The Vault

Relationship Between DCT II, DCT VI, and DST VII Transforms
Research Paper / Feb 2014




Yuriy A. Reznik 


InterDigital Communications 

9710 Scranton Rd, San Diego, CA 92121, USA 






Discrete Sine Transfonns of type VII (DST-VII) have recently 

received considerable interest in video coding. In this paper, 

we show that there exists a direct connection between DST­

VII and DCT-II transfonns, allowing their joint computation 

for certain transfonn sizes. This connection also yields fast 

algorithms for constructing DCT-VI and DCT-VII. 


Index Terms- KLT, DCT-II, DCT-VI, DST-VII, factor­

izations, video coding. 




The Discrete Cosine Transfonns of types II and IV are among 

most fundamental, well understood, and much appreciated 

tools in data compression. The DCT-II is used at the core of 

standards for image and video compression, such as JPEG, 

ITU-T H.26x-series, and MPEG 1-4 standards [1]. The 

DCT-IV is used in audio coding algorithms, such as ITU-T 

Rec. G.722.1, MPEG-4 AAC, and others [2]. Such transforms 

are very well studied, and a number of efficient technique ex­

ists for their computation [1, 3, 4, 5, 6, 7]. 


Much less known are so-called "odd" sinusoidal trans­

fonns: Discrete Cosine and Sine Transfonns of types V, VI, 

VII, and VIII. Existence of some of such transfonns was dis­

covered by A. Jain in 1979 [8]. A complete tabulation was de­

veloped in 1985 by Wang and Hunt [9]. However, not much 

work has followed. Surveys of related results can be found 

in [10, 3]. 


Recently, DST of types VI and VII have surfaced as useful 

tools in image and video coding. In 2010, Han, Saxena, and 

Rose have shown that DST-VII produce good approximations 

of Karhunen-Loeve Transform (KLT) for model of residual 

signals after Intra-prediction [11]. This was subsequently val­

idated in the course of experimental work on ISO/IEC/ITU-T 

High Efficiency Video Coding (HEVC) standard [12, l3]. 


The adoption of DST-VII in HEVC has prompted a dis­

cussion on the existence of fast algorithms for computing of 

such transfonns [12, 14]. This question was addressed in 

2011 by Chivukula and Reznik[15], who have established 

connection between DST-VII and DFT. 


978-1-4799-0356-6/13/$31.00 ©2013 IEEE 5642 


This paper offers an alternative solution by establishing a 

mapping between DST-VII, DCT-VI, and DCT-II. This map­

ping yields fast algorithms not only for DST-VIIVII, but also 

for DCT-VINII, as well as possible joint factorizations of 

such transfonns. The obtained mapping may also be of inter­

est from methodological standpoint, as it suggests additional 

connections between DST-VII, DCT-VI and KLT of residual 

and mixed signals. 


The rest of this paper is organized as follows. Section 2 

provides definitions. Section 3 establishes mapping between 

DCT-II, DCT-VI and DST-VII transfonns. Section 4 explains 

how this mapping can be used to construct fast algorithms. 

Discussion and concluding remarks are offered in Section 5. 




Let N be the length of data sequence. The matrices of Dis­

crete Fourier Transfonn (DFT) and Discrete Cosine and Sine 

transforms of types II, III, IV, VI, and VII will be defined as 



DFT: [FN ] mn 

DCT-II: [cy] mn 

DCT-III: [cyI] mn 

DCT-IV: [C'zn mn 

DCT-VI: [Ct�l] mn = 

DCT-VII: [Ct�l] mn = 

DST-VI: [stI] mn 

DST-VII: [SVII] = N mn 


e_j2n;;n, m, n E [O, N-l] 

cosm(2;�1)7r, m, nE[O, N-l] 


(2m+ l)n7r [0 N 1] cos 2N ' m, n E , -7r (2m+l)(n+ l) E [0 N-l] cos 4N ,m, n , 

m(2n+ l)7r [0 N] cos 2N+l ' m, n E , 

(2m+ l)n7r [0 N] cos 2N+l ' m, n E , 


. (m+l)(2n+ l)7r E [0 N-l] Sill 2N+l , m, n , . (2m+l)(n+ l)7r E [0 N-l] Sill 2N+l , m, n , 


In the above definitions, we have intentionally omitted 


nonnalization constants (such as y!2/N and Ai = [l/a��



conventionally used in definition of DCT-II) as they don't af­

fect factorization structures of the transfonns. Sub-indices N 

or N + 1 indicate lengths of the transfonns. We follow Wang 

and Hunt's convention of coupling N -point DST-VINII with 

N + I-point DCT-VINII [9]. 


As easily noticed, transfonns of types II and III, as well 

as VI and VII are closely related: 



N - N , N+l - N+l' N - N . 


















1--- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 

i 2N+1-point OCT-II i 

, I 1 Yo Yo 1 Xo 



















N -point 





YN., X2N•2 


Y, x" 


Y, x, 


Y, x, 




r - - - - - - - - - - - - - - - - -


(2N-;1 )



��i�t-oCT =ii --------------1 


I , 

� : 1 Xo 














r------ .. 


: N-point : :_����V�I_! 







i �t;'�I� : 

I _______ ! 






XN(2N+l)-3 2N+1-point 






- YN.2 







i �t;'�I� : 

1 I _______ ! 1 

I ' !...---------------------------------------------------: 




Fig. 1. Computing DCT-II of composite sizes: (a) split of 2N + I-point DCT-II into an N + I-point DCT-VI and N-point 

DST-VI; (b) computation of DCT-II of length N(2N + 1)_ 







In this section we prove the following statement. 


Theorem 1. The following holds: 


) (  


where Q2N+l is a matrix, such that when applied to a vector 

x, it produces the following sign alterations and reordering: 



( - I ) iXN+ Hi 


i = O, _ .. , N, 

i = O, . . .  , N  - 1, 




and IN and J N are N x N identity and order-reversal matri­

ces respectively. 


Proof Let us consider a 2N + I-long input sequence X 

Xo, . . .  , X2N, and apply DCT-II over it: 



7f(2n + I)k 


L Xn cos 2(2N + 1) , k = 0 ,  . . .  , 2N. n=O 




We first look at even output values (k = 2i, i = 0, ... ,N): 




7f(2n + 1)2i 

L Xn cos 2(2N + 1) n=O 




7f(2n + I)i 

L Xn cos 2N + 1 . n=O 


We split this sum as follows: 


ell � 7f(2n + I)i 

X2i - L Xn cos 2N + 1 n=O 



� 7f(2n + I)i L Xn cos 2N + 1 n=N+ l  




7f(2(2N - n) + I)i 

L Xn cos 2N + 1 n=N+ l  



( )  � 7f 2n + 1 i L X2N -n cos 2N + 1 ' n=O 


which implies that 




9-point OFT 


Yo Yo 


y, y, 


y, y, 


y, y, 


y, y, 


y, y, 


y, y, 


y, y, 


y, y, 








, , , , 

,,: :0 Xo 


_11 X2 


x, ��--_H�+,������--�����:� � 




--:3 � 






4 Xa 


x, -i-..,L-fH.�_+_<i--"-::..:=-=-=-'---'-'i;---__r--"-'--�___:_� x, 

:- : 


x, -i-���_+_�------��----� __ ���:�'�: � , 


x, �4T--����--����--�-4_,��ir�t+: Xs 


x, �4---��, ����� 

, , 

, " , 


: sln(t ) : : : ______________________________________ J : 







Fig. 2. Flow-graph of Winograd's factorization of DFT of length 9, and flow-graphs of 9-point DCT-II, 5 -point DCT-VI, and 

4-point DST-VII implied by mappings (1,3). 


where XC v I is a DCT-VI transform over the first N + 1 ele­

ments of input sequence x, and X,CV1 is a DCT-VI transform 

over the following input: 


, _ [ X2N-n, xn - 0 ,  

if n = 0 ,  . . .  , N - 1, 

ifn= N. 


We now turn our attention to the odd output values (k = 

2i + 1, i = 0 ,  . . .  , N - 1): 




7r(2n + 1)(2i + 1) 

� Xn cos 2( 2N + 1) n=D 


( )i+l � . 7r

(N -n)(2i + 1) 


-1 � X2N -n Sill --'----'-'---'-

n=D 2N + 1 


We split this sum as follows: 


XCII ( )i �


. 7r(N -n)(2i + 1) 

2i+ 1 + -1 � X2N -n Sill ---'--2N.,....,-'+-'-1---'-n=D 


(_l)i+l � . 7r

(N-n)(2i+l) � X2N-n Sill 2N + 1 n=N+l 


( )i 






. 7r(n + 1) (2i + 1) 

-1 XN-l-n Sill , 2N +1 n=D 


which implies that 


where XSVll is an N-point DST-VII transform over a se­



Xn = XN+l+n, n = 0 ,  . . .  , N - 1, 





VII and XS is an N -point DST-VII transform over a sequence 


Xn=XN-l-n, n=O, ... , N - l. 


By combining all these mappings we arrive at expression (1). 


We present a flowgraph of the resulting mapping between 

DCT-II, DCT-VI, and DST-VII transforms in Figure 1.a. Only 

2N additions, permutations and sign changes are needed to 

convert output of DCT-VI, and DST-VII into DCT-II. 





4. 1. Connection to DFT 


From (1) it follows that fast computation of DCT-VI and DST­

VII can be reduced to computing subsets of DCT-II. Accord­

ing to Heideman [4] it is also known that computing of DCT­

II of odd numbers is equivalent to computing same-length 

DFT. Considering 2N + I-point transforms, we can summa­

rize Heideman's result as follows: 





H ( [R(F2N+l)]rows D, ... ,N ) H (3) 2N+l - 1 ['2s(F )] 2, 2N+l rows N+l, ... ,2N 


where R (F2N+d and '2s (F2N+l) denote real and imaginary 

parts of the DFT transform matrix of size 2N + 1, and Hl and 

H2 are some permutation and sign-inversion matrices [4]. 


In combination with (1) this formula shows that an N + 1-

point DCT-VI, an N-point DST-VII and an 2N + I-point 

DCT-II can be computed by mapping to a 2N + I-point DFT. 

Since many algorithms for computing of DFT are readily 

available (see e.g. [16]), this automatically leads to to fast 

algorithms for computing DCT-VI and DST-VII. 




2N+1-point OCT-II 2N-point OCT-II 


(a) (b) 


Fig. 3. Conceptual illustration of decompositions of (a) 2N + I-point DCT-II (1) and (b) 2N -point DCT-II (4). 


4.2. Examples of fast algorithms for N = 4 


We use Winograd DFT module of length 9 shown in Fig­

ure 2.a. This particular factorization comes from [16]. By 

using this ftowgraph and mappings (3) and (1) we easily ob­

tain 9-point DCT-I1, 5-point DCT-VI, and 4-point DST-VII. 

This is shown in Figure 2.b. 


We note that all these algorithms are very efficient in 

terms of multiplicative complexity. Thus, obtained 9-point 

DCT-II requires only 8 non-trivial multiplications. In con­

trast, the least complex algorithms for computing DCT-ll of 

size 8 (nearest dyadic-size) requires 11 multiplications[7]. 


The obtained 4-point DST-VII is also very efficient: it 

uses only 5 multiplications. This factorization is immediately 

suitable for implementing an integer approximation of DST­

VII transform defined in HEVC standard [13]. 


Finally, factorization of a 5-point DCT-VI shown in Fig­

ure 2.b needs only 3 real multiplications and 2 shifts (multi­

plications by factors 1/2). 


4.3. Fast computing of transforms of length 2k N (2N + 1) 


It is known that a transform of a composite length N = pq, 

where p and q are co-prime, can be decomposed into a cas­

cade of p q-point transforms and q p-point transforms fol­

lowed by pq - p - q - 1 additions. This class of techniques 

is called Prime Factor Algorithms (PFA) [17, 18]. 


In Figure 1.b, we show how to compute DCT-II of length 

N(2N + 1). This factorization includes 2N + 1 N-point DCT­

II sub-transforms, and additionally N 2N + I-point DCT­

II transforms, which, in turn include N -point DST-VII as 

part of their ftowgraph. Hence, a system that implements 

and uses N-point DCT-II and DST-VII, can easily compute 

an N(2N + 1) transform by reusing them. Same principle 

more generally applies to computing transforms of lengths 

2k N(2N + 1). 


Embedded factorization structures including DST-VII 

blocks in ftowgraphs for DCT-II can be of interest to hard­

ware implementations, as it offers potential for reducing the 

area, cost, and power usage of a circuit responsible for com­

puting transforms. 






We notice that decomposition (1) looks very similar to the 

well-known split of even-sized DCT-I1 (see, e.g. [3]): 




where P2N is a certain permutation matrix. This split leads 

to recursive construction due to reappearance of DCT-ll in the 

upper part of decomposition. In contrast, our decomposition 

of 2N + I-point DCT-I1 (1) does not immediately lead to a 



In Figure 3 we offer conceptual illustration of both de­

compositions (1) and (4). Input data samples are denoted as 

YN, ... ,YI,ZO,XI, ... ,XN in a 2N + I-point case (a), and 

YN, ... , YI, Xl, ... , XN in 2N-point case (b). It is shown that 

the lower (right) portion of DCT-I1 transform becomes es­

sentially equivalent to DST-VII (or DCT-IV) transform over 

residual samples Y; = Yi - Xi, while the upper (right) por­

tion of DCT-I1 transform becomes essentially equivalent to 

DCT-VI (or DCT-II) transform over sums: x; = Xi + Yi, 

(i = 1, . . .  , N). In the 2N + I-point case, the upper trans­

form also absorbs the middle sample ZOo 


This illustration may be insightful for understanding 

meanings of the involved transforms. For instance, in sig­

nal processing, it is customary to think of DCT-II as an 

approximation of KLT for 1-st order Markov source with 

high correlation coefficient. Decomposition in Figure 3.a 

shows that DST-VII, as well as DCT-VI (with some permuta­

tions and sign changes) can be understood as approximations 

of KLT over residual or mixed signals with progressively 

increasing distances between samples. Similarly, decom­

position in Figure 3.b shows that in case of a 2N-sample 

arrangement, it is DCT-IV and DCT-II that can be understood 

as approximations of KLT over residual and mixed signals. 


The obtained relationship (1) may also be instrumental in 

showing that DST-VII-based coding of Intra-prediction resid­

ual is essentially equivalent to performing L -shaped DCT­

II, where one part of L -shape corresponds to boundary pix­

els, and the other part absorbs pixels predicted based on this 

boundary. A design of direction-adaptive transforms based on 

similar idea was proposed in [19]. 






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