FN={Wnm}. The N×N Sylvester Hadamard matrix is denoted by HN. The Sylvester Hadamard matrix is generated by the successive Kronecker products:


for N=4,8, In (25), we define H2=[1111]. We decompose a sparse matrix EN=PNF˜NWN in the following way:


where EN/2 is further decomposed by Lemma 1


where WN is the diagonal complex unit for the N-point DFT matrix. That is, we have WN=diag{W0,,WN1}.

Figure 3.

Butterfly data flow of DFT.

Similar to the development for DCT-II and DST-II, we first rewrite (26) using (27) as


[I2F˜N/2] in (28) can be recursively decomposed in the following way:


It is clear that the form of (29) is the same as that of (13), where we only need to change Ll to Pl and Dl to Wl for l{2,4,8,,N/2} to convert the DCT-II matrix into DFT matrix. Consequently, the butterfly data flow of the DFT matrix can be drawn in Fig. 3 using the baseline architecture of DCT-II.


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

We have derived recursive formulas for DCT-II, DST-II, and DFT. The derived results show that DCT-II, DST-II, and DFT 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. But our proposed method can be universally switching from DCT-II to DST-II, and DFT vice versa. Figs. 1-3 exhibit the similar recursive flow diagrams and let us motivate to develop universal hybrid architecture via switching mode selection. Moreover, the butterfly data flow graphs have log2N stages. From Fig.1, we can generate Figs. 2-3 according to the following proposed ways:

3.1. From DCT-II to DST-II

The N-point DCT-II of x is given by


The N-point DST-II of x is given by


Let CN and SN be orthogonal N×N DCT-II and DST-II matrices, respectively. Also, x=[x(0)x(1)x(N1)]T denotes the column vector for the data sequence x(n). Substituting m=Nk1, k=1,2,,N into (30), we have


Using the following trigonometric identity


(32) becomes


where CN=(Nk1) represents the reflected version of CN(k) and this can be achieved by multiplying the reversed identity matrix I¯N to CN. (34) can be represented in a more compact matrix multiplication form [13]:


where, MN=[M1IN/2],M1=[1001]

Then, the DST-II matrix is resulted from the DCT-II matrix. Note that compatibility property exists in the DCT-II and DST-II.

3.2. From DFT to DCT-II

The (m,n) elements of the DCT-II kernel matrix is expressed by


A new sequence x(1)(n) is defined by


For the sequence x(1)(n), we see that we can write

XNDCT(m)=cm2Nn=0N1x(1)(n)cosm(4n+1)π2N=cm2N n=0N1x(1)(n)cos2πm2N(2n+12)=cm2N(n=0N1x(1)(n)ej2πm(2n+1/2)/2N)=cm2N(ejπm/2Nn=0N1x(1)(n)ej2πmn/N)=cm2N(ejπm/2NFNx(1))E38

where R indicates a real part.

With the result above we have avoided computing a DFT of double size. We have


Now, the result can be put in the more compact matrix-vector form


Then, the DCT-II matrix is resulted from the DFT matrix.

3.3. From DCT-II and DST-II to DFT

We develop a relation between the circular convolution operation in the discrete cosine and sine transform domains. We need to measure half of the total coefficients. The main advantage of a proposed new relation is that the input sequences to be convolved need not be symmetrical or asymmetrical. Thus, the transform coefficients can be either symmetric or asymmetric [21].

From (30) and (31), it changes to coefficient for circular convolution (C) format. Thus, we have the following equations:


We can rewrite the DFT (24)


Multiplying (42) by 2ejπm/N, we can get


Comparing the first term of (41) with first one of (43), it can be seen that 2n=0N1x(n)(cos[m(2n+1)πN]) is decimated and asymmetrically extended of (41) with index m=0:N1. Similarly, 2n=0N1x(n)(sin[m(2n+1)πN]) is decimated and symmetrically extended of (41) with index m=1:N. It is observed that proper zero padding of the sequences, symmetric convolution can be used to perform linear convolution. The circular convolution of cosine and sine periodic sequences in time/spatial domain is equivalent to multiplication in the DFT domain. Then, the DFT matrix is resulted from the DCT-II and DST-II matrices.

3.4. Unified hybrid fast algorithm

Based on the above conversions from the proposed decomposition of DCT-II, we can form a hybrid fast algorithm that can cover DCT-II, DST-II, and DFT. The general block diagram of the proposed hybrid fast algorithm is shown in Fig. 4. The common recursive block of [P]N/2h1Lblockdiagonal()[I2Z2]Rblockdiagonal()[I2I2I2I2]QN/2h1 is multiplied repeatedly according to the size of the kernel with different transforms as like as bracket (((()))). The requiring computational complexity of individual DCT-II, DST-II, and DFT is summarized in Table 1 and Table 2. It can be seen that the proposed hybrid algorithm requires little more computations in addition and multiplication compared to Wang’s result [13]. However, the proposed scheme requires a much less computational complexity in addition and multiplication compared to those of the decompositions proposed by [11,13,18]. In addition, compared to these transforms, the proposed hybrid fast algorithm can be efficiently extensible to larger transform sizes due to its diagonal block-wise inverse operation of recursive structure. Moreover, the proposed hybrid structure is easily extended to cover different applications. For example, a base station wireless communication terminal delivers a compressed version of multimedia data via wireless communications network. Either DCT-II or DST-II can be used in compressing multimedia data since the proposed decomposition is based on block diagonalization it can significantly reduce its complexity due to simple structure[11,19, 22], for various multimedia sources. The DCT image coding can be easily implemented in the proposed hybrid structure as shown in Fig. 4(b). From (45), the DCT-II is obtained by taking a real part of multiplication result of ejπm/2N with FN={Wnm}. If the DCT-II is multiplied by I¯NCNMN, then we get DST. If the DCT and DST are convolved in time and frequency domain and multiplied by 2ejπm/N, the DFT matrix can be obtained. Thus, the proposed hybrid algorithm enables the terminal to adapt to its operational physical device and size.

Figure 4.

Recursive DCT-II/DST-II/DFT Structure Based on Jacket matrix.

3.5. Numerical simulations

As shown in [7] the coding performance DST outperforms DCT at high correlation values (ρ) and is very close to that of the KLT. Since the basis vectors of DCT maximize their energy distribution at both ends, hence the discontinuity appears at block boundaries due to quantization effects. However, since the basis vectors of DST minimizes their energy distribution at other ends, DST provides smooth transition between neighboring blocks. Therefore, the proposed hybrid transform coding scheme provides a consistent reconstruction and preserves more details, as shown in Fig. 6 with a size of 512 x 512 and 8 bits quantization.

Now consider an N×N block of pixels, X, containing xi,j,i,j=1,2,,N. We can write 2-D transformation for the kth block X as YS=TSQXkQT and YC=TCXk.

Depending on the availability of boundary values (in top- boundary and left-boundary) in images the hybrid coding scheme accomplishes the 2-D transform of a block pixels as two sequential 1-D transforms separately performed on rows and columns. Therefore the choice of 1-D transform for each direction is dependent on the corresponding prediction boundary condition.

  • Vertical transform (for each column vector): employ DST if top boundary is used for prediction; otherwise use DCT.

  • Horizontal transform (for each row vector): employ DST if left boundary is used for prediction; otherwise use DCT.

What we observed from numerical experiments is that the combined scheme over DCT-II only performs better in perceptual clarity as well as PSNR. Jointly optimized spatial prediction and block transform (see Fig. 5 (e) and (f)) using DCT/DST-II compression(PSNR 35.12dB) outperforms only DCT-II compression(PSNR 32.38dB). Less blocky artifacts are revealed compared to that of DCT-II. Without a priori knowledge of boundary condition, DCT-II performs better than any other block transform coding. The worst result is obtained using DST-II only.


4. Conclusion

In this book chapter, we have derived a unified fast hybrid recursive Fourier transform based on Jacket matrix. The proposed analysis have shown that DCT-II, DST-II, and DFT 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, and DFT. The proposed algorithm also uses the matrix product of recursively lower order diagonal sparse matrix and Hadamard matrix. The resulting signal flow graphs of DCT-II, DST-II, and DFT have a regular systematic butterfly structure. Therefore, the complexity of the proposed unified hybrid algorithm has been much less as its matrix size gets larger. This butterfly structure has grown by a recursive nature of the fast hybrid Jacket Hadamard matrix. Based on a systematic butterfly structure, a unified switching system can be devised. We have also applied the circulant channel matrix in our proposed method. Thus, the proposed hybrid scheme can be effectively applied to the heterogeneous transform systems having various matrix dimensions. Jointly optimized DCT and DST-II compression scheme have revealed a better performance (about 3dB) over the DCT or DST only compression method.

Figure 5.

Image Coding Results showing DCT-II only and jointly optimized DCT/DST-II compression (a) Original Lena image (b) zoomed original Lena image (c) DCT-II compressed Lena image(PSNR=32.38 dB) (d) Zoomed DCT-II compressed Lena image (e) DCT/DST-II compressed Lena image (PSNR=35.12 dB) (f) Zoomed DCT/DST-II compressed Lena image.



Appendix A

A Proof of Proposition 1

We use mathematical induction to prove Proposition 1. The lowest order BIJM is defined as


where C2=H22. Since


equation (4) holds for 2N = 8. Now we assume that the BIJM JN satisfies (4), i.e., JNJNT=N2IN. Since J2NJ2NT=(JNH2)(JNH2)T=(JNJNT)(H2H2T)=N2IN2I2=NI2N, this proposition is proved by mathematical induction that (4) holds for all 2N. If N=1, certainly J2J2T=I2.

Appendix B

A Proof of Lemma 1

According to the definition of an N×N matrix BN, BN is given as follows:


where ki=i+1. Since cos((2k+1)Φm)=2cos(2kΦm)cos(Φm)cos((2k1)Φm),we have


Using (47), BN can be decomposed as:


which proves (10) in Lemma 1.

Appendix C

A Proof of Equation (18)

By using the sum and difference formulas for the sine function, we can have


where ki=i+1,Φj=2j+1,i,j=0,1,,N1.

By taking (49) and into the right hand side of (18), we have


The left hand side of (18) matrix [A]N from [Y]N can be represented by


We can obtain (50) and (51) are the same and the expression of (18) is correct.



This work was supported by MEST 2012- 002521, NRF, Korea.


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Written By

Daechul Park and Moon Ho Lee

Submitted: 24 May 2014 Published: 03 June 2015