Proceedings ArticleDOI

# 2-D two-fold symmetric circular shaped filter design with homomorphic processing application

14 Mar 2010-pp 3694-3697
TL;DR: The proposed filter outperforms currently available filter design methods and is presented as a performance comparison, as well as a homomorphic processing image enhancement example to illustrate the effectiveness of this method.
Abstract: A design method of a linear-phased, two-dimensional (2-D), two-fold symmetric circular shaped filter is presented in this paper. Although the proposed method designs a non-separable filter, its implementation has linear complexity. The shape of the passband and the stopband is expressed in terms of level sets of second order trigonometric polynomials. This enables the transformation of the filter specifications to a Semi-Definite Program (SDP) of moderate dimension. The proposed filter outperforms currently available filter design methods. We present a performance comparison, as well as a homomorphic processing image enhancement example to illustrate the effectiveness of this method.

### 1. INTRODUCTION

• There are many different approaches in designing 2-D filters, each with its pros and cons.
• Their complexity is very low but it is not possible to control the filter characteristics such as passband shape and cut-off frequency.
• There have been continues interest in representing the passband and stopband by trigonometric polynomials that are positive in the region of interest.
• Initially first order trigonometric polynomials have been used to design two-fold symmetric filters [6].
• The resulting SDP formulation is of very high dimension.

### 2. TWO-FOLD SYMMETRIC 2-D FILTER DESIGN

• The frequency response of a zero-phased digital filter is a real valued function and its impulse response is symmetric about the origin h(n, n) = h(−n,−n) [10].
• The design objective is to find the coefficients of the matrix X and Y such that the desired frequency response is obtained.
• There are 2(n + 1)2 design variables, which is twice as that of the fourfold symmetric filter.
• Depending on the application and the data to be processed the filter specifications vary considerably.
• Generally filter specifications can be formulated as a minimum stopband attenuation problem as follows: min X,Y ,δs δs (4a) s.t. |H(Ω)−.

### 3. CIRCULAR SHAPED FILTER DESIGN

• Circular shape has not been attempted in [6] and the polynomials used in [8] do not produce the desired shape.
• The following least squares optimization problem can be used to find the coefficients of (7).
• Thus two trigonometric polynomials can be derived to represent the passband and the stopband.
• The filter performance is improved in two-fold symmetric filters.

### 4. SIMULATION

• The first step is to derive the second order trigonometric polynomials that represent the passband and the stopband.
• Semi-definite program was derived as described in Section 3 and the simulation was performed using optimization software YALMIP [11] and SDPT3 [12] in MATLAB.
• Fig. 1 shows the frequency response of the designed filter in log scale and the performance comparison with different design methods is given in Table 1.
• Highpass circular shaped filters can be designed analogously.

### 5. HOMOMORPHIC PROCESSING SYSTEM

• When images with large dynamic range such as natural scenes are recorded, image contrast can be significantly reduced.
• The reflectance component i(n1, n2) on the other hand is related to the contrast within the image and generally vary rapidly.
• To see these images more clearly visit http://ee.unsw.edu.au/∼z3265024/enhancement.html.
• To evaluate the effectiveness of the homomorphic process the original image given in Fig. 4 was blurred using a gaussian lowpass filter before applying the homomorphic process.

### 6. CONCLUSION

• A very general approach of designing two-fold symmetric circular shaped filters with complexity equal to that of a four-fold symmetric filter was presented in this paper.
• The main advantage is that it successfully produces the desired circular shape with minimum passband and stopband ripple compared to the currently available design methods.
• The coefficients of the polynomial T1(Ω) in Chebyshev recursion have an influence on the filter performance and a method of selecting optimal values for these coefficients is still open for research.

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2-D TWO-FOLD SYMMETRIC CIRCULAR SHAPED FILTER DESIGN WITH
HOMOMORPHIC PROCESSING APPLICATION
A. J. Seneviratne, H. H. Kha, H. D. Tuan
School of Electrical Engineering
and Telecommunication,
University of New South Wales,
Sydney, NSW 2052, Australia.
T. Q. Nguyen
Department of Electrical and
Computer Engineering,
University of California in San Diego,
9500 Gilman Dr., La Jolla CA 92093-0407 USA.
ABSTRACT
A design method of a linear-phased, two-dimensional (2-D), two-
fold symmetric circular shaped ﬁlter is presented in this paper. Al-
though the proposed method designs a non-separable ﬁlter, its imple-
mentation has linear complexity. The shape of the passband and the
stopband is expressed in terms of level sets of second order trigono-
metric polynomials. This enables the transformation of the ﬁlter
speciﬁcations to a Semi-Deﬁnite Program (SDP) of moderate di-
mension. The proposed ﬁlter outperforms currently available ﬁlter
design methods. We present a performance comparison, as well as a
homomorphic processing image enhancement example to illustrate
the effectiveness of this method.
Index Terms Two-fold symmetric, Two-dimensional Circular
shaped ﬁlter, Semi-deﬁnite program, Trigonometric polynomials,
1. INTRODUCTION
There are many different approaches in designing 2-D ﬁlters, each
with its pros and cons. The simplest method is in the form of a
separable ﬁlter [1] which is developed as a product of two 1-D ﬁl-
ters. They are favored in applications which require very high order
ﬁlters since their complexity is very low. However there is limited
freedom with the design variables and it is conﬁned to rectangu-
lar shape. Non-separable ﬁlters are preferred when designing dif-
ferently shaped ﬁlters but their design is much more challenging.
McClellan transform [2], [3] can be used to develop non-separable
ﬁlters using 1-D ﬁlters. Their complexity is very low but it is not
possible to control the ﬁlter characteristics such as passband shape
and cut-off frequency. This is mainly because there is no direct way
to ﬁnd the 1-D ﬁlter that, when transformed, will produce the desired
2-D ﬁlter.
Frequency sampling method [4] gives better control over the
passband shape and produces lower peak ripples. However the com-
plexity is very high and many difﬁculties such as singularities and
numerical instability arise in the implementation of these ﬁlters. The
greatest barrier in designing ﬁlters with desired characteristics is in
the handling the semi-inﬁnite constraints that arise due to the peak
passband and stopband ripple constraints. Classical minmax ﬁlters
[5] deal with this by approximating them by linear constraints calcu-
lated at samples of the frequency plane.
Recently SDP has been employed to deal with these semi-
inﬁnite constraints. There have been continues interest in represent-
ing the passband and stopband by trigonometric polynomials that are
positive in the region of interest. Initially ﬁrst order trigonometric
polynomials have been used to design two-fold symmetric ﬁlters [6].
However they did not consider the circular shape and they have not
presented a direct relationship between the ﬁlter speciﬁcations and
the coefﬁcients of the polynomials. The resulting SDP formulation
is of very high dimension.
SDP formulations of moderate dimension have been achieved
in four-fold symmetric ﬁlter designs [7], [8]. Although they have
linear complexity in implementation, the trigonometric polynomi-
als used do not approximate the passband and stopband shapes as
well as one would hope for. This problem has been addressed in
[9] where circular shaped passband and stopband has been success-
fully expressed by level sets of simple second order trigonometric
polynomials. They have clearly presented the relationship between
the desired ﬁlter characteristics and coefﬁcients of the trigonometric
polynomials.
The paper [9] is based on four-fold symmetric ﬁlters and we
will extend this to a more general case of two-fold symmetric ﬁl-
ters in this paper. Two-fold symmetric ﬁlters present more ﬂexibility
in the design variables and, therefore, the design procedure becomes
more challenging. However since the optimization is performed over
greater number of variables the ﬁlter performance can be improved
in terms of frequency selectivity. Circular ﬁlters are of great impor-
tance in two-dimensional signal processing but they have not been
given proper attention so far. Thus design and implementation of
two-fold symmetric circular shaped ﬁlter are discussed in this pa-
per followed by an example of homomorphic processing application.
Standard mathematical notations are used in this paper, except that
A
A
A refers to the trace of a matrix and the elements of a matrix are
given as A
A
A =[a
ij
]
n
i,j=0
.
2. TWO-FOLD SYMMETRIC 2-D FILTER DESIGN
The frequency response of a zero-phased digital ﬁlter is a real val-
ued function and its impulse response is symmetric about the origin
h(n, n)=h(n, n) [10]. Thus the Z-transform of a (2n +1)×
(2n +1), zero-phase, two-fold symmetric, two-dimensional ﬁlter
can be written as follows
H(z
1
,z
2
)=h
nn
+
n
l=1
h
(n+l)n
(z
l
1
+ z
l
1
)
+
n
l=n
n
i=1
h
(n+l)(n+i)
(z
l
1
z
i
2
+ z
l
1
z
i
2
). (1)

Its Fourier transform can be obtained by evaluating on the unit circle
{(z
1
,z
2
)=(e
1
,e
2
), Ω:=(ω
1
2
) [π, π] × [0]},
H(Ω)=
n
i=0
n
l=0
x
il
cos(
1
+
2
)+
n
i=0
n
l=0
y
il
cos(
1
2
), (2)
= X
X
X,M
1
M
1
M
1
(Ω) + Y
Y
Y,M
2
M
2
M
2
(Ω), (3)
where M
1
M
1
M
1
(Ω)= [cos(
1
+
2
)]
n
i,l=0
, M
2
M
2
M
2
(Ω)= [cos(
1
2
)]
n
i,l=0
,
X
X
X =[x
il
]
n
i,l=0
and Y
Y
Y =[y
il
]
n
i,l=0
.
The design objective is to ﬁnd the coefﬁcients of the matrix X
X
X
and Y
Y
Y such that the desired frequency response is obtained. There
are 2(n +1)
2
design variables, which is twice as that of the four-
fold symmetric ﬁlter. Although this makes the design of a two-fold
symmetric ﬁlter more challenging, the ﬁlter performance can be im-
proved in terms of frequency selectivity since the optimization is
performed over a larger number of variables. In this paper the ﬁlter
order is taken as n. Depending on the application and the data to be
processed the ﬁlter speciﬁcations vary considerably. Generally ﬁlter
speciﬁcations can be formulated as a minimum stopband attenuation
problem as follows:
min
X,Y
X,Y
X,Y
s
δ
s
(4a)
s.t. |H(Ω) 1|≤δ
p
, Ω Ω
p
(4b)
|H(Ω)|≤δ
s
, Ω Ω
s
, (4c)
where δ
p
and δ
s
are the passband ripple and the stopband attenuation
respectively. The constraints given in (4b) and (4c) are semi-inﬁnite
constraints and pose the greatest challenge in designing the desired
ﬁlter. In the next section we consider an effective method of trans-
forming this optimization problem in to an SDP of linear complexity.
3. CIRCULAR SHAPED FILTER DESIGN
The passband and the stopband of a circular shaped ﬁlter can be
speciﬁed as follows:
Ω
p
= {(ω
1
2
) [π, π] × [0]:ω
2
1
+ ω
2
2
ω
2
p
}, (5)
Ω
s
= {(ω
1
2
) [π, π] × [0]:ω
2
1
+ ω
2
2
ω
2
s
}. (6)
It is very difﬁcult to represent the circular shape in the form of
trigonometric polynomials. Circular shape has not been attempted in
[6] and the polynomials used in [8] do not produce the desired shape.
However it has been proven in [9] that the passband and the stopband
of the form (5) and (6) can be exactly described by level sets of sec-
ond order trigonometric polynomials of the following form,
T (Ω) = a(cos 2ω
1
+ cos 2ω
2
)+b(cos ω
1
+ cos ω
2
)+c. (7)
Coefﬁcients a,b and c in Equation (7) have to be found such that
its level sets closely represents the circular shape given in (5) and
(6). Since the trigonometric polynomial of the form of (7) has been
proven to be a successful method of representing the circular shape,
the same mask will be used in this paper. The following least squares
optimization problem can be used to ﬁnd the coefﬁcients of (7).
min
x=(a,b,c)
T
,||x||=1
ω
α
/
2
0
x
T
cos 2ω
1
+ cos 2
ω
2
α
ω
2
1
cos ω
1
+ cos
ω
2
α
ω
2
1
1
cos 2ω
1
+ cos 2
ω
2
α
ω
2
1
cos ω
1
+ cos
ω
2
α
ω
2
1
1
T
xdω
1
, (8)
with α ∈{p, s}. The normalized eigenvector of the minimum eigen-
value of the following matrix will give the solution x =(a, b, c)
which minimizes the optimization problem deﬁned in (8).
ω
α
2
0
cos2ω
1
+cos2
ω
2
α
ω
2
1
cosω
1
+cos
ω
2
α
ω
2
1
1
cos2ω
1
+cos2
ω
2
α
ω
2
1
cosω
1
+cos
ω
2
α
ω
2
1
1
T
1
. (9)
Generally the minimum eigenvalue of the matrix (9) is zero or nearly
zero. This conﬁrms the fact that the second order trigonometric poly-
nomials of the form (7) exactly describes the circular shaped pass-
band and the stopband. Thus two trigonometric polynomials can be
derived to represent the passband and the stopband.
T
α
(Ω)= a
α
(cos 2ω
1
+cos 2ω
2
)+b
α
(cos ω
1
+cos ω
2
)+c
α
, (10)
with α ∈{p, s}. The next step is to deﬁne a family of trigonometric
polynomials using the Chebyshev recursion.
If T
0
(Ω) =1,
T
1
(Ω) =A + B cos ω
1
+ C cos ω
2
+ D cos(ω
1
+ ω
2
)
+ E cos(ω
1
ω
2
), (11)
with predeﬁned coefﬁcients A,B,C,D and E, then T
j
(Ω),j =2, 3, ...
can be derived using Chebyshev recursion as follows,
T
j
(Ω) =2T
(j1)
(Ω)T
1
(Ω) T
(j2)
(Ω). (12)
Here it should be noted that more complicated class of trigonometric
polynomials of the form {cos(
1
+
2
),j,k =n, .., 1, 0, 1, ...n}
are used in this paper than that used in [9] to facilitate the two fold
symmetry. However our results show that this leads to improvement
of the performance of the ﬁlter. Using the set of trigonometric poly-
nomials derived using (11) and (12) a moment matrix is deﬁned as
follows,
Ψ
Ψ
Ψ(Ω) =
1
T
1
(Ω)
.
.
.
T
m
(Ω)
1
T
1
(Ω)
.
.
.
T
m
(Ω)
T
,m=[(n 2)/2]. (13)
Since T
α
(Ω) 0, Ω Ω
α
and since Ψ
Ψ
Ψ(Ω) is a positive deﬁnite
matrix, T
α
(Ω)Ψ
Ψ
Ψ(Ω) 0, Ω Ω
α
. This result can be used to de-
ﬁne cone constraints which represent the passband and the stopband
of the ﬁlter,
C
α
={(X, Y
X, Y
X, Y ),X
X
X R
(n+1)×(n+1)
,Y
Y
Y R
(n+1)×(n+1)
:
X
X
X,M
1
M
1
M
1
(Ω) + Y
Y
Y,M
2
M
2
M
2
(Ω)≡
ˆ
X
α
ˆ
X
α
ˆ
X
α
,T
α
(Ω)Ψ
Ψ
Ψ(Ω),
ˆ
X
α
ˆ
X
α
ˆ
X
α
0 ∈{p, s}}. (14)
Thus the semi-inﬁnite program given by (4a),(4b) and (4c) can be
transformed in to a semi-deﬁnite program as follows:
min
X,Y
X,Y
X,Y
s
δ
s
(15a)
s.t.
X
X
X (1 δ
p
)E
E
E,Y
Y
Y
C
p
(15b)
X
X
X +(1+δ
p
)E
E
E,Y
Y
Y
C
p
(15c)
X
X
X (δ
s
)E
E
E,Y
Y
Y
C
s
(15d)
X
X
X +(δ
s
)E
E
E,Y
Y
Y
C
s
, (15e)
where E
E
E R
(n+1)×(n+1)
has zero entries except E
E
E(0, 0) = 1.
The passband deﬁned by (5) can be sampled, and for each sample
3695

Table 1. Lowpass circular shaped ﬁlters with (ω
p
s
)=(0.4π,0.6π)
Speciﬁcations Our method [9] [8] [3]
Filter order 12 12 25 25
Filter complexity 30 30 96 625
δ
p
0.0106 0.0124 0.0173 0.0257
δ
s
0.0019 0.0036 0.017 0.0248
two constraints can be derived using (15b) and (15c). Similarly two
constraints can be derived for each sample of the stopband (6) by
evaluating expressions (15d) and (15e). Then the coefﬁcients of the
matrix X
X
X and Y
Y
Y have to be found such that it minimizes δ
s
subject to
the constraints derived by sampling the passband and the stopband.
By considering equations (3) and by expanding the expression
given in (14), the following interesting result can be derived to rep-
resent the frequency response of the passband and the stopband.
H(Ω) = T
p
2m
j=0
a
j
T
j
(Ω)+(1 δ
p
) Ω Ω
p
(16)
H(Ω) = T
s
2m
j=0
a
j
T
j
(Ω) δ
s
Ω Ω
s
. (17)
It is clear from equations (16) and (17) that the passband and stop-
band can be described by 2m +1waveforms each, which are of the
form {T
α
T
j
∈{p, s},j =0, .., 2m}. The number of waveforms
required to describe the passband and stopband of the two-fold sym-
metric ﬁlters presented here is the same as that of the four-fold sym-
metric ﬁlters presented in [9] and therefore the digital complexity of
both cases are the same(2n +6). However due to the selection of
a more complicated polynomial for T
1
(Ω) in Chebyshev recursion,
the ﬁlter performance is improved in two-fold symmetric ﬁlters.
4. SIMULATION
In this section we will consider the simulation of a circular shaped
lowpass ﬁlter with ω
p
=0.4π, ω
s
=0.6π and of order 24. The
ﬁrst step is to derive the second order trigonometric polynomials that
represent the passband and the stopband. The eigenvector of the
minimum eigenvalue of the matrix given in (9) was calculated for
ω
α
= ω
p
and ω
α
= ω
s
.
Minimum eigenvalues were zero and the corresponding eigen-
vectors were (a
p
,b
p
,c
p
)=(0.0488, 0.6108, 0.7903) and
(a
s
,b
s
,c
s
)=(0.0922, 0.8269, 0.5547) which was then sub-
stituted in equation (10). Select δ
p
as 0.01 and the coefﬁcients of
equation (11) as (A, B, C, D, E)=(0.5, 0.25.0.25.0.125, 0.25).
Semi-deﬁnite program was derived as described in Section 3 and
the simulation was performed using optimization software YALMIP
[11] and SDPT3 [12] in MATLAB. Fig. 1 shows the frequency
response of the designed ﬁlter in log scale and the performance
comparison with different design methods is given in Table 1.
Highpass circular shaped ﬁlters can be designed analogously.
The simulation result of a circular shaped highpass ﬁlter with ω
p
=
0.6π, ω
s
=0.4π and of order 36 is shown in Fig. 2. Performance
comparison of different design methods is given in Table 2. In the
case of both lowpass and highpass ﬁlter designs, it is clearly evident
that the ﬁlter design method presented in this paper has the lowest
complexity and achieves the lowest δ
p
and δ
s
values.
1
0.5
0
0.5
1
1
0.5
0
0.5
1
140
120
100
80
60
40
20
0
20
ω
1
ω
2
|H(Ω)| (dB)
Fig. 1. Lowpass circular shaped ﬁlter in log scale.
1
0.5
0
0.5
1
1
0.5
0
0.5
1
120
100
80
60
40
20
0
20
ω
1
ω
2
|H(Ω)| (dB)
Fig. 2. Highpass circular shaped ﬁlter in log scale.
5. HOMOMORPHIC PROCESSING SYSTEM
When images with large dynamic range such as natural scenes are
recorded, image contrast can be signiﬁcantly reduced. Homomor-
phic processing can be used to reduce the dynamic range and to in-
crease the contrast of such images. An image is formed by recording
the light reﬂected from an object which is illuminated by some light
source. Based on this the model f (n
1
,n
2
)=i(n
1
,n
2
)r(n
1
,n
1
)
can be used to represent the image, where i(n
1
,n
2
) and r(n
1
,n
2
)
represent the illumination and reﬂection respectively [10].
Since they are combined multiplicatively the components
f(n
1
,n
2
). The illumination component i(n
1
,n
2
) is related to the
dynamic range of the image and is generally slow varying. So it
can be separated by lowpass ﬁltering log f (n
1
,n
2
). The reﬂectance
component i(n
1
,n
2
) on the other hand is related to the contrast
Table 2. Highpass circular shaped ﬁlters with (ω
p
s
)=(0.6π,0.4π)
Speciﬁcations Our method [9] [8] [3]
Filter order 18 39 25 25
Filter complexity 42 84 96 625
δ
p
0.0003 0.0062 0.0171 0.0248
δ
s
0.0014 0.0026 0.0178 0.0257
3696

within the image and generally vary rapidly. Highpass ﬁltering of
log f(n
1
,n
2
) can be used to extract the reﬂectance component.
Fig. 3 depicts the complete homomorphic processing system.
The dynamic range can be reduced by decreasing α and the contrast
can be increased by increasing β. This process was implemented
using circular shaped lowpass ﬁlter with (ω
p
s
)=(0.4π, 0.8π)
and highpass ﬁlter with (ω
p
s
)=(0.8π, 0.4π). It was applied
on youtube video with (α, β)=(0.99, 1.5) and the results are
given in Figs. 4 and 5. To see these images more clearly visit
http://ee.unsw.edu.au/z3265024/enhancement.html. It can be seen
that the background details are more highlighted in the enhanced im-
age and people standing at the back of the court can be clearly seen.
To evaluate the effectiveness of the homomorphic process the
original image given in Fig. 4 was blurred using a gaussian lowpass
ﬁlter before applying the homomorphic process. Then we recovered
the sharpness and the contrast of the original image by changing
the α and β values of the homomorphic process. The PSNR value
between the blurred image and the original image, and that between
the enhanced image and original image was calculated. Results are
given in Fig. 6. This effectively shows that the PSNR value can be
considerably improved by the homomorphic process implemented
using the ﬁlters discussed in this paper.
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Fig. 3. Homomorphic system for image enhancement.
Fig. 4. Original image.
Fig. 5. Enhanced image.
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ϮϬ
ϯϬ
ϰϬ
ϱϬ
ϲϬ
ϳϬ
ϴϬ
ϵϬ
Ϭϯ Ϭϰ Ϭϱ Ϭϲ Ϭϳ Ϭϴ ϭ
W^EZͰ;ĚͿ
sĂƌŝĂŶĐĞ
ůƵƌƌĞĚ/ŵĂŐĞ ŶŚĂŶĐĞĚ/ŵĂŐĞ
Fig. 6. PSNR improvement using homomorphic processing.
6. CONCLUSION
A very general approach of designing two-fold symmetric circular
shaped ﬁlters with complexity equal to that of a four-fold symmet-
ric ﬁlter was presented in this paper. The main advantage is that it
successfully produces the desired circular shape with minimum pass-
band and stopband ripple compared to the currently available design
methods. The coefﬁcients of the polynomial T
1
(Ω) in Chebyshev
recursion have an inﬂuence on the ﬁlter performance and a method
of selecting optimal values for these coefﬁcients is still open for re-
search.
7. REFERENCES
[1] T. Chen and P. Vaidyanathan, “Recent developments in multidimen-
sional multirate systems, IEEE Trans. on Circuits and Systems for
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[2] C. Chen and J. Lee, “McClellan transform based design techniques for
two-dimensional linear-phase FIR ﬁlters, IEEE Trans. on Circuits and
Systems I: Fundamental Theory and Applications, vol. 41, pp. 505–
517, 1994.
[3] R. Mersereau, W. Mecklenbrauker, and T. Quatieri, “McClellan trans-
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[4] S. Bagchi and S. Mitra, “The nonuniform discrete Fourier transform
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433, 1996.
[5] J. Hu and L. Rabiner, “Design techniques for two-dimensional digital
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257, 1972.
[6] B. Dumitrescu, “Trigonometric polynomials positive on frequency do-
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[7] T. Hung, H. Tuan, B. Vo, and T. Nguyen, “SDP for 2-D ﬁlter de-
sign: General formulation and dimension reduction techniques, Proc.
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[8] T. Hung, H. Tuan, and T. Nguyen, “Design of diamond and circular ﬁl-
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[9] H. Tuan, K. Fanian, and T. Nguyen, “2-D nonseparable ﬁlters of com-
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[11] Y. Lofberg, “Yalmip: A toolbox for modeling and optimization
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