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A dyadic shift-invariant spectrum for the discrete cosine transform
LXGITAL SKXGL PROCESSING 2, 65-68
( 1992)
A Dyadic Shift-Invariant Spectrum for the Discrete Cosine Transform Nasir Ahmed and Bruce Armstrong Department of Electrical and Computer Albuquerque, New Mexico 87131
Engineering,
1. INTRODUCTION
University
of New Mexico,
b(m) I4 = {x(4)x(5)x(6)x(7)x(0)x(1)x(2)x(3)},
The discrete cosine transform (DCT) was introduced in 1974 by Ahmed et al. [l] . It is now widely used for a variety of data compression applications [ 21. In particular, in the area of digital image compression for transmission, it has been accepted as an industry standard [ 3 ] . The main objective of this communication is to introduce a magnitude spectrum for the DCT that is invariant to dyadic shifts of the data sequence. A corresponding phase spectrum is also proposed.
etc. (2d)
3. SIMILARITYTRANSFORMATION Given a data sequence x(m) , 0 < m G N - 1, its DCT is defined as [ 41
2. DYADICSHIFT If y (m),, 0 < m < N - 1, denotes the sequence that is obtained by subjecting a given data sequence x(m) , 0 < m < N - 1, to a dyadic shift of size I, then [4] y(mh
= x(m @ 0,
O
(1)
where @ denotes modulo 2 addition. To illustrate, suppose N = 8; then (1) yields the following dyaditally shifted sequences: {y(m)
{y(m)
h
= {r(a)x(3)x(o)x(l)x(s)x(7)x(4)x(5)} {y(m)
(2a)
= LX(O) fi
+
x cos (2m + l)kK 2N
’
O
(4)
(2b)
I3
= (~(3)~(2)x(l)~(O)~(7)~(6)~(5)~(4))
(3)
where X (112)is the kth DCT coefficient. Again, given the DCT sequence X(k), 0 < k < N - 1, the data sequence can be recovered using the following inverse DCT (IDCT) :
x(m)
I1
= {x(l)~(O)~(3)~(2)~(~)~(4)~(7)~(6)}
O
(2~)
For the purposes of discussion, we consider the case N =8in(l)andZ=lin(l).ThentheDCTin(3)can be expressed in matrix form as 1051-2004/92 $1.50 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.
l/E
D(3)
cos 34 cos 64 -cos 74 -co.5 44 -cos (fJ -cos 24 -cos 54
= ;
l/fi cos f#l cos 24 cos 39 cos 44 cos 54 cos 64 cos 74
llfi
cos 74 -cos 2f$ - cos 54 cos 44 cos 34 --OS 64 -cos cp
where 4 = 7r/ 16, and D (3) denotes the (8 X 8) DCT matrix. Since D (3) is an orthonormal matrix, we have
D-‘(3)
= D(3)‘,
WkL l/i2
l/E
cos 54 --OS 64 -cos 4 -cos 44 cos 74 cos 24 cos 34
- cos 5f$ --OS 64 cos 4 -cos 44 -cos 74 cos 24 -cos 34
(6)
= D(3)‘X(3),
cos -cos cos -cos cos -cos
24 34 44 54 6d 74
cos 64 cos 74 -cos 44 cos rp -cos 24 cos 54
7
(5)
where
where D-l ( 3) and D ( 3) ’ denote the inverse and the transpose of D (3)) respectively. Thus, the IDCT in (4) can be expressed in matrix form as x(3)
-co; c$ - cos’ 34
-cos 74 -cos 24 cos 54 cos 44 -cos 34 - cos 64 cos 4
M(3)
=
(7)
-0 10 0 10000000 00010000 00100000 00000100 00001000 00000001 00000010
0
0
0
o-
where x(3)‘=
[x(O)x(l)~(~)x(3)x(4)x(5)~(6)~(7)1
and X(3)’
By substituting (9) in (8) and using the IDCT tion in (7) we get
rela-
Y (3), = D(3)M(3)D(3)‘X(3),
(10)
= [X(0)X(1)X(2)X(3)
X(4)X(5)X(6)X(7)1. where the matrix Next, y(mh
the DCT is
of the dyatically
shifted
product
sequence
A(3), Y(3),
= D(3)M(3)D(3)’
(11)
(8)
= Do,,
is refered to as a similarity transformation corresponding to the DCT. This transformation relates the DCT of a dyadically shifted (size 1) sequence to the DCT of the original sequence.
where
Y (3); = [Y(O),Y(l),Y(2),Y(3),
Y(4),Y(5),Y(6),Y(7),1 and ~(3);
= IY(~),Y(~),Y(~),Y(~),
Y(~),Y(~),Y(~),Y(~),I. From (2a) it follows that y(3),
= M(3)x(3),
(9)
4. DYADIC SHIFT-INVARIANT MAGNITUDE SPECTRUM In(ll),wesubstituteforD(3)andM(3)using(5) and (9)) respectively, which results in the similarity transformation 0 0 0 0 a7 0 0 0
0 0
0
az) 0 (% - a,) 0 0
“0” 0 0
-(cllo+
0
(%
0 - 4 0 0 0 0
-a5
0
-ho+
cy2)
(12)
where ffl
= cos 34 cos f#J;
ff2
a3
= cos 34 cos 74;
a 4 = cos I$ cos 54
a5 = 2 cos 24 cos 64;
= cos 54 cos 7f#J
ff6 = cos26c$ - cos22~
a7 = -2 cos24c#L Substitution of ( 12) in (10) leads to the relationships between the DCT coefficients dically shifted (size 1) sequence y(m), , 0 and the DCT coefficients of the original x(m), 0
following of a dya6 m < 7, sequence
[X2(3) + X2(5)], andX2(4) are invariant to subjecting x (m) , 0 G m G 7, to a dyadic shift of size 1. It is straightforward to use the above approach to show that these five quantities are also invariant to subjecting x ( m) , 0 G m G 7, to dyadic shifts of size 1, where 1 = 2, 3,. . . , 7. Thus we can now define the following dyadic shift-invariant DCT magnitude spectrum: PO’
pk = [X2(k)
+ X2(N
+ (a, - a,)X(7)
Y(7),
=
-
Y(2),
= agX(2)
+ &,X(6)
(13d)
$. = 0 or ir,
Y(6),
= +X(2)
- a,X(G)
(13e)
& = tan-‘[X(
Y(3),
=
Y(5),
= -(a,
Y(4),
= &,X(4).
-
a,)X(l)
(a,
(lab)
where N = 8. Next, the following be defined:
= (a, + %)X(l)
-(a3
=
(16)
IX(N/2)1,
(13a)
Y(l),
(a3
- k)]““, l
Pn/2
Y(O), = X(O)
IX(O)1
(13c)
(Y2)x(7)
+
- %)X(3)
- (al + a,)X(5)
(13f)
+ a2)X(3)
+ (a3 - (-u,)X(5)
(13g) (13h)
Using straightforward trignometric manipulations, it can be shown that the (Y,, 1~ m G 7, in ( 12) have the following properties:
DCT phase spectrum can also
N - k)/X(k)], l
4 N,2 = Oor r,
(17)
where N = 8. It follows that given the DCT magnitude and phase spectra, we can recover the given data sequence x ( m) , OGmG7.
5. CONCLUDING REMARKS =
ffl&2
cky?L4
(14b)
a25 + lx2 6 = 1 a7
2’1
(14c) (14c)
*
Combining (13) and (14) we obtain dyadic shift invariants:
From
X2(O),
the following
Y2(0),
= X2(O)
Y2(1),
+ Y2(7),
= X2(1)
+x2(7)
Y2(2),
+ Y2(6),
= X2(2)
+X2(6)
Y2(3),
+ Y2(5),
= X2(3)
+ X2(5)
Y2(4),
= X2(4).
(15)
we conclude
[X2(1)
that
+ X2(7)1,
the
[X”(2)
(15) five
quantities
+ X2(6)1,
The authors have developed closed-form expressions for the elements of the similarity transformationA(n),n=log2N,whereA(3)in(12)isaspecial case for N = 8. The related analysis that follows leads to the conclusion that the magnitude and phase spectra in ( 16) and ( 17)) respectively, are valid for any N = 2”. Work is ongoing to study the effectiveness of these spectra for securing data compression in applications which can be modeled as first-order Markov processes, e.g., digital image processing.
REFERENCES 1. Ahmed, N., Natarajan, T., and Rao, K. R. Discrete cosine form. IEEE Tram. Comput. C-23 (Dec. 1974)) 90.
trans-
2. Elliott, D. G., and Rao, K. R. Fast Tran.sforms:Algorithms, Analysis, Applications. Academic Press, New York, 1982. 3. Video compression makes big gains. IEEE Spectrum 28 (Oct. 1991)) 16. 4. Ahmed, N., and Rao, K. R. Orthogonal Transforms for Digital Signal Processing. Springer-Verlag, New York, 1975.
NASIR AHMED was born in Bangalore, India, in 1940. He received the B.S. degree in electrical engineering from the University of Mysore, India, in 1961, and the MS. and Ph.D. degrees from the University of New Mexico in 1963 and 1966, respectively. From 1966 to 1968 he worked as Principal Research Engineer in the area of information processing at the Systems and Research Center, Honeywell, Inc., St. Paul, Minnesota. He was with Kansas State University, Manhattan, from 1968 to 1983. Since 1983 he has been a professor in the electrical and computer engineering department at the University of New Mexico, Albuquerque. He became the Chairman of this department in July 1989. In August 1985 he was
awarded one of the twelve Presidential Professorships at the University of New Mexico. He is the leading author of Orthogonal Transforms for Digital Signal Processing ( Springer-Verlag, 1975) and Discrete-Time Signals and Systems (Reston, 1983)) and coauthor of Computer Science Fundamentals (Merrill, 1979). He is also the author of a number of technical papers in the area of digital signal processing. Dr. Ahmed was an associate editor for the IEEE Transactions on Acoustics, Speech, and Signal Processing (19821984) and is currently an associate editor for the IEEE Transactions on Electromagnetic Compatibility (Walsh Functions Applications). BRUCE ARMSTRONG received the B.S. and M.S. degrees in electrical engineering from the University of New Mexico in Albuquerque New Mexico. Currently he is a doctoral student in electrical engineering at the University of New Mexico. He has worked as an electrical engineer for 21 years at the BDM Corporation, Albuquerque, New Mexico, in electromagnetic coupling, EMP susceptibility and hardening, and adaptive modeling. His current research interests include adaptive beam forming, fuzzy logic control, and video lossless and lossy data compression. Mr. Armstrong is a student member of the IEEE and the IEEE Computer Society.
( 1992)
A Dyadic Shift-Invariant Spectrum for the Discrete Cosine Transform Nasir Ahmed and Bruce Armstrong Department of Electrical and Computer Albuquerque, New Mexico 87131
Engineering,
1. INTRODUCTION
University
of New Mexico,
b(m) I4 = {x(4)x(5)x(6)x(7)x(0)x(1)x(2)x(3)},
The discrete cosine transform (DCT) was introduced in 1974 by Ahmed et al. [l] . It is now widely used for a variety of data compression applications [ 21. In particular, in the area of digital image compression for transmission, it has been accepted as an industry standard [ 3 ] . The main objective of this communication is to introduce a magnitude spectrum for the DCT that is invariant to dyadic shifts of the data sequence. A corresponding phase spectrum is also proposed.
etc. (2d)
3. SIMILARITYTRANSFORMATION Given a data sequence x(m) , 0 < m G N - 1, its DCT is defined as [ 41
2. DYADICSHIFT If y (m),, 0 < m < N - 1, denotes the sequence that is obtained by subjecting a given data sequence x(m) , 0 < m < N - 1, to a dyadic shift of size I, then [4] y(mh
= x(m @ 0,
O
(1)
where @ denotes modulo 2 addition. To illustrate, suppose N = 8; then (1) yields the following dyaditally shifted sequences: {y(m)
{y(m)
h
= {r(a)x(3)x(o)x(l)x(s)x(7)x(4)x(5)} {y(m)
(2a)
= LX(O) fi
+
x cos (2m + l)kK 2N
’
O
(4)
(2b)
I3
= (~(3)~(2)x(l)~(O)~(7)~(6)~(5)~(4))
(3)
where X (112)is the kth DCT coefficient. Again, given the DCT sequence X(k), 0 < k < N - 1, the data sequence can be recovered using the following inverse DCT (IDCT) :
x(m)
I1
= {x(l)~(O)~(3)~(2)~(~)~(4)~(7)~(6)}
O
(2~)
For the purposes of discussion, we consider the case N =8in(l)andZ=lin(l).ThentheDCTin(3)can be expressed in matrix form as 1051-2004/92 $1.50 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.
l/E
D(3)
cos 34 cos 64 -cos 74 -co.5 44 -cos (fJ -cos 24 -cos 54
= ;
l/fi cos f#l cos 24 cos 39 cos 44 cos 54 cos 64 cos 74
llfi
cos 74 -cos 2f$ - cos 54 cos 44 cos 34 --OS 64 -cos cp
where 4 = 7r/ 16, and D (3) denotes the (8 X 8) DCT matrix. Since D (3) is an orthonormal matrix, we have
D-‘(3)
= D(3)‘,
WkL l/i2
l/E
cos 54 --OS 64 -cos 4 -cos 44 cos 74 cos 24 cos 34
- cos 5f$ --OS 64 cos 4 -cos 44 -cos 74 cos 24 -cos 34
(6)
= D(3)‘X(3),
cos -cos cos -cos cos -cos
24 34 44 54 6d 74
cos 64 cos 74 -cos 44 cos rp -cos 24 cos 54
7
(5)
where
where D-l ( 3) and D ( 3) ’ denote the inverse and the transpose of D (3)) respectively. Thus, the IDCT in (4) can be expressed in matrix form as x(3)
-co; c$ - cos’ 34
-cos 74 -cos 24 cos 54 cos 44 -cos 34 - cos 64 cos 4
M(3)
=
(7)
-0 10 0 10000000 00010000 00100000 00000100 00001000 00000001 00000010
0
0
0
o-
where x(3)‘=
[x(O)x(l)~(~)x(3)x(4)x(5)~(6)~(7)1
and X(3)’
By substituting (9) in (8) and using the IDCT tion in (7) we get
rela-
Y (3), = D(3)M(3)D(3)‘X(3),
(10)
= [X(0)X(1)X(2)X(3)
X(4)X(5)X(6)X(7)1. where the matrix Next, y(mh
the DCT is
of the dyatically
shifted
product
sequence
A(3), Y(3),
= D(3)M(3)D(3)’
(11)
(8)
= Do,,
is refered to as a similarity transformation corresponding to the DCT. This transformation relates the DCT of a dyadically shifted (size 1) sequence to the DCT of the original sequence.
where
Y (3); = [Y(O),Y(l),Y(2),Y(3),
Y(4),Y(5),Y(6),Y(7),1 and ~(3);
= IY(~),Y(~),Y(~),Y(~),
Y(~),Y(~),Y(~),Y(~),I. From (2a) it follows that y(3),
= M(3)x(3),
(9)
4. DYADIC SHIFT-INVARIANT MAGNITUDE SPECTRUM In(ll),wesubstituteforD(3)andM(3)using(5) and (9)) respectively, which results in the similarity transformation 0 0 0 0 a7 0 0 0
0 0
0
az) 0 (% - a,) 0 0
“0” 0 0
-(cllo+
0
(%
0 - 4 0 0 0 0
-a5
0
-ho+
cy2)
(12)
where ffl
= cos 34 cos f#J;
ff2
a3
= cos 34 cos 74;
a 4 = cos I$ cos 54
a5 = 2 cos 24 cos 64;
= cos 54 cos 7f#J
ff6 = cos26c$ - cos22~
a7 = -2 cos24c#L Substitution of ( 12) in (10) leads to the relationships between the DCT coefficients dically shifted (size 1) sequence y(m), , 0 and the DCT coefficients of the original x(m), 0
following of a dya6 m < 7, sequence
[X2(3) + X2(5)], andX2(4) are invariant to subjecting x (m) , 0 G m G 7, to a dyadic shift of size 1. It is straightforward to use the above approach to show that these five quantities are also invariant to subjecting x ( m) , 0 G m G 7, to dyadic shifts of size 1, where 1 = 2, 3,. . . , 7. Thus we can now define the following dyadic shift-invariant DCT magnitude spectrum: PO’
pk = [X2(k)
+ X2(N
+ (a, - a,)X(7)
Y(7),
=
-
Y(2),
= agX(2)
+ &,X(6)
(13d)
$. = 0 or ir,
Y(6),
= +X(2)
- a,X(G)
(13e)
& = tan-‘[X(
Y(3),
=
Y(5),
= -(a,
Y(4),
= &,X(4).
-
a,)X(l)
(a,
(lab)
where N = 8. Next, the following be defined:
= (a, + %)X(l)
-(a3
=
(16)
IX(N/2)1,
(13a)
Y(l),
(a3
- k)]““, l
Pn/2
Y(O), = X(O)
IX(O)1
(13c)
(Y2)x(7)
+
- %)X(3)
- (al + a,)X(5)
(13f)
+ a2)X(3)
+ (a3 - (-u,)X(5)
(13g) (13h)
Using straightforward trignometric manipulations, it can be shown that the (Y,, 1~ m G 7, in ( 12) have the following properties:
DCT phase spectrum can also
N - k)/X(k)], l
4 N,2 = Oor r,
(17)
where N = 8. It follows that given the DCT magnitude and phase spectra, we can recover the given data sequence x ( m) , OGmG7.
5. CONCLUDING REMARKS =
ffl&2
cky?L4
(14b)
a25 + lx2 6 = 1 a7
2’1
(14c) (14c)
*
Combining (13) and (14) we obtain dyadic shift invariants:
From
X2(O),
the following
Y2(0),
= X2(O)
Y2(1),
+ Y2(7),
= X2(1)
+x2(7)
Y2(2),
+ Y2(6),
= X2(2)
+X2(6)
Y2(3),
+ Y2(5),
= X2(3)
+ X2(5)
Y2(4),
= X2(4).
(15)
we conclude
[X2(1)
that
+ X2(7)1,
the
[X”(2)
(15) five
quantities
+ X2(6)1,
The authors have developed closed-form expressions for the elements of the similarity transformationA(n),n=log2N,whereA(3)in(12)isaspecial case for N = 8. The related analysis that follows leads to the conclusion that the magnitude and phase spectra in ( 16) and ( 17)) respectively, are valid for any N = 2”. Work is ongoing to study the effectiveness of these spectra for securing data compression in applications which can be modeled as first-order Markov processes, e.g., digital image processing.
REFERENCES 1. Ahmed, N., Natarajan, T., and Rao, K. R. Discrete cosine form. IEEE Tram. Comput. C-23 (Dec. 1974)) 90.
trans-
2. Elliott, D. G., and Rao, K. R. Fast Tran.sforms:Algorithms, Analysis, Applications. Academic Press, New York, 1982. 3. Video compression makes big gains. IEEE Spectrum 28 (Oct. 1991)) 16. 4. Ahmed, N., and Rao, K. R. Orthogonal Transforms for Digital Signal Processing. Springer-Verlag, New York, 1975.
NASIR AHMED was born in Bangalore, India, in 1940. He received the B.S. degree in electrical engineering from the University of Mysore, India, in 1961, and the MS. and Ph.D. degrees from the University of New Mexico in 1963 and 1966, respectively. From 1966 to 1968 he worked as Principal Research Engineer in the area of information processing at the Systems and Research Center, Honeywell, Inc., St. Paul, Minnesota. He was with Kansas State University, Manhattan, from 1968 to 1983. Since 1983 he has been a professor in the electrical and computer engineering department at the University of New Mexico, Albuquerque. He became the Chairman of this department in July 1989. In August 1985 he was
awarded one of the twelve Presidential Professorships at the University of New Mexico. He is the leading author of Orthogonal Transforms for Digital Signal Processing ( Springer-Verlag, 1975) and Discrete-Time Signals and Systems (Reston, 1983)) and coauthor of Computer Science Fundamentals (Merrill, 1979). He is also the author of a number of technical papers in the area of digital signal processing. Dr. Ahmed was an associate editor for the IEEE Transactions on Acoustics, Speech, and Signal Processing (19821984) and is currently an associate editor for the IEEE Transactions on Electromagnetic Compatibility (Walsh Functions Applications). BRUCE ARMSTRONG received the B.S. and M.S. degrees in electrical engineering from the University of New Mexico in Albuquerque New Mexico. Currently he is a doctoral student in electrical engineering at the University of New Mexico. He has worked as an electrical engineer for 21 years at the BDM Corporation, Albuquerque, New Mexico, in electromagnetic coupling, EMP susceptibility and hardening, and adaptive modeling. His current research interests include adaptive beam forming, fuzzy logic control, and video lossless and lossy data compression. Mr. Armstrong is a student member of the IEEE and the IEEE Computer Society.