Noise-Produced Patterns in Images Constructed from Magnetic Flux Leakage Data

0.59 MB·English

by  Anastasiya V. Pimenova

Math. Model.Nat. Phenom.
Vol. 10,No. 1,2015,pp. 10-20
Noise-Produced Patterns in Images Constructed
5
1
0 from Magnetic Flux Leakage Data
2

n
a
J Denis S. Goldobina,b,c,1, AnastasiyaV. Pimenovaa, Jeremy Levesleyb,

3
1 Peter Elkingtond, MarkBacciarellid

]
n a InstituteofContinuousMediaMechanics,UB RAS, Perm 614013,Russia
a b Department ofMathematics,UniversityofLeicester, LeicesterLE1 7RH, UK
-
a c DepartmentofTheoreticalPhysics,Perm StateUniversity,Perm 614990,Russia
t
a d Weatherford, East Leake, LoughboroughLE126JX, UK
d
.
s
c
i
s Abstract. Magnetic flux leakage measurements help identify the position, size and shape of
y
corrosion-related defects in steel casings used to protect boreholes drilled into oil and gas reser-
h
p voirs. Images constructed from magneticflux leakage datacontain patterns related to noiseinher-
[
ent in the method. We investigate the patterns and their scaling properties for the case of delta-

1 correlated input noise, and consider the implications for the method’s ability to resolve defects.
v
1 Theanalyticalevaluationofthenoise-producedpatternsismadepossiblebymodelreductionfacil-
8 itatedbylarge-scaleapproximation. Withappropriatemodification,theapproachcanbeemployed
0
to analyze noise-produced patterns in other situations where the data of interest are not measured
3
0 directly, but are related to the measured data by a complex linear transform involvingintegrations
.
1 withrespect tospatial coordinates.
0
5
1 Keywords: magneticflux leakage, noise-producedpatterns, corrosivedefects
:
v AMSsubject classification: 78A30,78M34,60G60
i
X
r
a
1. Introduction
A common challenge in engineering is the need to infer properties of interest from indirect mea-
surements. The direct measurement of the property of interest may be possible but expensive or
1Correspondingauthor.E-mail:[email protected]
1
D.S. Goldobin, A.V.Pimenova,etal. MFL: Noise-Produced Patterns
Figure1: Ferromagneticlayerofnon-uniformthicknessinthefield ~ and thecoordinateframe
H
inefficientcomparedtoanindirectmeasurement—anexamplebeingthedeterminationofmethane
hydrate saturation profiles in seafloor sediments, which can be done efficiently and cheaply by
indirect means by measuring water salinity profiles from which saturations can be determined
(e.g., see [1, 2]). The challenge is particularly widespread in methods employing acoustic and
electromagneticwavescattering(e.g., see[3,4,5]), andinthedeformationofstaticfieldsduetoa
medium’ssusceptibilitytothesefields. Anexampleofthelatteristhemagneticfluxleakage(MFL)
methodusedtoassesscasingintegrityinboreholesdrilledintorockformationsfortheexploration
and evaluation of oil and gas resources (e.g., see [6, 7, 8]). In these cases the profiles of interest
are complex transforms of the measured data involving integrations with respect to spatial coor-
dinates. These transforms can produce non-trivial and unexpected responses in the reconstructed
(output)dataduetonoiseinherenttothemeasurement. Evenforuncorrelatedinputdatathenoise-
produced patterns in reconstructed data can be correlated, smooth and indistinguishable from the
actual patterns in an idealistic noise-free situation. (In some cases, the appearance of smooth,
noise-producedpatternscanbetreatedasaresultofnoiseaccumulationinthereconstructeddata.)
Therefore noise-produced patterns cannot simply be filtered out by means of some data process-
ing procedure (which would be the case for uncorrelated noise). Such noise-induced patterns can
significantlyinfluencetheutilityofthereconstructionprocedure, and knowledgeof theproperties
ofthesepatternsisthereforeimportant,andservesasanvaluableguideinthedevelopmentofdata
analysisalgorithmsand infuturedevicedesign.
Inthispaperwesuggestandimplementanapproachtocomprehensivelyanalysenoise-produced
patterns in MFL-reconstructed data for different device designs. The problem set-up corresponds
to measurementsmadewithmodern devicesdesigned fortheMFL inspectionofwellborecasings
described for examplein [7]. Our approach is based on the mathematical methodologydeveloped
in [8] within the framework of large-scale approximation, which is relevant for the case of corro-
sivedamage. Such model reduction provides the opportunityfor a comprehensiveanalysis whose
results remain qualitatively valid for the general case beyond large-scale approximation. In the
analysis of inaccuracies, it is generally sufficient to have estimates of errors. A similar approach
can beemployedforvariousdatareconstructiontasks.
Thepaperisorganizedasfollows. InSection2weintroducethebasicmathematicalmodelwe
2
D.S. Goldobin, A.V.Pimenova,etal. MFL: Noise-Produced Patterns
use. In Section3weevaluatethepatternsinthereconstructeddataproducedbyδ-correlated noise
in the measured magnetic field data. In Section 4 we analyse the scaling properties of the noise-
produced patterns for various device designs, compare the cases of different designs, and discuss
the role of the defect geometry for their resolvability against the background of noise-produced
patterns. Conclusionsaredrawn inSection 5.
2. Magnetic field in the presence of large-scale defects
The system under consideration is shown in Figure 1; it comprises a ferromagnetic layer subject
to an external magnetic field parallel to the layer middle-plane; the x-axis is directed along ~,
H
and the z-axis is perpendicular to the layer middle-plane. Within the framework of large-scale
approximation, the magnetic field within the ferromagnetic layer is nearly homogeneous along
the z-axis and is parallel to the layer-middle plane, and thus can be represented by a gradient of
potential φ(x,y), which is a function of two coordinates (x,y) only: H~(in) = φ(x,y). To the
−∇
leadingorder, thispotentialobeystheequation[8]
e−σ(x,y) φ(x,y) = 0, (2.1)
2 2
∇ · ∇
(cid:0) (cid:1)
wheresubscript‘2’indicatesthat operatesinthetwo-dimensionalspaceof(x,y). Thefunction
2
∇
σ(x,y)describes therelativelossoftheferromagneticmaterial;thelayerthicknessprofile
2ζ(x,y) = 2ζ e−σ(x,y),
0
where2ζ isthethicknessoftheundamagedlayer.
0
Outsideofthelayerthemagneticfield isthe3-dgradient ofaharmonicpotential;
H~(ext) = ϕ(x,y,z), (2.2)
−∇
∆ϕ(x,y,z) = 0, (2.3)
withtheboundaryconditionon theferromagnet surface
ϕ(x,y,z = ζ(x,y)) = φ(x,y)
and ( ϕ) = ~ . Withinthe framework oflarge-scale approximation,to theleading order,
z→±∞
−∇ H
onecan translatetheboundaryconditionforϕfrom thelayersurfaceto z = 0:
ϕ(x,y,0) = φ(x,y). (2.4)
By virtue of Eq.(2.3), the potential ϕ of the external field can be represented in the basis of
exponentialfunctions
ϕ(x,y,z) = x+ dk dk A(k ,k )e−k|z|ei(kxx+kyy), k = k2 +k2. (2.5)
−H x y x y x y
ZZ q
3
D.S. Goldobin, A.V.Pimenova,etal. MFL: Noise-Produced Patterns
Here A(k ,k )e−k|zm| is the Fourier transform of the deformation of the magnetic potential field,
x y
ϕ + x, measured on the plane elevated by (z ζ ) above the layer surface. While the field
m 0
H −
ϕ(x,y,z ) can be found from measurements of H(ext)(x,y,z ) or/and H(ext)(x,y,z ) by plain
m x m y m
integrationwithrespecttoxory,evaluationofϕ(x,y,z )frommeasurementsofH(ext)(x,y,z )
m z m
is non-trivial; the relationshipbetween ϕ(x,y,z ) and H(ext)(x,y,z ) is non-local. For this rela-
m z m
tionship,onecan rely onEq.(2.5)and itsz-derivative;
H(ext)(x,y,z) = dk dk kA(k ,k )e−k|z|ei(kxx+kyy), k = k2 +k2. (2.6)
z x y x y x y
ZZ q
Thus, one can introduce a non-local operator ˆ which transforms ϕ(x,y,z ) (a function of two
m
L
variables(x,y))intoH(ext)(x,y,z ) (anotherfunctionoftwovariables(x,y)):
z m
∂
ϕ(x,y,z ) = ˆϕ(x,y,z ). (2.7)
m m
− ∂z L
This transform is correctly defined for any harmonic function vanishing for z , and in the
Fourier space it is a multiplication by k = ~k . For trivial conditions on the b→oun±d∞aries limiting
the system in the x- and y-directions, the inv|er|seoperator ˆ−1 is well defined; the operator ˆis a
L L
homeomorphism.
The aim of our work is to establish smooth large-scale patterns in reconstructed data pro-
duced by the noise in measurements. Hence, the approximate calculation of these patterns will
suffice, and we may make a further model reduction for Eq.(2.1) and the system under con-
sideration. Specifically, we make this reduction for z small compared to the scale of defects,
m
φ(x,y) ϕ(x,y,z ). Hence, Eq.(2.1)yieldsan equationforcalculation ofσ(x,y)from measur-
m
≈
able derivatives of φ(x,y): φ σ = ∆ φ. To the leading order, H~(ext)(z ) ~; therefore,
2 2 2 m
∇ ·∇ ≈ H
Eq.(2.1)can beapproximatelyrewrittenas
(ext)
∂σ ∂H
z
= ∆ ϕ(x,y,z ) = . (2.8)
2 m
H∂x − − ∂z
(cid:12)
(cid:12)z=zm
(cid:12)
(cid:12)
(cid:12)
3. Delta-correlated noise in measurements
Let usconsidernoiseinthemeasurementdata;
f(x,y) = f (x,y)+aξ(x,y), (3.1)
0
where f(x,y)is themeasured field, f (x,y)is the actual field withoutdistortionby noisein mea-
0
surements, and aξ(x,y) is the measurement noise with amplitude a and ξ(x,y) normalized as
specified below in the text. We assume that the noise at the neighbouring measurement points
to be uncorrelated (bias in measurements can be removed by calibration of the measuring sen-
sor), that is ξ(x,y) can be assumed to be δ-correlated. We choose the following normalization:
ξ(x,y)ξ(x′,y′) = 4δ(x x′)δ(y y′). Hereandhereafter, ... meansaveragingoverthenoise
h i − − h i
4
D.S. Goldobin, A.V.Pimenova,etal. MFL: Noise-Produced Patterns
realizations. The amplitude a of normalized noise is determined by the variance of the measure-
menterror ε(meas) = f(meas) f ;
f − 0
4
(ε(meas))2 = a2 ξ2 = a2 , (3.2)
h f i h i h h
x y
where h and h are the distances between the measurement points in the x- and y-directions,
x y
respectively.
3.1. Measuring f(x,y) = ∂H /∂x
z
Letusfirstconsiderthecaseofmeasuringf(x,y) = ∂H(ext)/∂x, whichisroutinelyperformedby
z
devicesin commonusage[7]. Accordingto Eq.(2.8), thereconstructed profile
1 x x′
σ(x,y) = dx′ ˆ dx′′f(x′′,y)
L
H Z Z
1 x x′
= ˆ dx′ dx′′f(x′′,y). (3.3)
L
H Z Z
Here one can change the order of operation of the integration with respect to x and the operator
ˆ, which is differentiation with respect to z. Hence, the purely noise-produced pattern in the
L
reconstructed profileσ(x,y)is
1 x x′
ε (x,y) = ˆ dx′ dx′′ε (x′′,y). (3.4)
σ f
L
H Z Z
In contrastto thenoisein f(x,y),which isδ-correlated, thenoise-producedpatternε (x,y)is
σ
aresultoftwointegrationsandtheactionofnon-localoperator ˆ;therefore,thispatterncanhavea
L
smoothcomponentwithafinitecorrelationlengthwhichwillnotbeeasilyrecognisableagainstthe
background of the actual noise-free profile σ (x,y). In Figure 2b one can see the profile ε (x,y)
0 σ
and its smooth component calculated for f(x,y) = aξ(x,y) plotted in Figure 2a. We need to
evaluatethecharacteristicmagnitudeofthesesmoothedprofiles onthescaleL L .
x y
×
To calculate the characteristic magnitude of ε , one can consider the average value of ε over
σ σ
theareaS = L L ;
x y
×
1
ε (x,y) = ε (x,y)dxdy. (3.5)
σ σ
L L
x y ZZS
As ξ(x,y) is δ-correlated, its integral is a Gaussian random variable by virtue of the central limit
theorem. It is more convenient to evaluate the magnitude of ˆ−1ε (x,y) and then assess how ˆ
σ
L L
influencesthemagnitudeofits(L L )-scalecomponent. Accordingtothecentrallimittheorem,
x y
×
thearea-average
1
ˆ−1ε (x,y) = ˆ−1ε (x,y)dxdy (3.6)
σ σ
L L L L
x y ZZS
5
D.S. Goldobin, A.V.Pimenova,etal. MFL: Noise-Produced Patterns
isaGaussian randomvariablewithvariance
2
2 1
ˆ−1ε = ˆ−1ε (x,y)dxdy
L σ L2 L2 L σ
(cid:28)(cid:16) (cid:17) (cid:29) x y *(cid:18)ZZS (cid:19) +
1 Lx Ly x1 x′1
= dx dy dx′ dx′′ε (x′′,y )
L2 L2 2 1 1 1 1 f 1 1
x yH *Z0 Z0 Z0 Z0
Lx Ly x2 x′2
dx dy dx′ dx′′ε (x′′,y )
× 2 2 2 2 f 2 2
Z0 Z0 Z0 Z0 +
1 Lx Ly x1 x′1
= dx dy dx′ dx′′
L2 L2 2 1 1 1 1
x yH Z0 Z0 Z0 Z0
Lx Ly x2 x′2
dx dy dx′ dx′′ ε (x′′,y )ε (x′′,y )
× 2 2 2 2h f 1 1 f 2 2 i
Z0 Z0 Z0 Z0
4a2 Lx Ly x1 x′1
= dx dy dx′ dx′′
L2 L2 2 1 1 1 1
x yH Z0 Z0 Z0 Z0
Lx Ly x2 x′2
dx dy dx′ dx′′δ(x′′ x′′)δ(y y )
× 2 2 2 2 1 − 2 1 − 2
Z0 Z0 Z0 Z0
4a2 Lx x1 x′1 Lx x2 x′2
= dx dx′ dx′′ dx dx′ dx′′L δ(x′′ x′′)
L2 L2 2 1 1 1 2 2 2 y 1 − 2
x yH Z0 Z0 Z0 Z0 Z0 Z0
4a2 L L5
= y x . (3.7)
L2 L2 2 20
x yH
According to Eq.(2.6), the action of the operator ˆ in Fourier space is multiplication by the
L
wavenumber k = (k2 + k2)1/2. Hence, for the mode with scale L L —i.e., k = π/L and
x y x × y x x
k = π/L —action of ˆis equivalent to multiplicationby the factor π(L−2 +L−2)1/2. Let a be
y y L x y σ
the characteristic magnitude of the noise-produced profile on the scale L L . Then Eq.(3.7)
x y
×
yieldstherepresentation
π a 1 L
a(∂Hz/∂x) = (ε )2 x L2 +L2
σ σ ≈ √5 √S L x y
y
q H q
(cid:10) (cid:11)
π (ε(meas))2 h h
= h f i x y 1 Lx L2 +L2. (3.8)
q 2√5 √S L x y
y
H q
Hereweexplicitlydistinguishthefactor1/√S whichfeaturesthe“reference”convergencelawfor
the average valueof a noisy variable with respect to the length scale L , L . One can observe not
x y
simply a slower convergence of the amplitude of the noise-produced pattern but its independence
on the scale length for L L . The error is the same on all spatial scales, suggesting that
x y
∼
measurement(∂H /∂x) isequallyapplicablefor detectionofdefects ondifferent spatialscales.
z
6
D.S. Goldobin, A.V.Pimenova,etal. MFL: Noise-Produced Patterns
(a)
(b)
(c)
(d)
(e)
Figure2: Samplerealizationofnoise(a/ )ξ(x,y)(a),andcorrespondingreconstructedprofilesσ
H
(leftcolumn),andprofilesσwithsmall-scaleoscillationsfiltered-out(rightcolumn). (b):f(x,y) =
(∂H /∂x), (c):f(x,y) = H ,(d): f(x,y) = H ,(e): f(x,y) = H . Thesmoothedprofiles area
z z x y
consequenceofnoiseand shouldnot bemisinterpretedas real defects.
7
D.S. Goldobin, A.V.Pimenova,etal. MFL: Noise-Produced Patterns
3.2. Measuring f(x,y) = H , H or H
z x y
Alternatively to the previous case, some devices measure H = ∂ϕ/∂z, H = ∂ϕ/∂x and/or
z x
− −
H = ∂ϕ/∂y. In Figure 2 one can see sample profiles ε (x,y) for the cases of measuring
y σ
−
differentcomponentsofH~. SimilarlytoEq.(3.4)forthecaseoff = ∂H /∂x, Eq.(2.8)yields
z
1 x
ε(Hz)(x,y) = ˆ dx′ε(Hz)(x′,y), (3.9)
σ − L f
H Z
1 x x′
ε(Hx)(x,y) = ˆ2 dx′ dx′′ε(Hx)(x′′,y), (3.10)
σ − L f
H Z Z
1 x y
ε(Hy)(x,y) = ˆ2 dx′ dy′ε(Hy)(x′,y′). (3.11)
σ − L f
H Z Z
Area-averages of ε(Hz), ε(Hx) and ε(Hy) are Gaussian random variables, thevariances ofwhich are
σ σ σ
related tothefollowingvariances:
2 1 Lx Ly x1
ˆ−1ε(Hz) = dx dy dx′
L σ L2L2 2 1 1 1
(cid:28)(cid:16) (cid:17) (cid:29) x yH Z0 Z0 Z0
Lx Ly x2 4a2 L L3
dx dy dx′ ε(Hz)(x′,y )ε(Hz)(x′,y ) = y x , (3.12)
× 2 2 2 f 1 1 f 2 2 L2L2 2 3
Z0 Z0 Z0 D E x yH
2 1 Lx Ly x1 x′1
ˆ−2ε(Hx) = dx dy dx′ dx′′
L σ L2 L2 2 1 1 1 1
(cid:28)(cid:16) (cid:17) (cid:29) x yH Z0 Z0 Z0 Z0
Lx Ly x2 x′2 4a2 L L5
dx dy dx′ dx′′ ε(Hx)(x′′,y )ε(Hx)(x′′,y ) = y x , (3.13)
× 2 2 2 2 f 1 1 f 2 2 L2 L2 2 20
Z0 Z0 Z0 Z0 D E x yH
2 1 Lx Ly x1 y1
ˆ−2ε(Hy) = dx dy dx′ dy′
L σ L2 L2 2 1 1 1 1
(cid:28)(cid:16) (cid:17) (cid:29) x yH Z0 Z0 Z0 Z0
Lx Ly x2 y2 4a2 L3L3
dx dy dx′ dy′ ε(Hy)(x′,y′)ε(Hy)(x′,y′) = y x . (3.14)
× 2 2 2 2 f 1 1 f 2 2 L2 L2 2 9
Z0 Z0 Z0 Z0 D E x yH
Finally,thecharacteristicmagnitudesofpatternson thescaleL L are
x y
×
a(Hz) π h(ε(fmeas))2ihxhy 1 Lx L2x +L2y , (3.15)
σ ≈ q √3 √S L L
y p x
H
a(Hx) π h(ε(fmeas))2ihxhy 1 LxL2x +L2y , (3.16)
σ ≈ q 2√5 √S L L2
y y
H
π (ε(meas))2 h h
a(Hy) h f i x y 1 Lx + Ly . (3.17)
σ ≈ q 3 √S L L
H (cid:18) y x(cid:19)
8
D.S. Goldobin, A.V.Pimenova,etal. MFL: Noise-Produced Patterns
Table1: Dependenceoftheform-factorongeometryofdefects
γ γ γ γ
(∂Hz/∂x) (Hz) (Hx) (Hy)
π 2 π 2π
L L π
x y
∼ √10 3 √5 3
r
1/2
π L π π L πL
x x y
L L
x y
≪ 2√5 L √3 2√5L 3 L
(cid:18) y(cid:19) y x
π L 3/2 π L π L3 2πL
L L x x x x
x ≫ y 2√5 L √3L 2√5L3 3 L
(cid:18) y(cid:19) y y y
It is noteworthy that, in contrast to the case of f = ∂H /∂x, Eqs.(3.15)–(3.17) demonstrate the
z
decay ofthenoise-producedpatternson largescaleaccording tothe“reference” law 1/√S.
∝
4. Discussion: Scaling of noise-produced patterns, form-factor,
and optimization of measurement techniques
In this section we compare 4 possible cases of f = (∂H /∂x), H , H and H , and discuss the
z z x y
roleoftheshapeofdefects fortheirresolution.
Let us consider the discrepancy between the scaling laws of amplitude of the noise-produced
patterns a(∂Hz/∂x) and a(Hj): for L L L, amplitude a(∂Hz/∂x) L0 is independent of L,
σ σ x y σ
∼ ∼ ∝
while a(Hj) L−1. For the former case, the profile reconstruction technique is equally accurate
σ
∝
for recognition of both small- and large-scale defects as the noise-produced patterns are of the
same magnitude on all scales. One can rely on this technique, given the noise in the input signal
(measurements) can be kept small enough. For the case of f = H , the noise-produced patterns
j
decay for longer scales and, therefore, the reconstruction procedure is more accurate. Even for
strongnoisetherecognitionofsufficientlybroaddefectscanbeaccurate. However,simultaneously
with good resolution of large-scale defects, reliable identification and characterization of narrow
defects become problematic or even impossible (it is impossible as well within the framework of
the virgin equation system, without large-scale approximation). Comparing Eqs.(3.8) and (3.15),
one can find the “reference” scale L ; measuring f = (∂H /∂x) is preferable for defects small
x∗ z
compared to this scale, whilefor a larger scale of defects the reconstructionis moreaccurate with
f = H ;
z
2√5 (ε(meas))2
L = h f=Hz i . (4.1)
x∗ √3 v (ε(meas) )2
u
uh f=∂Hz/∂x i
t
Accuracy of the reconstruction procedure is influenced not only by defect scale but also by its
9
D.S. Goldobin, A.V.Pimenova,etal. MFL: Noise-Produced Patterns
shape. Onecan rewriteEqs.(3.8)and (3.15)–(3.17)as
(ε(meas))2 h h
a(∂Hz/∂x) = γ h f i x y , (4.2)
σ (∂Hz/∂x)q
H
(ε(meas))2 h h
a(Hj) = γ h f i x y , (4.3)
σ (Hj)q √S
H
withform-factorγ
π L L L
γ = x x + y , (4.4)
(∂Hz/∂x) 2√5LysLy Lx
π L L2
γ = x 1+ y , (4.5)
(Hz) √3Lys L2x
π L L2
γ = x x +1 , (4.6)
(Hx) 2√5L L2
y (cid:18) y (cid:19)
π L L
γ = x + y . (4.7)
(Hy) 3 L L
(cid:18) y x(cid:19)
In Table 1 the behaviour of the form-factor for defects with similar dimensions in the x- and y-
directions (L L ), defects stretched along the y-axis (L L ) and along the x-axis (L
x y x y x
∼ ≪ ≫
L ) is summarized. Notice, for L L , the form-factor is diminished for f = (∂H /∂x) and
y y x z
≫
f = H ,and increased forf = H . ForL L , theform-factorisincreased forall cases off.
x y x y
≫
5. Conclusion
We have developed an approach for the analytical calculation of noise-induced patterns produced
when employing a complex non-local transformation for data reconstruction. This approach has
been applied to the magnetic flux leakage (MFL) method for inspection of wellbore casing in-
tegrity; with the MFL method, the ferromagnetic layer thickness profile is reconstructed from
measurements of the magnetic field abovethe layer. Within the framework of large-scale approx-
imation, Eqs.(3.8), (3.15)–(3.17) have been derived for the device designs based on measuring
(∂H /∂x), H , H and/or H , respectively; these equations form one of the principle results of
z z x y
our work. In particular, thecase of (∂H /∂x) has been found to beequally effectivefor all scales
z
ofdefect, whileuseof H , H and H providemorereliableresolutionofbroaderdefects, butare
z x y
inaccurate for the detection of narrow defects. The procedures of reconstruction of the ferromag-
neticlayerthicknessfromH ,H andH arenotequallysensitivetonoise. Moreimportantly,for
z x y
thesethreecases, thesensitivitydepends differentlyonthedefect shape. Thisdependenceischar-
acterisedbytheform-factor,which,asonecanseefromEqs.(4.5)–(4.7),dependsonthelong-axis
orientationforscar-shapedefects. Thus,thecombinedusageoftheseprocedurescanbebeneficial
fortheaccuracy ofrecognitionofdefects.
10