Approximating Spline filter: New Approach for Gaussian Filtering in Surface Metrology
ABSTRACT This paper presents a new spline filter named approximating spline filter for surface metrology. The purpose is to provide a new approach of Gaussian filter and evaluate the characteristics of an engineering surface more accurately and comprehensively. First, the configuration of approximating spline filter is investigated, which describes that this filter inherits all the merits of an ordinary spline filter e.g. no phase distortion and no end distortion. Then, the approximating coefficient selection is discussed, which specifies an important property of this filterthe convergence to Gaussian filter. The maximum approximation deviation between them can be controlled below , moreover, be decreased to less than 1% when cascaded. Since extended to 2 dimensional (2D) filter, the transmission deviation yields within − . It is proved that the approximating spline filter not only achieves the transmission characteristic of Gaussian filter, but also alleviates the end effect on a data sequence. The whole computational procedure is illustrated and applied to a work piece to acquire mean line whereas a simulated surface to mean surface. These experimental results indicate that this filtering algorithm for 11200 profile points and 2000 × 2000 form data, only spends 8ms and 2.3s respectively.
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ABSTRACT: A fast and reliable convolution algorithm to calculate the mean line of a roughness profile using the Gaussian filter according to ISO 11562 has been derived. The algorithm is based on a recurrence relation for the weighting function. This greatly speeds up the calculation, making the algorithm nearly comparable to algorithms using the fast Fourier transform (FFT) in a usual way. The algorithm has been implemented as a short C function to be used with any evaluation program. The application of this function to a measured profile is given for demonstration and compared with the results obtained by the ordinary FFT filter algorithm.Precision Engineering. 01/1996;  International Journal of Machine Tools and Manufacture 01/2001; 41(13):21532161. · 2.26 Impact Factor

Article: Form filtering by splines
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ABSTRACT: The recommended use of a Gaussian filter as a form filter turns out to be problematic, because the resulting mean line often appears outside the surface profile. In contrast to this spline filters seem to be much better suited to meet the requirements for a form filter. It will be shown how a spline filter can be designed and calculated. The efficiency of the spline filter in comparison with a Gaussian filter will be demonstrated by an example. In addition the transfer function of the spline filter will be compared with that of the Gaussian filter.Measurement 01/1996; · 1.13 Impact Factor
Page 1
I.J. Image, Graphics and Signal Processing, 2009, 1, 916
Published Online October 2009 in MECS (http://www.mecspress.org/)
Approximating Spline filter: New Approach for
Gaussian Filtering in Surface Metrology
Hao Zhang
Harbin Institute of Technology, Harbin, China
Email: Zhanghaowo79@163.com
Yibao Yuan
Harbin Institute of Technology, Harbin, China
Email: Yibaoyuan2008@yahoo.cn
Abstract—This paper presents a new spline filter named
approximating spline filter for surface metrology. The
purpose is to provide a new approach of Gaussian filter and
evaluate the characteristics of an engineering surface more
accurately and comprehensively. First, the configuration of
approximating spline filter is investigated, which describes
that this filter inherits all the merits of an ordinary spline
filter e.g. no phase distortion and no end distortion. Then,
the approximating coefficient selection is discussed, which
specifies an important property of this filter
convergence to Gaussian
approximation deviation between them can be controlled
below , moreover, be decreased to less than 1% when
cascaded. Since extended to 2 dimensional (2D) filter, the
transmission deviation yields within −
proved that the approximating spline filter not only achieves
the transmission characteristic of Gaussian filter, but also
alleviates the end effect on a data sequence. The whole
computational procedure is illustrated and applied to a
work piece to acquire mean line whereas a simulated
surface to mean surface. These experimental results indicate
that this filtering algorithm for 11200 profile points and
2000 ×× 2000 form data, only spends 8ms and 2.3s
respectively.
Index Terms—surface metrology,
approximating spline filter, Gaussian filter, form filter
rthe
filter. The maximum
. It is
4.36%
0.63%1.
?
48%
+
profile filter,
I. INTRODUCTION
In conventional surface metrology, surface assessment
is always on profile as the curve of intersection. In recent
years, with improvement of manufacturing techniques
and demand of quality control, three dimensional (3D)
analysis of surface geometry has become more and more
important. Thereunto, mean line is the reference line
about which the profile deviations are measured, mean
surface is the 3D reference surface about which the
topographic deviations are measured[1]. Both the mean
line and mean surface is established by applying a
filtering process to the measured surface. Filters selected
in this establishment have become critical for numerical
_________________________________________________________
Manuscript received January 18, 2009; revised June 19, 2009;
accepted August 22, 2009.
characterization and parameters determination.
The profile filter of Gaussian is the most widely used
filter described in ISO11562 [2], [3]. Gaussian filter is
superior to 2RC filter by two advantages, phasecorrected
property, which is a simpler way to employ by many fast
algorithms [4],[5]. However conventional Gaussian filter
has indelible end effect, even the GR2 haven't resolved it
entirely but with larger computation cost [6]. In fact,
according to the point of view in digital processing,
convolution is the primary reason of end effect. The
output data are the convolution results between the filter
transfer function and input data, then the end effect
generates during this process.
In order to overcome end effect, Krystek proposed a
new spline filter algorithm to substitute for Gaussian
Filter [7]. This method adopts numerical fitting and
matrix equation solution to obtain the mean line, with
result in abandoning the filter's convolution [8],[9].
T.Goto proposed the robust spline filter which is less
influenced by outliers and also implemented through
matrix equation [10].
Both of them can obtain a kind of mean line to analyze
the surface profile. These filters are all designed for
overcoming certain limitations existing in the surface
measurement such as end effect, computing efficiency
and robustness. However, they, besides Numada [11], all
overlook the most important thing, which is the
substitution likelihood for ISO standard of Gaussian.
Above all, Gaussian filter is the ISO11562 standard with
a distinguishing feature of at least spacefrequency
product whose cutoff is sufficiency for profile filter. How
to not only preserve Gaussian filter's transmission
characteristic, but also restrain end distortion becomes a
stubborn problem.
In this paper, the new kind of spline filter which not
only inherits the ordinary spline filter quality but
approximates to Gaussian filter is brought forward. i.e., in
bosom of measured data, the filtering results conform to
the ISO standard, at the same time, the end effect is
restrained by numerical fitting technique. In addition, to
3D surface evaluation including enormous amount of
computation due to large quantity of sampling data, it
also yields a fast filtering process.
Copyright © 2009 MECS I.J. Image, Graphics and Signal Processing, 2009, 1, 916
Page 2
10 Approximating Spline filter: New Approach for Gaussian Filtering in Surface Metrology
II. APPROXIMATING SPLINE FILTER
The spline filter based on variation differential theory
had been proposed by Schoenberg[12] and Reinsh[13] for
50 years. Poggio applied to the variation spline of
Tikhonov, regularization to image process and stated it as
leading to a Gaussianlike convolution filter[14],[15].
Given a set of measured value sampled
filter is defined as the function:
, the spline
kz
{}
2
2
1
( )
m
()
m
n
kk
k
d s x
dx
zs x dxMin
εμ
=
⎧
⎨
⎩
⎫
⎬
⎭
=−+→
∑
∫
(1)
where
output data, n is the number of measured points, μ is the
LaGrange constant, 2
is order degree of spline. The
differential order m is higher than the spline filter cutoff
is more steplike [16]. Actually, in ordinary condition of
convenient calculation, m is given 2, and the spline is
cubic order. This spline definition can be thought as the
compromise between approximation to data and bending
energy of spline through
μ constant. The functions are
discussed in this literature considering the general case of
equally spaced nodes and a finite number of input data,
for the natural condition of measured profile data.
In order to resolve the problem of end effect existing in
filtering results by Gaussian filter, the spline filter has
been proposed for profile measurement [8]. However,
these spline filters differ greatly from Gaussian
transmission. In fact, for digital instruments, excluding
disadvantage of end effect, Gaussian filter is appropriate
filter for surface profile evaluation, and sufficient to
separate profiles into long wave and shortwave
components. Therefore, (1) is expected to be
reconstructed to perform better realization of Gaussian
filter. In Johannes’ paper [17], a first order differential is
added into the second item of (1) to achieve this pursuit.
So a new type of spline filter named approximating spline
filter in this paper is constructed. The approximating
spline filter is expressed by
is the index for the sampled data,
k
()
k
s x
is the
1
m−
{}
2
2
2
2
1
( )
()
n
kk
k
d s x
dx
zs x
εμ
=
⎧⎛
⎪
⎨⎜
⎪⎩
⎞
⎟
⎠
=−+
⎝
∑
∫
+
2
( )
dx
ds x
dx Min
τ
⎫⎪
⎬
⎛
⎜
⎝
⎞
⎟
⎠ ⎪ ⎭
→
(2)
where τ is the Gaussian approximation coefficient.
Regulating τ properly, the solution of (2) can be close to
the result filtered by Gaussian filter, and (2) can be
named the spline realization of Gaussian filter.
Adapting to digital processing, (2) must be discretized
as the following equation:
{}
{
11
kk
==
where is the difference operator[16], and
∇
∑
(3)
}
2
222
()()
nn
kkkk
zsssMin
εμτ=−+∇+∇→
∑∑
(4)
/ 2
0
{( 1)
−
}
m
mii
mk
kmi
i
sCs+−
⎢
⎣
⎥
⎦
=
∇=⋅
where are polynomial coefficients.
i
m
C
To achieve the minimum of function (3), an equation
similar to partial difference with respect to
implemented by
ε∂
=
∂
There are two types of boundary conditions, which are
the nonperiodic data and periodic data [10], thereunto,
the former is also called nature boundary condition.
Depending on the different boundary conditions, we can
deduce different filters that also adapt to different tasks.
For most instances, engineering surfaces possess arbitrary
and nonperiodic distribution, for which the nature
boundary condition is needed, that is
1
s
∇=∇
Utilizing equation of (5) and (6) ， the following
equations are constructed:
{
11
1
s
ε∂
= −−+
∂
+
∂
= −−+
∂
∂
= −−+
∂
+
∂
= −−+
∂
For some special cases such as calibration specimens,
their surface present strongly regular and period
distribution, and such a surface should be disposed with
period boundary condition as shown as:
k
s
=
Hence, it introduces another partial difference
equation by (5) and (8)
ε∂
= −−+
∂
We observe that both (7) and (9) can be written as a
kind of uniform matrix equation such as:
()
IQ S
μ+
where is identity matrix,
IZ is the vector of measured
data, and S is the output result data.
coefficient matrix, with (11) and (12) corresponding to
the nature boundary condition and period boundary
condition respectively.
121
2 5 241
146 24
Q
τ
− −
⎜
⎜
⎜
⎝
ks
is
0
ks
(5)
22
0
n
s
= (6)
}
{
}
{
}
321
2243
2
21
42
1132
1
1
21
2()2(2) (1)
2()2 (4)
(52 )
τ
(2)
2()2 (
μ
)
2()2 (4)
(52 )
τ
(2)
2()2(2)
kkkk
k
nnnn
n
nn
nnnn
n
zssss
zsss
s
ss
zsss
s
zsss
s
ss
zsss
s
ε∂
∂
μττ
μτ
τ
ε
τ
ε
μτ
τ
ε
μτ
−−−−
−
−
−−
= −−+−+++
−++
−+
∇ − ∇
−++
−+
−+
{}
(1)ns
τ
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎩
++
(7)
k
s+n
(8)
42
2() 2 (
μ
)0
kkkk
k
zsss
s
τ
∇ − ∇=
(9)
Z
=
(10)
is welldeduced
Q
1
O
146 2
4
1
4
+
1
15 2
2
− −
2
1
τ
τ
τ
τ
τ
τ
ττ
τ
τ
τ
τ
τ
τ
τ
+
+− −
+
− −
O
⎛
⎜
⎜
⎜
⎜
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
− −− −
+
O
− −
O
=⎜
⎜
+− −
− −− −
O
(11)
Copyright © 2009 MECS I.J. Image, Graphics and Signal Processing, 2009, 1, 916
Page 3
Approximating Spline filter: New Approach for Gaussian Filtering in Surface Metrology 11
6 2
4
1
4
+
114
6 2
4
− −
O
4
+
O
11
6 24
O
1
O
146 2
4
1
4
+
1
116 2
4
− −
416 2
+
Q
4
τ
τ
τ
τ
τ
τ
τ
ττ
ττ
τ
τ
τ
τ
τ
ττ
+− −− −
⎛
⎜
⎜
⎜
⎜
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
− −− −
− −
=⎜
⎜
⎜
⎜
⎜
⎝
− −+− −
− − − −
− −
O
Figure 1. Transmission characteristics of digital filters.
Figure 2. Deviation of transmission characteristics.
(12)
III. APPROXIMATING COEFFICIENT
The weighting function and transmission
characteristics of Gaussian filter are given by
ISO11562[2]:
1
( )
h t
αλ
2
( /
t
)
c
c
e
π αλ−
=
2
( / )
λ
(/ )
c
c
He
π αλ−
λ λ=
where α is a constant, ,λ is wavelength，and
off wavelength. When
transmission, then
0.4697
α =
the transmission for the cutoff wavelength is 50% since
the short wave and long wave potions of the surface
profile are separated and can be recombined without
altering the surface profile[2]”
The transmission characteristic of the approximating
spline filter has been derived from (9)[17]:
(4) (1 (6
kkk
zss
μτ μ
−−
=−+++
Using the Ztransform, we get
Z z
the Ztransform of the sampled data
Ztansform of the solution data s and with filter G(z) is
given by
c λ is cut
c
λλ
. “It is of importance that
=
, the filter has 50%
211
) )
τ μ
(4
=
)
kkk
s
S z
ss
τ μ
( ) ( )
G z Z z
and
μ
, with
( )
S z
+
+−++
( )
( )
2
+
kz
k
the
221
1
( )
()
exp(
( )
G ω of this regularization filter
1
(1 cos ) 4 (1 cos )
τμωμ
⋅−+
In (13), there are many different approximation forms
to Gaussian filter obtained by different value τ . Our
purpose is to find a knowable expression as an optimal
choice. `”The closest approximation to the Gaussian
function Fourier spectrum is obtained in the case of
1/
τμ
=
[17]”. In addition, it is obviously reduced to
ordinary spline filter when
τ = .
We have known that the Gaussian filter has 50%
transmission characteristic in the cutoff frequency, this
requirement is also been adopted to spline function, that
is
1
1 2(1 cos)
c
μω
+⋅−+
For spatial signal,
frequency
2/
d
ωπλ
=
, is sampling interval which is
assumed unit value in this paper for convenience,
and
n
λ = . When
c
λλ
=
,
(4) (
τ μ
)
jω
) 1 (6
+ +
yields the discrete
( )
G z
2
G z
zzzz
)
μ τ μ
−−
=
+−+++
Substitution of by
Fourier transform
z
−
in
2
( )
1 2
+
G ω=
−
ω
(13)
0
2
1
2
4 (1 cos
μ
−
the
)
digital
c
ω
=
(14)
angle
d
2
c
cn
π
ω =
(15)
From (14) and (15), LaGrange constant μ can be
derived by
1
2
1 2(1 cos)
c
n
It's observed that (16) is a quadratic equation in one
variable with respect to μ , therefore, μ can be solved
as
( 5 1)
μ
=
2
1
2
2
n
4 (1 cos
μ
−
)
c
ππ
μ
=
+⋅−+
(16)
2
4
64sin
cn
π
−
⎛
⎜
⎝
⎞
⎟
⎠
(17)
From this μ , as shown as Fig.1, the frequency
amplitude of an approximating spline filter matches with
that of Gaussian filter well. Fig.4 shows that the
maximum of the amplitude
approximating spline and Gaussian is only 4.26%, which
is even less than many Gaussian filter simplified
algorithms. On the contrary, the maximum of amplitude
deviation between ordinary spline and Gaussian filter is
10.6%, which is illustrated graphically in Fig.2. So it is
clear that approximating spline, not original spline, is
selected as the approximation to Gaussian filter.
deviation between
Approximating spline has no phase shift error, because
the odd order spline is symmetry to a vertical axis whose
property is the same as Gaussian function, and even order
spline also has this property only with a displacement
constant product.
Copyright © 2009 MECS I.J. Image, Graphics and Signal Processing, 2009, 1, 916
Page 4
12 Approximating Spline filter: New Approach for Gaussian Filtering in Surface Metrology
IV. CASCADES OF APPROXIMATING SPLINE FILTER
For some special conditions, approximation with
4.26% deviation isn't enough for practice. This filter
algorithm must be improved to fit higher accuracy
Gaussian results. If single step of approximating spline
filter is regarded as a basic process prototype, quadratic
cascades of them can attain better approximating effect.
In these situations, μ must be recalculated, because the
quadratic algorithm should also stand by 50%. The
quadratic algorithm is constructed as
(
G ωτ
)1 2 (1 cos )
μ
⎡
=+⋅−
⎣
The approximating coe
ar e also calculated by the method introduced, the similar
as (17) in previous section, but
(
μ
=
yright © 2009 MECS I.J. Image, Graphics and Signal Processing, 2009, 1, 916 Cop
Figure 3. Transmission characteristics of digital filters.
Figure 4. Deviation amplitude between approximating spline filter and
Figure 5. Approximation deviation of each cascaded spline filter.
2
2
4 (1 cos )
μ
−
ωω
−
⎤
⎦
+
nt and lagrange constant fficie
(18)
)
2
2
4 2 −
4
3 1
−
64sin
cn
π
⎛
⎜
⎝
⎞
⎟
⎠
and
betw
1/
τ
een the q
μ
=
Furthermore, if n cascaded spline filter is executed,
the transmission function is defined as
( )1 2 (1 cos )
G ω τμω
⎡
=+⋅−
⎣
where
(
μ
=
2
4 (1 cos )
μ
−
n
ω
−
⎤
⎦
+
(19)
)
2
4
4 23 1
−
64sin
n
cn
π
−
⎛
⎜
⎝
⎞
⎟
⎠
(20)
It's found that if the spline order n is more higher, the
approximation deviation between multiple approximating
ssian filter is more lower, that is,
when, this spline filter approaches Gaussian
transmission characteristic infinitely. In Fig.4, the 30
cascades can approximate to Gaussian filter with
fluctuant but maximum 0.52% deviation.
Fig 5 shows this approximation trend for each
cascading order spline through 1 to 500, in which the
maximum approximation deviations are almost under 1%,
except order 1 and 3 to 11, and generally incline to zero.
The stricter explanation of this trend is referred to central
limit theorem [4]. From above analysis, it is turned out
spline filter and Gau
n → ∞
that quadratic cascaded approximating spline filter is the
optimal choice compromising between efficiency and
precision.
as of old. In Fig.3, amplitude transmission
uadratic cascaded approximating spline and
Gaussian filter is compared, and the maximum deviation
depicted in Fig.3 between them is 0.92%. This result is
considerable satisfying compared with relative deviation
tolerance (5%～+5%) in ISO11562. See Fig.3, their
transmission characteristics is superposition each other
ultimately.
V. TWO DIMENSIONAL FILTER
3D surface analysis presents the geometrical form of
an area on a work piece more particularly than the profile
of 2D. The mean surface, similarly to mean line for
profile, established by the form filter, is a reference
surface for 3D surface evaluation. A roughness surface is
obtained by subtracting the mean surface from the
primary surface.
Extending the weighting function of Gaussian filter to
2D, we obtain the 2D Gaussian filter for 3D surface
measurement
⎧
⎪
=⋅−
⎨
⎪
⎩
whose transmission characteristic is given as
2
2
1
( , )
h x y
exp
xcyc xc yc
xy
π
ββλ λλλ
⎫
⎪
⎬
⎪
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦⎭
⎛
⎜
⎝
⎞
⎟
⎟
⎠
⎛
⎜
⎝
⎞
⎟
⎠
+⎜
(21)
Page 5
Approximating Spline filter: New Approach for Gaussian Filtering in Surface Metrology 13
2
2
(,) exp
yc
xc
xy
xy
H
λ
λ
λ
λ
λ λπβ
−
⎧
⎪
⎨
⎪
⎩
⎫
⎪
⎬
⎪
⎭
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
⎛
⎜
⎝
⎞
⎟
⎟
⎠
⎛
⎜
⎝
⎞
⎟
⎠
= +⎜
Figure 6. The transmission characteristic of 2D approximating spline
filter.
Figure 7. The deviation between the transmission characteristics of
approximating spline filter and Gaussian. approximating spline filter and Gaussian.
between the transmission characteristics of
(22)
where
and y axis. When
amplitude transmission is appealed for 50%, then
ln2/
βπ=
.
In general, (22) can be reduced to the product form of
two independent Gaussian filters:
⎧
⎪
=−
⎨
⎪
⎩
⎧
⎛
⎪
=−
⎨⎜
⎝
⎪
⎩
=⋅
which clearly indicates that it is a simple realization of
2D Gaussian filter and that it can be achieved by
employing two 1D process to xcoordinate and y
coordinate respectively.
The 3D surface evaluation, the total number of pending
data is so enormous that original spline filter requires
maximum of 1h for computation, hence, we should pay
more attention to the implementing of efficient
processing algorithm [16]. On the other hand, the data
corresponding one direction section of 3D is down
sampling generally in contrast to profile, it need to
alleviate the end effect more necessarily for preserving
limit valid data. Here, we select the 2D approximating
xc
λ and
yc
λ is the cutoff wavelength along x axis
λλ=
,
xxc
y λ = ∞ or
yyc
λλ=
,
x λ = ∞ , the
2
2
(,)exp
yc
xc
xy
xy
H
λ
λ
λ
λ
λ λ πβ
⎫
⎪
⎬
⎪
⎭
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
⎛
⎜
⎝
⎞
⎟
⎟
⎠
⎛
⎜
⎝
⎞
⎟
⎠
+⎜
2
2
exp exp
()()
yc
xc
x
xy
HH
λ
λ
λ
λ
πβ πβ
−
λλ
⎧
⎪
⎨
⎪
⎩
⎫
⎪⎬
⎪ (23)
⎫
⎪
⎪
⎭
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
⎞
⎟ ⎬
⎠
⋅
⎭
y
spline filter as the form filter to extract mean surface. The
similar to 2D Gaussian filter, a 2D approximating spline
filter can also be obtained by extending respectively two
1D approximating spline filter along xdirection and y
direction, whose results are composed for a whole 3D
reference surface. According to above discussion, we
have known the approximating filter can approximate to
the transmission characteristic of Gaussian filter with
admissible tolerance. The 2D approximating spline filter
can also approach the 2D Gaussian filter with high
accuracy.
Fig. 6 shows the transmission characteristic of 2D
approximating spline filter of quadratic cascades. Fig.7
describes the deviation between the 2D approximating
spline filter and Gaussian filter, the deviation is limited in
0.63%1.48%
+
?
, which is acceptable outcome for most
assessments of 3D surfaces.
VI. THE ALGORITHM
In previous sections, we present the configuration of
approximating spline filter,
approximation property with Gaussian filter particularly,
in this section, an efficient fast algorithm to implement
approximating spline filter will be introduced. Observing
(10), it is distinctly that this filter algorithm to profile data
will be solved by matrix technique, which is discussed by
three step as follows:
a) Compute μ by (17), and defined (
still a positive definite matrix and diagonal dominant.
b) can be disposed based on Cholesky
decomposition. QR
, then
=
c) Assuming DRSY
=
, the algorithm can be divided
into
R YZ
=
and
RSD−
=
Y .
d) If cascaded algorithm needed, repeat step c) with
designed cascaded order iteration, else end. Note μ is
calculated depend on the cascaded order by (20).
It has related that 2D filter for 3D surface evaluation
can also be realized by this spline filter, who will perform
respectively the algorithm related above along the x
coordinate, then along xcoordinate in successive order,
and achieve the approximating results to 2D Gaussian
filter.
In fact, this algorithm is similar to Krystek's[7]. This
algorithm denotes a complete matrix calculation without
the convolution between data and discrete filter, and
doesn't need a data preparation stack. Therefore there is
almost no end distortion during this process. This good
property will be testified by the experiments of next
section.
and explain the
. as
'
)
IQQ
μ+=
'
Q
'
Q
'
TDR
T
R DRSZ
=
.
T
1
VII. EXPERIMENTS AND DISCUSSION
Copyright © 2009 MECS I.J. Image, Graphics and Signal Processing, 2009, 1, 916
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