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Image Processing & Communications, vol. 21, no. 4, pp.5-12
DOI: 10.1515/ipc-2016-0019
5
RECONSTRUCTED QUANTIZED COEFFICIENTS MODELED WITH
GENERALIZED GAUSSIAN DISTRIBUTION WITH EXPONENT 1/3
ROBERT K RUPI ŃSKI
West-Pomeranian University of Technology in Szczecin, Chair of Signal Processing and Multimedia Engineering,
ul. 26-Kwietnia 10, 71-126 Szczecin, Poland, e-mail:[email protected]
Abstract.
DCT, WHT (Walsh-Hadamard Transform) and DST (Dis-
Generalized Gaussian distribution (GGD) in-
crete Sine Transform) could be modeled with GGD and it
cludes specials cases when the shape parame-
was discussed by Clarke [5]. Zero-mean GGD was ap-
ter equals p = 1 and p = 2. It corresponds
plied to the tangential wavelet coefficients for compress-
to Laplacian and Gaussian distributions respec-
ing three-dimensional triangular mesh data by Lavu et
tively. For p → ∞, f (x) becomes a uniform
al. [18]. Sharifi et al. [27] applied GGD to 16 frequency
distribution, and for p → 0, f (x) approaches
subbands of the original and the difference frames of a
an impulse function.
Chapeau-Blondeau et
video sequence. Achim et al. [1] modeled the ultrasound
al. [4] considered another special case p = 0.5.
image wavelet coefficients by the generalized Laplacian
The article discusses more peaky case in which
density. The image segmentation algorithm based on the
GGD p = 1/3.
wavelet transform with the application of GGD was presented in [31]. GGD and asymmetric GGD (AGGD)
Key words. Generalized Gaussian distribution,
were fitted to certain regular statistical properties of natu-
maximum likelihood estimation, quantization,
ral images to get the natural scene statistics (NSS) model
reconstruction.
in [33]. Wang et al. [32] applied GGD to approximate an
atmosphere point spread function (APSF) kernel to pro-
1
Introduction
pose the efficient method to remove haze from a single
image. Song et al. [28] constructed a GGD based model
Generalized Gaussian distribution has been widely used to to introduce more facial details into the initial image synmodel distributions ranging from a highly peaked to a uni- thesis. The statistical properties of the stereoscopic image
form one. It has been also applied in many different areas, of the reorganized discrete cosine transform (RDCT) subex. watermarking [9]. It is very often applied to model the band coefficients were modeled with GGD to propose the
transform coefficients such as discrete cosine transform stereoscopic image quality assessment in [20].
(DCT) or wavelet ones. The coefficients of the transforms
Many methods have been designed to estimate the paUnauthenticated
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R. Krupiński
Tab. 1: The shape parameters of GGD for "Cameraman"
Ci,j
i=0
1
2
3
4
5
6
7
j=0
–
0.32
0.41
0.42
0.39
0.40
0.49
0.51
1
0.29
0.27
0.32
0.38
0.41
0.46
0.52
0.49
2
0.26
0.29
0.34
0.40
0.41
0.46
0.52
0.54
3
0.27
0.31
0.31
0.37
0.40
0.46
0.53
0.53
4
0.31
0.31
0.33
0.42
0.38
0.44
0.53
0.59
5
0.36
0.32
0.39
0.40
0.45
0.47
0.47
0.52
6
0.35
0.37
0.37
0.42
0.46
0.49
0.50
0.54
7
0.39
0.37
0.40
0.44
0.49
0.52
0.54
0.55
Tab. 2: The shape parameters of GGD for “Cameraman”,
where at least 95% of DCT coefficients were none-zero
after dequantization
Ci,j
i=0
1
2
3
4
5
6
j=0
–
0.32
0.41
0.42
0.39
0.40
0.49
1
0.29
0.27
0.32
0.38
0.41
0.46
–
2
0.26
0.29
0.34
0.40
0.41
–
–
3
0.27
0.31
0.31
0.37
0.40
–
–
4
0.31
0.31
0.33
0.42
–
–
–
5
0.36
0.32
0.39
–
–
–
–
6
0.35
0.37
–
–
–
–
–
7
0.39
0.37
–
–
–
–
–
Fig. 1: The "Cameraman" image
pixels. DCT was performed for a block 8×8. The estimation of the shape parameters of GGD was applied to DCT
rameters of GGD. A review of the different approaches coefficients. Ci,j denotes a coefficient in i row and j colto the shape parameter estimation problems can be found umn in the block 8 × 8 of DCT coefficients. The indexes
in [30]. The disadvantage of this distribution is that these of DCT coefficients Ci,j vary i, j ∈< 0, 7 >, where C0,0
methods are complex. Therefore, authors proposed the corresponds to DC coefficient. Table 2 was derived from
approximated approach [15, 17].
Table 1 in order to check which distributions are avail-
Chapeau-Blondeau et al. [4] showed that by restricting able to the decoder. If, after dequantization, at least 95%
the power of distribution to a value of 0.5 the calculations of DCT coefficients were none-zero, the distribution was
can be simplified and the equations can be presented in a taken into account. Otherwise, the shape parameters were
closed form. This allowed to improve the image recon- skipped, which resulted in the reduced table.
struction based on the quantized DCT coefficients [14].
The article extends this approach to more peaky distribution, where the power coefficient of GGD goes toward
0. By assuming a source signal with GGD with a power
parameter p = 1/3, equations for the centroid reconstruction in a closed form can be obtained, whereas for a GGD
model it cannot be done. The maximum likelihood (ML)
It can be noticed that some distributions are appropriate to model with GGD with a power parameter p = 1/3.
It depends on the source image where for the "Lenna"
and "Barbara" images the distributions were more close
to GGD 0.5 [14].
The article is organized in the following manner. In
method of discrete GGD p = 1/3 is derived, which re- Section 2 the continuous GGD p = 1/3 is presented and
quires the estimation of only one parameter.
in Section 3 the discrete GGD p = 1/3 is discussed. In
Table 1 contains the results collected for the estimation Section 4 the biased reconstruction of quantized coeffiof the shape parameters of GGD for the “Cameraman” cients assuming GGD p = 1/3 is introduced. The experiimage. The image was monochromatic, 256 × 256 size in
mental results are presented in Section 5.
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Image Processing & Communications, vol. 21, no. 4, pp. 5-12
2
Continuous generalized Gaussian
density function with exponent
1/3
takes the form
N
1 X
|xi |1/3
3N i=1
λ=
!−3
(5)
Probability density function of the continuous random where N denotes the number of observations.
variable of GGD is [3, 5, 7, 8]
p
λ·p
e−[λ·|x|]
f (x) =
2 · Γ p1
0
λ=0.5
λ=1
λ=2
0.1
(1)
0.08
tz−1 e−t dt, z > 0 [21], p is the shape
f(x)
where Γ(z) =
R∞
0.12
parameter and λ is connected to the variance of the distri-
0.06
0.04
bution.
0.02
The special case of the density function of GGD with
0
−10
exponent p = 1/3 of the continuous random variable is
f (x) =
λ −[λ·|x|]1/3
e
12
−5
0
x
5
10
(2)
Fig. 2: Density function of GGD with exponent p = 1/3
of the continuous random variable for three selected paThe cumulative distribution is obtained by integrating rameters λ
Equation (3)
Fig. 2 depicts density function of GGD with exponent
Zx
F (x) =
f (z)dz
(3)
p = 1/3 of the continuous random variable for three selected parameters λ.
−∞
which results in the cumulative GGD p = 1/3
1 −g
· (2 + 2g + g 2 ),
for x ≤ 0
4e
F (x) =
1 −h
1 − 4 e · (2 + 2h + h2 ), for x > 0
Probability density function of the continuous zeromean random variable is usually assumed for the coef(4)
ficients before quantization available to the encoder.
1/3
g = (−λ · x) ,
1/3
h = (λ · x) .
The advantage of fixing the power parameter is that the
where
cumulative GGD p = 1/3 can be presented in a closed
3
Discrete generalized Gaussian
density function with exponent
1/3
form.
Many methods to estimate parameters has been de- In JPEG and MPEG reconstruction [10–12, 23, 25, 26]
signed. The most common approach is to use the max- the coefficients available to the decoder are reconstructed
imum likelihood estimators. The maximum likelihood to the bin center. The reconstructed values are yi = i · Q,
function and estimators are discussed in [6, 19, 29, 34]. where i is both the bin index and the quantized value. The
The maximum likelihood estimator for continuous GGD parameter Q is the quantization factor (the length of the
p = 1/3 can be obtained by finding the maximum like- interval). The value yi represents a reconstructed value,
lihood function of Equation (2) and maximizing it with which is also the bin center.
respect to λ. After certain transformations the estimator
Integrating function f (x) (Equation (2)) over the interUnauthenticated
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R. Krupiński
val (Q · i − 0.5 · Q, Q · i + 0.5 · Q)
Pi =
λ
12
(i+0.5)·Q
Z
e−[λ·|x|]
1/3
form
dx
N0 · e−A · Q
+
P0
(6)
N
(i−0.5)·Q
+
gives the probability density function of the discrete random variable of GGD 1/3

 Pi = 41 e−Ci · (2 + 2Ci + Ci2 )−
− 14 e−Bi · (2 + 2Bi + Bi2 ) i 6= 0

P0 = 1 − 12 e−A · (2 + 2A + A2 ) i = 0
1
e−Bi · Bi3 − e−Ci · Ci3
1X
= 0,
λ i=1
Pi
(8)
where N0 denotes the number of observations equal zero
and N1 denotes the number of observations not equal
(7)
zero. The equation N = N0 + N1 must be held. The estimated λ parameter is received only from the discrete ob-
servations (the quantized values available to the decoder)
1/3
A = (0.5 · λ · Q) ,
without prior knowledge of the coefficients before quantiwhere Bi = (λ · (|yi | + 0.5 · Q))1/3 ,
1/3
zation available to the encoder. Therefore, it is expected to
Ci = (λ · (|yi | − 0.5 · Q)) .
Fig. 3 depicts the density function of GGD with expo- restore GGD p = 1/3 of the continuous random variable
nent p = 1/3 and λ = 1 for discrete random variable available to the encoder before the quantization process.
Pi (Equation (7)) and continuous random variable f (x)
(Equation (2)). It should be noted that the distribution of
coefficients available to the decoder is discrete and consists of scaled deltas in the bin centers.
4
The reconstruction of coefficients
The reconstructed coefficients can be biased based on
the assumed model. Different models have been applied
0.25
P
i
in [2, 16, 24]. Based on the observation that the DCT co-
f(x)
0.2
efficients have a peak at zero and decrease exponentially,
Ahumada et al. [2] made adjustments to the reconstructed
0.15
coefficients. The reconstruction for Laplace distribution
with centroid was presented in [16, 24].
0.1
According to the equation for the centroid reconstruc-
0.05
tion of the distribution in [22], the equation for the cen0
−50
0
x
50
Fig. 3: Density function of GGD with exponent p = 1/3
and λ = 1 for discrete random variable Pi (Equation (7))
and continuous random variable f (x) (Equation (2))
The estimator of discrete GGD 1/3 of the maximum
likelihood method can be found similarly as for the Laplacian discrete source [24] and GGD 0.5 [14]. The maxi-
troid reconstruction of GGD 1/3 takes the form
yˆi = sgn(yi ) ·
1 Li
·
λ Mi
(9)
where
Li =
e−Ci · (Ci5 + 5 · Ci4 + 20 · Ci3 +
+60 · Ci2 + 120 · Ci + 120)−
−e−Bi · (Bi5 + 5 · Bi4 + 20 · Bi3 +
+60 · Bi2 + 120 · Bi + 120),
Mi = e−Ci · (2 + 2Ci + Ci2 ) − e−Bi · (2 + 2Bi + Bi2 ).
The value reconstructed with centroid minimizes the
mum likelihood function of Equation (7) is set and then Mean Square Error (MSE) in the interval (Q · i − 0.5 ·
it is maximized with respect to λ. After certain transfor- Q, Q · i + 0.5 · Q).
mations the ML estimator of discrete GGD 1/3 takes the
Another advantage of fixing the power parameter p =
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Image Processing & Communications, vol. 21, no. 4, pp. 5-12
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1/3 is that the reconstruction equation can be presented in
10
λ=0.1
λ=2.3
λ=4.5
a closed form.
2
10
Experiments
∆RMSE
5
In the first simulation, the performance of maximum like-
0
10
−2
10
lihood estimator is evaluated (Equation (8)). The input
sequence is generated with the GGD generator [13] with
−4
10
0
200
400
600
800
1000
N
p = 1/3. Then the sequence is quantized and dequan-
tized, which corresponds to lossy compression. The aim Fig. 5: Difference between RMSE of the estimators Equaof the estimator is to reproduce the initial distribution be- tion (5) and Equation (8) for the quantization factor Q =
fore the quantization only on the dequantized coefficients. 20
The sequence range N ∈< 31, 1000 > and the quantization steps Q ∈< 2, 20 > are considered. The simulation
performs better than the estimator based on the contin-
was repeated 1000 times. Relative Mean Square Error uous distribution. Figures show the positive values that
confirms the expectations. The higher value of λ, the es-
(RMSE) was calculated from the equation
RM SE =
timator (8) is getting better in terms of RMSE. It can be
M
1 X (λ̂ − λ)2
M i=1
λ2
(10) also noticed that the higher value of quantization factor Q,
the estimator (8) is getting better in terms of RMSE.
where λ̂ is a value estimated by the model and λ is a real
In the next simulation, the set of DCT coefficients is
value of a lambda parameter. M denotes the number of generated with the GGD generator and the shape paramrepetitions.
eter p = 1/3. The input sequence is created xi with calculating the inverse of DCT. These DCT coefficients are
2
10
quantized and dequantized, and the restored sequence yi
is calculated with the inverse of DCT. MSE is calculated
0
10
∆RMSE
for the lossy transformation.
−2
10
M SE =
−4
10
10
0
5
10
Q
15
(11)
where yˆi is reconstructed sequence and xi is the input sig-
λ=0.1
λ=2.3
λ=4.5
−6
N
1 X
(yˆi − xi )2
N i=1
20
nal.
The reconstructed DCT coefficients are modified with
Fig. 4: Difference between RMSE of the estimators Equa- Equation (9) for the λ parameter estimated from Equation (5) and Equation (8) for the sequence length N = tions (5) and (8). Then the restored sequence yˆ is calcui
1000
lated with the inverse of DCT and MSE.
Figures 4 and 5 depict the difference between RMSE
Figures 6 and 7 depict the difference between MSE of
of the estimators Equation (5) and Equation (8). It is ex- normally reconstructed signal yi and modified reconstrucpected that the estimator based on the discrete distribution tion yˆi ((5) or (8)). It can be noticed that the modified
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R. Krupiński
reconstruction yˆi gives smaller MSE than MSE of nor- transformation. The reconstructed DWT coefficients are
mally reconstructed signal yi (the positive values in the
modified with Equation (9) for the λ parameter estimated
figures). It should be noted that the estimator (8) gives from Equations (5) and (8). Then the restored sequence
yˆi is calculated with the inverse of DWT and MSE.
smaller MSE than the estimator (5).
1.5
1.5
λd=0.1
λd=0.1
λc=0.1
λc=0.1
λ =2.3
d
λ =2.3
λc=2.3
d
λc=2.3
λd=4.5
1
λd=4.5
1
λc=4.5
∆MSE
∆MSE
λc=4.5
0.5
0.5
0
0
5
10
Q
15
20
0
0
5
10
Q
15
20
Fig. 6: Difference between MSE of normally recon- Fig. 8: Difference between MSE of normally reconstructed signal yi and modified reconstruction yˆi (λc (5) structed signal y and modified reconstruction yˆ (λ (5)
i
i
c
or λd (8)) for the sequence length N = 1000 (IDCT)
or λd (8)) for the sequence length N = 1000 (IDWT)
1.6
1.4
1.2
1.5
λc=0.1
λd=0.1
λd=2.3
λc=0.1
λc=2.3
λd=2.3
λd=4.5
λc=2.3
λc=4.5
1
λd=4.5
λc=4.5
0.8
∆MSE
∆MSE
1
λd=0.1
0.6
0.4
0.5
0.2
0
0
200
400
600
800
1000
N
0
0
200
400
600
800
1000
N
Fig. 7: Difference between MSE of normally reconstructed signal yi and modified reconstruction yˆi (λc (5) Fig. 9: Difference between MSE of normally reconstructed signal yi and modified reconstruction yˆi (λc (5)
or λd (8)) for the quantization factor Q = 20 (IDCT)
or λd (8)) for the quantization factor Q = 20 (IDWT)
In the last simulation, the set of detailed Discrete
Wavelet Transform (DWT) coefficients is generated with
Figures 8 and 9 depict the difference between MSE of
the GGD generator and the shape parameter p = 1/3. The normally reconstructed signal yi and modified reconstrucinput sequence is created xi with calculating the inverse tion yˆi ((5) or (8)). It can be noticed that the modified
of DWT, whereas the approximation DWT coefficients are reconstruction yˆi gives smaller MSE than MSE of norset to zero. These DWT coefficients are quantized and de- mally reconstructed signal yi (the positive values in the
quantized, and then the restored sequence yi is calculated
figures). It should be noted that the estimator (8) gives
with the inverse of DWT. MSE is calculated for the lossy smaller MSE than the estimator (5).
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Image Processing & Communications, vol. 21, no. 4, pp. 5-12
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Summary
[6] Deutch, R. (1965). Estimation theory. PrenticeHall, Englewood Cliffs, N.J.
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ficients, the modified reconstruction equation is defined.
(2000). DCT-domain watermarking techniques for
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