Reversible integer color transform with bitconstraint
ABSTRACT In color image processing, the RGB color coordinate is usually transformed into another one (e.g., YIQ or KLA) for system fitting or other purposes. Most of the color transforms are done by 3×3 matrices. However, these matrices are always not fixedpoint. In this paper, we use a systematic algorithm to convert every 3×3 color transform into a reversible integertointeger transform. The resulted transform can be implemented with only fixedpoint processor and no floatingpoint processor is required. Moreover, with the use of laddertruncation technique, we can make least bit of the output the same as that of the input, and the long bitlength problem that always occurs for other integer transforms can be avoided. We derive the integer color transforms of RGBtoKLA, IV_{1}V_{2}, YCrCb, DCT, and YIQ successfully.

Conference Paper: Avoidance of singular point in reversible KLT.
[Show abstract] [Hide abstract]
ABSTRACT: In this report, permutation of order and sign of signals are introduced to avoid singular point problem of a reversible transform. When a transform is implemented in the lifting structure, it can be "reversible" in spite of rounding operations inside the transform. Therefore it has been applied to lossless coding of digital signals. However some coefficient values of the transform have singular points (SP). Around the SP, rounding errors are magnified to huge amount and the coding efficiency is decreased. In this report, we analyze the SP of a three point KLT for RGB color components of an image signal, and introduce permutation of order and sign of signals to avoid the SP problem. It was experimentally confirmed that the proposed method improved PSNR by approximately 15 [dB] comparing to the worst case.Proceedings fo the Picture Coding Symposium, PCS 2010, Nagoya, Japan, 810 December, 2010; 01/2010  SourceAvailable from: Sejung YangJournal of Electronic Imaging 07/2009; 18:033010. · 0.85 Impact Factor
 SourceAvailable from: unisi.it[Show abstract] [Hide abstract]
ABSTRACT: The feasibility of lossless compression of encrypted images has been recently demonstrated by relying on the analogy with source coding with side information at the decoder. However previous works only addressed the compression of bilevel images, namely sparse black and white images, with asymmetric probabilities of black and white pixels. In this paper we investigate the possibility of compressing encrypted grey level and color images, by decomposing them into bitplanes. A few approaches to exploit the spatial and crossplane correlation among pixels are discussed, as well as the possibility of exploiting the correlation between color bands. Some experimental results are shown to evaluate the gap between the proposed solutions and the theoretically achievable performance.01/2008;
Page 1
REVERSIBLE INTEGER COLOR TRANSFORM WITH BITCONSTRAINT
SooChang Pei, JianJiun Ding
Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C
Email address: pei@cc.ee.ntu.edu.tw
ABSTRACT
In color image processing, the RGB color coordinate is
usually transformed into another one (e.g., YIQ or KLA)
for system fitting or other purposes. Most of the color
transforms are done by 3×3 matrices. However, these ma
trices are always not fixedpoint. In this paper, we use a
systematic algorithm to convert every 3×3 color transform
into a reversible integertointeger transform. The resulted
transform can be implemented with only fixedpoint proc
essor and no floatingpoint processor is required. More
over, with the use of laddertruncation technique, we can
make least bit of the output the same as that of the input,
and the long bitlength problem that always occurs for
other integer transforms can be avoided. We derive the
integer color transforms of RGBtoKLA, IV1V2, YCrCb,
DCT, and YIQ successfully.
1. INTRODUCTION
In color image processing, there are varieties of color sys
tems. Most of them use three components to represent a
color. Some popular ones are RGB (red, blue, and green),
KLA (KarhunenLoeve average, highest ability for color
decorrelation), IHS (intensity, hue, saturation), IV1V2
(equivalent to IHS) [1][2], YCrCb, DCT, and YIQ. Since
there are a variety of color systems, in image processing,
we usually have to transform one color system into an
other one. The transformation is done by a 3×3 matrix. For
example, the transformation of RGBtoKLA is [2]:
⎡
=
−
77447. 022661. 0
Since the entries of color transforms are usually not
binary fixedpoint, we should use the floatingpoint proc
essor to implement them. When using the fixedpoint
processor, we should approximate them by binary fixed
point matrices. Unfortunately, the approximated matrices
are always irreversible. For example, suppose that we use
the way of rounding to approximate KKLA by a binary
fixedpoint matrix, i.e., RKLA = round(AKLA2K)⋅2−K and
SKLA = round(AKLA
it can be shown that
SKLA⋅ ⋅RKLA ≠ I no matter how large K is.
. (1)
⎥
⎦
⎥
⎤
⎢
⎣
⎢
−
−−
59063. 0
56194. 019322. 080429. 0
57912. 060238 . 054933. 0
KLA
A
− −12K)⋅2−K. From computer experiments,
(2)
Thus if we use the fixedpoint processor to implement the
color transform, the reversibility property is always lost. It
affects the performance of many image processing appli
cations. For example, we usually hide the watermark in
formation in the least bit. If we can not recover the origi
nal image, even if only the least bit is wrong, the water
mark information will be destroyed.
If we want to preserve the reversibility property, we
should use floatingpoint processor, which is more time
consuming and inefficient. To overcome this problem,
some integer color transforms that were used to approxi
mate the noninteger transforms were developed [2][3][4].
In this paper, we introduce a general algorithm that
can convert every color transforms into a reversible inte
ger transform. We also use it to derive the integer RGB
toKLA, IV1V2, YCrCb, DCT, and YIQ transforms suc
cessfully. They satisfy:
[Goal 1] The integer color transform should be reversible.
[Goal 2] No floatingpoint processor is required for both
the forward and the inverse transforms.
[Goal 3] Accuracy: If y and z are the transform results of
the original and the integer transforms, then z ≈ σy.
[Goal 4] The least bit should be constrained. For example,
we may constrain that the least bit of the output must be
the same as that of the input.
The advantages of the proposed algorithm are general
and all the above four goals can be achieved. The existing
integer transforms are hard to achieve Goals 3 and 4 at the
same time. In this paper, with the laddertruncation and
the related error analysis techniques, we can make the
least bit of the output the same as that of the input and at
the same time the accuracy is acceptable.
2. THE GENERAL ALGORITHM FOR DERIVING
THE INTEGER TRANSFORMS
First, we normalize the original 3×3 color transform A0 as
A such that det(A) = ±1.
A = σ⋅A0.
 )det(
=
0
A
σ
Note that, if y = Ax and y1 = A0x, then y = σy1. Thus, if
the difference of scaling is ignored, the performances of A
and A0 are in essence the same.
Then we do row and column permutations for A:
C = P1AP2
. (3)
3 / 1
−
(4)
0780391349/05/$20.00 ©2005 IEEE
Page 2
where P1 and P2 are permuting matrices, i.e., for each row
and column, only one entry is 1 and others are 0. Then,
applying the lifting scheme [5] and the triangular matrix
scheme [6] with several modifications, we decompose C
into three triangular matrices and one diagonal matrix:
where
123
TT DTC =
⎤
⎢
=
100
(5)
, s = sign[det(C)], (6)
⎥
⎦
⎥
⎥
⎢
⎣
⎢
⎡
010
00s
D
, t1 = (c22−1)/c21, t2 = −(t1z2+z1),
⎥
⎦
⎥
⎥
⎤
⎢
⎣
⎢
⎢
⎡
=
100
10
1
3
21
t
tt
1
T
t3 = −z2,
⎡
z
cmn’s (m, n = 1∼3) are the entries of C,
⎤
⎢
=
1
65
tt
t6 = (c32−t1c31),
⎤
⎢
=
100
t8 = s(c13+z1c11+ z2c12).
We then use fixedpoint values gn’s to approximate tn’s:
( )∑
−∞=
r
gn ≈ tn, n = 1 ∼ 8, Max(r) = b.
Then we derive the reversible integer transform B that
approximates A from:
21231
PGGDG PB =
⎤
⎢
=
100
(7)
[
1
[ ]
2
, (8)
⎥⎦
⎤
⎢⎣
⎡
−
−
k
⎥⎦
⎤
⎢⎣
⎡
=
⎥⎦
⎤
⎢⎣
−
33
23
c
1
32 31
2221
2
1
3
2
c
cc
cc
z
, t4 = c21, t5 = c31,
⎥
⎦
⎥
⎥
⎢
⎣
⎢
⎡
01
001
t
4
2
T
(9)
, t7 = s(c12−t1c11)− t6t8,
⎥
⎦
⎥
⎥
⎢
⎣
⎢
⎡
010
1
87
3
tt
T
(10)
, hn,r = 0 or 1, (11)
−
=
b
r
rnnn
htsigng2
,
(12)
where
, , (13)
TT
⎥
⎦
⎥
⎥
⎢
⎣
⎢
⎡
10
1
3
21
g
gg
1
G
⎥
⎦
⎥
⎥
⎤
⎢
⎣
⎢
⎢
⎡
=
1
01
g
001
g
65
4
g
2
G
.
⎥
⎦
⎥
⎥
⎤
⎢
⎣
⎢
⎢
⎡
=
100
010
1
87
gg
3
G
[Corollary 1] B is a binaryvalued matrix that approxi
mates A. Moreover, if we define B′ ′ = P2G1
then it is no hard to prove that B′ ′B = I. Moreover, since
G1
inverse transform matrix B1 is also a binaryvalued matrix
and the first two goals listed in Sec. 1 are satisfied.
1G2
1G2
1DP1,
1, G2
1, and G3
1 are all binaryvalued matrices, thus the
Then we try to satisfy Goal 3 and Goal 4 in Sec. 1.
Note that, if in (11) the least bit we use for approximating
tn’s is 2−b, and we use 2−h1 and 2−h2 to denote the least bit
of x and z (z = Bx), then
h2 = h1 + 3b.
Thus the bitlength of z is obviously longer than that of x.
For example, if the least bit of gn’s is 1/32 and kn’s = 0,
then h2 = h1 + 15. It may not be economical. To solve the
problem, we can convert the triangular matrices G1, G2,
and G3 in (13) into laddertruncation operations.
From (12), the multiplication of z = Bx can be divided
into the following process
(1) x1 = P2
(4) x4 = G3x3, (5) z = P1
If the laddertruncation operations are applied, the 2nd, 3rd
and 4th Steps will be modified, and the process of the
forward integer color transform is:
(Step 1): x1 = P2
(Step 2):
][ ]
{
1
112
gQxx
r
+=
[ ]
{
2
312
gQxx
r
+=
where Qr is the truncation operation, which throws the bits
that are less than 2−r:
∞
−
=
⎥⎦
n
n
(Step 3): x3[1] = x2[1],
3
x
[ ][ ]
{
33
523
xgQxx
r
+=
(Step 4):
[ ][ ]
{
11
734
gQxx
r
+=
x4[2] = x3[2], x4[3] = x3[3].
(Step 5): z = P1
(14)
Tx, (2) x2 = G1x1, (3) x3 = G2x2,
TDx4. (15)
Tx. (16)
[ ]
2
[ ]
3
1
x
[ ]
3
}
121
xgx
+
}
,
, x2[3] = x1[3], (17)
(dn’s = 0 or 1). (18)
∑
−∞=
∑
−∞=
−
⎤
⎢⎣
⎡
r
n
n
n
nr
ddQ22
[ ]
2
[ ]
1
2
x
[ ]
2
x
g
[ ]
1
{}
242
g
+
xgQ
2
[ ]
3
3
x
r
+
[ ]
2
x
=
,
(19)
}
6
+
.
}
[ ]
2
83
,
(20)
(21)
TDx4.
The process of the inverse integer color transform is:
(Step 1): x4 = DP1z.
(Step 2): x3[3] = x4[3], x3[2] = x4[2],
[ ][ ]
{
11
3743
xgQxx
r
−=
(Step 3): x2[1] = x3[1],
[ ]
2
2
x
=
[ ][ ]
{
33
2532
xgQxx
r
−=
(Step 4): x1[3] = x2[3],
[ ]
2
1
x
=
[ ][ ][ ]
{
11
1121
xgQxx
r
−=
(Step 5): x = P2x1.
[Corollary 2] If 2−h1 is the least bit of x, b > 0 (b is the
least bit used for approximating tn’s, see (11)), and r satis
fies
h1 ≤ r ≤ h1 +b,
then the least bit of the output z is
2−r (independent of b).
Therefore, for our algorithms, the least bit of the output is
determined by how many bits are preserved by the trunca
tion operation Q. It is independent of b. That is, in (11), no
matter how many bits we use for approximating tn’s, the
least bit of the output is remained to be r.
(22)
[ ]
2
x
[ ]
1
x
2
[ ]
3
{
r
[ ]
2
2
Q
r
[ ]
3
1
x
}
g
38
−
x
Q
x
g
+
[ ]
2
+
[ ]
2
2
+
. (23)
,
(24)
}
,
(25)
(26)
[ ]
1
}
243
x
.
x
.
}
g
6
−
g
[ ]
3
{
}
13
2
g
(27)
(28)
[Corollary 3] The proposed integer color transform re
quires 8 multiplications and the original transform re
quires 9 multiplications. Thus the integer color transform
does not increase the computing complexity.
Page 3
Then we discuss the problem of accuracy. Note that, in
Steps 2, 3, and 4, the truncation operation Qr is equivalent
to adding a small number:
Qr{a} = a+ τ where −2−r−1 < τ ≤ 2−r−1.
If the input of Qr is not known, τ can be treated as a ran
dom variable uniformly distributed in (−2−r−1, 2−r−1) and
E[τ] = 0, E[τ 2] = 4−r/12,
where E means the expected value. Thus the process in
(16)∼(21) can be rewritten as:
xP (G[GD{GPz
21231
=
where ∆ ∆1, ∆ ∆2, and ∆ ∆3 are 3×1 random vector:
∆1[3] = ∆2[1] = ∆3[2] = ∆3[3] = 0,
and ∆1[1], ∆1[2], ∆2[2], ∆2[3], and ∆3[1], are random vari
ables whose statistical characters are the same in (30). We
can compare (31) with the original noninteger color trans
form. If y = Ax, then
,
xPTT DTPy
21231
=
T
1
P∆G DGPyz
+=−
(
GGGDP
1231
−+
Suppose that b is large enough such that
∇1 = G1−T1 ≈ 0, ∇2 = G2−T2 ≈ 0, ∇3 = G3−T3 ≈ 0. (35)
TTTGGG
+∇+∇+=
TTTT
∇+∇+∇≈
Therefore,
[Corollary 4] The error of the integer color transform can
be approximated by:
11231
P∆T DTPyz
+≈−
PT DTP
21231
∇+
PTTDP
21231
∇+
Thus the error comes from the six terms:
(1) the truncation in Step 2, which causes P1
(2) the truncation in Step 3, which causes P1
(3) the truncation in Step 4, which causes P1
(4)quantization T1 into G1, which causes P1
(5)quantization T2 into G2, which causes P1
(6)quantization T3 into G3, which causes P1
(29)
(30)
(31)
}∆]∆)∆
321
TT
+++
(32)
(33)
TT
3
T
123
T
1123
D∆P∆DG
+
In this section, we show some integer color transforms we
derived. Before deriving the integer color transform, we
normalize the original color transform such that det(A) =
±1. Suppose that the least bit of the input data is 1 (i.e., in
Corollary 2, h1 = 0). We choose the values of b and r as
b = 10, r = 0.
It is easy to see that (27) is satisfied. Thus the least bit of
the output for the following integer transforms is
2−0 = 1.
Therefore, the least bits of both the input and the output
data are 1. We also suppose that the input signal is uni
form distributed in [0, 255] such that
E(x[n]) = 127.5, E(x2[n]) = 2552/3, E2(x[n]) = 127.52.(42)
Use (42) together with (30), (37), (39), and the fact that
yTy = xTATAx, we can calculate the NRMSE.
. (34)
)
xPTTT
T
2123
T
( )( )()
1231
T
12233
123123
TTTTTT
−∇
−
. (36)
123123123
T
3
T
12
DT
3
P
TT
D∆P
∇
∆DT
+
+
xPTx
T
2123
T
1
TT
. (37)
x
TT
TDT3T2∆ ∆1.
TDT3∆ ∆2.
TD∆ ∆3.
TDT3T2∇ ∇1P2
TDT3∇ ∇2T1P2
TD∇ ∇3T2T1P2
Tx.
Tx.
Tx.
There are some things to be noticed.
(a) Since if (27) is satisfied the least bit of z is independ
ent of b, thus in (11) we can choose a large value of b.
If b is very large, ∇1, ∇2, and ∇3 will be very small and
the last three terms in (37) can be ignored, i.e.,
1231
P∆T DTPyz
+≈−
Also notice that in this case the error is independent of
the input x.
(b) From (37), it can be seen that the entries of T1, T2, and
T3 affect the error of approximation. Especially, from
(38), the entries of T2 and T3 have larger effects than
. (38)
3
T
123
T
1
T
D∆P∆DT
+
those of T1. Thus, to make the error small, the values
of t4, t5, t6, t7, and t8 should be as small as possible.
(c) After the integer transform is designed, we can use (37)
together with the following equation to estimate the
normalized root mean square error (NRMSE):
−
=
][
)](
y
)
y
[(
yzyz
T
T
E
E
NRMSE
−
. (39)
where E means the expected value. In (4), we can vary the
permuting matrices P1 and P2 iteratively and calculate the
NRMSE of the integer transform we obtain. Since there
are 3! = 6 choices for each of P1 and P2, there are at least
36 possible integer color transforms we can obtain. We
can choose the optimal one that can minimize the NRMSE.
In addition to P1 and P2, in (4), we can use AT, A− −1, or
(AT) − −1 instead of A to search the optimal color transforms.
3. EXAMPLES
(40)
(41)
(1) RGB to KLA
original: . (43)
⎥
⎦
⎥
⎤
⎢
⎣
. 0
⎢
. 1
⎡
=
−−
. 1
−−
8800. 01539 3376
8373. 0 2879. 01984
8629. 08975 . 0 8185. 0
A
From (4)∼(11), the parameters of gn’s we obtain are:
g1 = 215/1024, g2 = 1313/1024,
g4 = 221/256, g5 = 857/1024,
g7 = 149/1024, g8 = 7/1024,
⎤
⎢
=
100
⎢
⎣
0
NRMSE = 0.187%,
where P1 and P2 are found iteratively to minimize the
NRMSE. Thus, from (16)∼(21), the process of the integer
RGB to KLA transform is:
(Step 1): x1[1] = x[3], x1[2] = x[1], x1[3] = x1[2].
(Step 2) :x2[1]=x1[1]+Q0{215x1[2]/1024 + 1313x1[3]/1024}
x2[2] = x1[2]+Q0{107x1[3]/512}, x2[3] = x1[3],
g3 = 107/512,
g6 = 1047/1024,
(44)
, , , (45)
⎥
⎦
⎥
⎥
⎢
⎣
⎢
⎡−
010
001
D
⎥
⎦
⎥
⎥
⎤
⎢
⎢
⎡
=
01
001
100
1
P
⎥
⎦
⎥
⎥
⎤
⎢
⎣
⎢
⎢
⎡
=
001
100
010
2
P
(46)
Page 4
(Step 3): x3[1] = x2[1], x3[2] = x2[2]+Q0{221x2[1]/256},
x3[3]=x2[3]+Q0{857x2[1]/1024 + 1047x2[2]/1024}
(Step 4) x4[1]=x3[1]+Q0{149x3[2]/1024 − 7x3[3]/1024},
x4[2] = x3[2], x4[3] = x3[3].
(Step 5): z[1] = x4[2], z[2] = x4[3], z[3] = x4[1].
(2) RGB to IV1V2
⎡
=
−
6 / 16 / 1
• parameters of the integer color transform in (16)∼(26):
g1 = 115/256, g2 = 47/256,
g4 = 209/512, g5 = 1,
g7 = 119/1024, g8 = 557/1024,
⎤
⎢
=
100
⎢
⎣
001
NRMSE = 0.175%.
(3) RGB to YCrCb
⎡
=
−−
5354. 0 2734. 0
• parameters of the integer color transform in (16)∼(26):
g1 = 289/1024, g2 = 343/1024,
g4 = 347/512, g5 = 137/256,
g7 = 201/1024, g8 = 79/512,
⎤
⎢
=
100
⎢
⎣
00
NRMSE = 0.288%.
(4) RGB to DCT
⎡
=
−
8165. 04082. 0
• parameters of the integer color transform in (16)∼(26):
g1 = 53/128, g2 = 325/1024,
g4 = 181/256, g5 = 209/512,
g7 = 53/128, g8 = 325/1024,
⎤
⎢
=
100
⎢
⎣
00
NRMSE = 0.267%.
(5) RGB to YIQ
⎡
=
−
8259. 03332. 0
• parameters of the integer color transform in (16)∼(26):
g1 = 139/1024, g2 = 227/512,
(47)
original:
⎥
⎦
⎥
⎤
⎢
⎣
⎢
−−
0
6/26/ 16/ 1
111
A
. (48)
g3 = 341/1024,
g6 = 141/256,
(49)
, , , (50)
⎥
⎦
⎥
⎥
⎢
⎣
⎢
⎡
010
001
D
⎥
⎦
⎥
⎥
⎤
⎢
⎢
⎡
=
010
100
1
P
⎥
⎦
⎥
⎥
⎤
⎢
⎣
⎢
⎢
⎡
=
010
100
001
2
P
(51)
original:
. (52)
⎥
⎦
⎥
⎤
⎢
⎣
⎢
−
. 0
−
8087
1310 . 06777 . 08087 . 0
1844. 0 9495 . 04836 . 0
A
g3 = 99/1024,
g6 = 125/1024,
(53)
, , , (54)
⎥
⎦
⎥
⎥
⎢
⎣
⎢
⎡−
010
001
D
⎥
⎦
⎥
⎥
⎤
⎢
⎢
⎡
=
1
010
001
1
P
⎥
⎦
⎥
⎥
⎤
⎢
⎣
⎢
⎢
⎡
=
100
001
010
2
P
(55)
original:
(56)
⎥
⎦
⎥
⎤
⎢
⎣
⎢
−
. 04082
7071 . 007071 . 0
5774 . 05774 . 05774 . 0
A
g3 = 115/512,
g6 = 245/1024,
(57)
, , , (58)
⎥
⎦
⎥
⎥
⎢
⎣
⎢
⎡−
010
001
D
⎥
⎦
⎥
⎥
⎤
⎢
⎢
⎡
=
1
010
001
1
P
⎥
⎦
⎥
⎥
⎤
⎢
⎣
⎢
⎢
⎡
=
001
100
010
2
P
(59)
original:
. (60)
⎥
⎦
⎥
⎤
⎢
⎣
⎢
. 0
−
. 0
−
4927
5085. 04327. 09412
1800 . 09270. 04722 . 0
A
g3 = 81/256,
g4 = 443/1024,
g7 = 205/512,
⎡−
=
0
NRMSE = 0.297%.
Notice that the NRMSEs of all the integer color transform
is no more than 0.3%. (In contrast, the NRMSEs of the
integer RGBtoKLA and YCrCb, transforms shown in
[2][3] are more than 10%). We can achieve higher accu
racy because the least bit 2−b in (11) used for approximat
ing tn’s can be chosen very small and at the same time the
least bit of the output is not affected.
g5 = 423/512,
g8 = 31/256,
⎡
=
00
g6 = 57/128,
(61)
, , , (62)
⎥
⎦
⎥
⎥
⎤
⎢
⎣
⎢
⎢
10
010
001
D
⎥
⎦
⎥
⎥
⎤
⎢
⎣
⎢
⎢
1
010
001
1
P
⎥
⎦
⎥
⎥
⎤
⎢
⎣
⎢
⎢
⎡
=
100
001
010
2
P
(63)
4. CONCLUSIONS
In this paper, we introduced a systematic way that ap
proximates a 3×3 color transform matrix by a reversible
integer transform. When doing fixedpoint approximation,
the goals of (1) binary matrix entry, (2) reversibility, (3)
good approximation accuracy, and (4) constraint for the
least bit are hard to satisfy at the same time. However,
with the algorithm introduced in Sec. 2, we are easy to
achieve all the above four goals. In Sec. 3 we successfully
convert several wellknown color transforms into the re
versible integer transforms. With them, we can use the
fixedpoint processor instead of the floatingpoint one to
do color transformation. It is helpful for improving the
efficiency of digital image processing.
5. REFERENCES
[1] W. K. Pratt, Digital Image Processing, 2nd ed., Wiley,
New York, 1991.
[2] B. Deknuydt, J. Smolders, L. V. Eychen, and A. Ooster
linck, “Color space choice for nearly reversible image
compression”, SPIE vol. 1818, Visual Communications
and Image Processing, p. 13001311, 1992.
[3] M. J. Gormish, E. L. Schwartz, A. Keith, et al, “Lossless
and nearly lossless compression for high quality images“,
Proceedings of the SPIE, vol. 3025, pp. 6270, Feb. 1997.
[4] P. Hao and Q. Shi., “Comparative study of color trans
forms for image coding and derivation of integer re
versible color transform”, Int. Conf. Pattern Recogni
tion, vol. 3, pp. 224227, Sept. 2000.
[5] S. Oraintara, Y. J. Chen, and T. Q. Nguyen, ‘ Integer Fast
Fourier Transform’, IEEE Trans. Signal Processing, vol.
50, p. 607618, 2002.
[6] P. Hao and Q. Shi., “Matrix Factorizations for Re
versible Integer Mapping”, IEEE Trans. Signal Proc
essing, vol. 49, no. 10, pp. 23142324, Oct. 2001.
[7] M. D. Adams, F. Kossentini, and R. K. Ward, ‘General
ized S transform’, IEEE Trans. Signal Processing, vol. 50,
no. 11, pp. 28312842, Nov. 2002.
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