NONDESTRUCTIVE TESTING HANDBOOK -
Electromagnetic Testing
Manual de Ensaio Não Destrutivo - Ensaio Eletromagnético
- Parte 1. Melhoria do Sinal
- Filtro Wavelet Compactação e Filtragem
- Filtro Adaptativo
- Recuperação de Sinal
- Remoção de Ruídos
- Transformação Discreta de Cosseno
- Deconvolução
- Filtragem Wiener
- Deconvolução Cega
- Parte 2. Classificação dos Sinais
- Extração de Característica
- Transformada de Fourier Discreta
- Transformada Discreta de Cosseno
- Transformada Wavelet Discreta
- Analise do Principal Componente
- Coeficientes da Codigicação Preditiva Linear
- Avaliação de Característica
- Algorítimo de Classificação
- Clusterização K-Means
- Algorítimo K-Means
- Redes Neurais
- Parte 3. Caracterização dos Sinais
- Redes de Função de Base Radial
- Sumário
1 MELHORIA DO SINAL
Part 1. Signal Enhancement
Signal and image processing techniques
are valuable for the accurate and
consistent interpretation of signals in
nondestructive testing. Signal processing
performs important functions in data
analysis — ranging from simple noise
filtering for enhancing the ratio of signal
to noise to automated signal classification
for improving discontinuity detectability.
This chapter focuses on some of the more
advanced signal processing techniques.
Classical texts can provide a fundamental
understanding of the subject.!? A
schematic diagram for the overall
approach used in nondestructive test
signal analysis is shown in Fig. 1.
Ficure 1. Overall approach for signal analysis
in nondestructive testing.
Techniques of signal processing can be
broadly classified into procedures for
(1) enhancement, (2) restoration,
(3) classification and (4) characterization.
Signal enhancement techniques are
used to minimize high frequency noise
and artifacts in a signal. These techniques
generally do not require a precise
understanding of the factors that
contribute to the distortion. Techniques
for enhancing the ratio of signal to noise
can range from simple averaging and low
pass filtering! to more sophisticated techniques such as wavelet shrinkage
denoising. In general, the noise contained
in a signal can be attributed to several
sources, including instrumentation, probe
wobble and variations in liftoff and
surface roughness. Signals can be
enhanced using simple standard linear
low pass filters, band stop filters and band
pass filters.? These filters are often
implemented in either hardware or
software and available as special features
in the instrument. However, these filters
are effective only when the signal is
stationary. A signal is considered
stationary when its statistical properties
such as mean or variance do not vary
with time. Nondestructive test signals that
contain time localized discontinuity
indications are, as a rule, nonstationary.
Such problems are addressed using
techniques such as wavelet shrinkage
denoising,* described next.
1.1 FILTRO WAVELET COMPACTAÇÃO E FILTRAGEM
Wavelet Shrinkage
Denoising Filter
Consider a noisy signal y; represented by
the discrete time sequence:
where i = 0, 1, ..., n-1; subscript i is the
time index of the signal; n is the length of
the time sequence; x; is the desired signal;
and z; represents conventional white
noise (indicating that the noise is
uncorrelated!7) with standard deviation
o. The discrete wavelet transform
decomposes a signal y into a weighted
sum of basis functions yy:
wherev andk are integer values. The basis
functions y,,; are derived by using
dilations and translation operations from
a single function y, referred to as the
mother wavelet:
where v andk are the dilation and
translation parameters respectively. (Here,
the terms dilation and translation should
not be confused with the morphological
operations called erosion and dilation.)
Dilation is a scaling operation that
compresses the time axis (that is, the X
axis and not the Y axis) of a signal.
Translation involves shifting a signal in
time. As in the case of fourier series
coefficients,!.? the wavelet transform
coefficients c,, are determined by
projecting the signal onto the wavelet
basis set y,;. The wavelet transform W of
Eq. 1 can be expressed as:
for orthonormal basis functions (such as
fourier bases), where w; represents W(zj).
Solving for x; yields:
Because ow; is in general unknown,
denoising can be accomplished by
removing the noise contribution from
each wavelet coefficient by applying a
data dependent soft threshold. (R03)
Wavelet shrinkage has found increasing
use in nondestructive test applications.
Programs for fast implementation of the
discrete wavelet transform are available
commercially. The result of implementing
the wavelet shrinkage denoising filter on a
typical eddy current signal in Fig. 2a is
shown in Fig. 2b. Although the resulting
signal is relatively noise free, it contains
low frequency trends, which are
undesirable.
Ficure 2. Eddy current signals: (a) raw signal; (b) wavelet
denoised signal.
In many inspection processes, the
statistical properties of noise can vary
spatially, making conventional invariant
position filters not very useful. In such
situations, adaptive filtering procedures
are needed for noise removal.
1.2 FILTRO ADAPTATIVO
Adaptive Filtering
Adaptive filtering approaches make use of
statistical correlation properties of noise
and target signals. An adaptive noise
cancellation scheme is shown in Fig. 3.45
The reference signal d; and the primary
input signal u;, obtained from adjacent
positions of the transducer, are applied to
the filter H(z) and the filter output is
represented by y;. Assuming that the noise
is uncorrelated with the input signal, an
adaptive filter can be designed to cancel
the noise by minimizing the square error.
Ficure 3. Overall schematic diagram of adaptive noise
cancellation.
The mean square error is given
by the
expectation E of the squared error:
If it is assumed that the reference signal
d; contains both signal and noise (that is,
dj = s; + n; where s; and n; are the
discontinuity signal and grain noise
components respectively), then:
If n; is uncorrelated with s; and y; the last
term reduces to zero:
and:
where p,, is the noise power.
When the filter is optimized, the error
minimum p, and the filter output
yi = si;
is completely noise free.
Figure 4a shows a magnetic flux
leakage signal obtained during tests of gas
transmission pipelines. The measurements
are corrupted by a periodic noise caused
by helical variations introduced by the
manufacturing process. The result of
applying adaptive filtering (Fig. 3) is
shown in Fig. 4b. (R05)
Ficure 4. Results obtained from application of adaptive noise cancellation algorithm: (a) raw
magnetic flux leakage data; (b) output after noise cancellation.
1.3 RECUPERAÇÃO DE SINAL
Signal Restoration
Signal restoration procedures are used
when the distortion processes that
introduce specific artifacts in the signal
are known and can be expressed in the
form of a mathematical function. Two
classes of signal restoration procedures are
discussed below: (1) low frequency trends
and (2) the effect of the transducer
footprint. These distortions can be
eliminated by using restoration
procedures such as detrending and
deconvolution, which are described next.
1.4 REMOÇÃO DE RUíDOS
Detrending
Noise and trends are common forms of
distortion that are often present in eddy
current and magnetic flux leakage signals.
Trends are low frequency changes in the
signal levels caused by several factors
including instrument drift and gradual
variations in probe orientation. In the
case of eddy current nondestructive
testing, low frequency trends are
introduced in the signal because of
gradual variations in probe liftoff. The raw
eddy current signal in Fig. 2a shows
typical distortion introduced by slowly
varying trends. A
commonly used
technique for eliminating such artifacts is
based on zero phase high pass filters,
which can be implemented by using the
discrete cosine transform.
1.4.1 TRANSFORMAÇÃO DISCRETA DE COSSENO
Discrete Cosine Transform
The discrete cosine transform is a special
case of the discrete fourier transform!
where the basis functions consist of
cosines instead of complex exponentials.
The discrete cosine transform of an
N-dimensional discrete time signal x is
given by Eq. 10:
where B(k) denotes the N transformed
values and where the coefficients for c(k)
are given by:
and:
for 1 < k < N-1.
To get rid of the low frequency trends,
the low frequency coefficients in the
transformed signal B(k) are set to zero and
the signal is reconstructed. This technique
results in removing the low frequency
trends and the mean without affecting the
phase information, crucial in eddy current
signals. Figure 5 shows the results of
discrete cosine transform based
detrending of the eddy current signal in
Fig. 2a.
Ficure 5. Eddy current signal after detrending of signal in
Fig. 2a.
1.5 DECONVOLUÇÃO
Deconvolution
A second source of distortion is the
blurring of a signal because of the point
spread function or impulse response of
the transducer. Deconvolution techniques
can be used to eliminate transducer
responses from signals and thereby
estimate the true response. One of the
most popular restoration techniques uses
the wiener filter.
1.5.1 FILTRAGEM WIENER
Wiener Filtering
The wiener filter models the measured
signal y(t) as the output of a linear time
invariant system corrupted by noise n(t): (R06)
where h(t) represents the impulse response
of the measurement system or the point
spread function, x(t) represents the true
signal and the symbol* is the
convolution operator. The fourier
transform of Eq. 13 yields:
The estimation of the true function
X(f) from Y(f) is performed using the
wienerfilter A(f), characterized in the
frequency domain by: (R06)
where H*(f) is the complex conjugate of
H(f) and Q is related to the ratio of signal
to noise. Wiener filters® use a constrained
least squares minimization procedure to
estimate x(t) from the measurement y(t).
Although this technique is well
established, it requires knowledge of H(f).
In cases where the distortion kernel H(f)
is not available, blind deconvolution
algorithms can be used as described
below.
1.5.2 DECONVOLUÇÃO CEGA
Blind Deconvolution
Blind deconvolution techniques are
particularly attractive because they do not
require specification of the distortion
kernel. These algorithms iteratively
estimate the kernel (in this case the point
spread function of the probe) from the
available data. Another advantage
associated with blind deconvolution
algorithms is the ease with which
constraints can be added. This property
can be used to constrain the size of the
kernel based on the size of the probe
used. The blind deconvolution technique
based on the richardson-lucy algorithm (R07) uses maximum likelihood principles and
obtains high quality reconstructed images
even in the presence of noise. Consider
the experimental (convolved) eddy
current image data c(x,y) obtained using a
raster scan of a test object. Let f(x,y)
represent the true image that would be
obtained with a point sensor.
where P(ylx) is the conditional probability
of an event y, and the probability P(x) of
an event X. In the context of
deconvolution, P(x) and P(y) are identified
with the unknown f(x,y) to be estimated
and the convolved (or measured) image
c(x,y), respectively. Also, the conditional
probability P(ylx) is identified as the
kernel function or the point spread
function centered at (x,y), that is, g(x,y).
where P(ylx) is the conditional probability
of an event y, and the probability P(x) of
an event X. In the context of
deconvolution, P(x) and P(y) are identified
with the unknown f(x,y) to be estimated
and the convolved (or measured) image
c(x,y), respectively. Also, the conditional
probability P(ylx) is identified as the
kernel function or the point spread
function centered at (x,y), that is, g(x,y).
wherei is the iteration number. All
quantities in Eq. 17 are two-dimensional
and depend on two spatial variables m
and n. Given the point spread function
g(m,n) and an initial guess of the original
image f(m,n), the reconstructed image can
be obtained by iteratively applying Eq. 17
until convergence.
The inverse iterative equation, derived
by reversing the role of the reconstructed
image and the point spread function in
Eq. 17, is given by:
The inverse iterative equation is also
referred to as a richardson-lucy operation.
The blind deconvolution algorithm
consists of a two-step procedure. At the
kth iteration, the point spread function
g*(y,x) is calculated from Eq. 19 by
performing a specific number of
richardson-lucy operations, given the
knowledge of the reconstructed image
f*(m,n) obtained from Eq. 20 after the
(k-1)th iteration:
Here, k denotes the iteration number and
i denotes the number of richardson-lucy
operations during each iteration. The new
estimate of the deconvolved image f*(m,n)
is obtained by performing the same
number of richardson-lucy operations
(Eq. 20), given the point spread function
gk(m,n) obtained from Eq. 19 above:
These two steps are repeated until
convergence is achieved.
Some typical results of applying this
algorithm to data from an eddy current
pancake coil probe are presented in Fig. 6.
Figure 7 shows the results of blind
deconvolution on experimental eddy
current measurements, illustrating how
the process of deconvolution can help in
resolving two closely spaced
discontinuities.
Ficure 6. Images from pancake coil probes: (a) true image;
(b) gaussian point spread function kernel; (c) measured,
convolved image; (d) result of blind deconvolution.
Ficure 7. Eddy current test with two closely
spaced discontinuities: (a) raw image;
(b) result of blind deconvolution algorithm.
2. CLASSIFICAÇÃO DOS SINAIS
Part 2. Signal Classification
Signal classification techniques often rely
on pattern recognition for interpreting
nondestructive test data. Pattern
recognition techniques help to classify
signals into one of a known set of classes.
Such techniques may be used, for
example, to discriminate between
multiple types of discontinuities or
between discontinuities and benign
sources. In the case of steam generator
tube tests, for example, such techniques
could be used to distinguish eddy current
signals from those caused by cracks, tube
supports and antivibration bars. The
parameters of the classifier are generally
determined by using a data bank of
signals from expected discontinuity types.
The collection of signals, referred to as the
training database, is used for training the
classification algorithm. Most
classification techniques use a two-step
procedure.
- In feature extraction, characteristic
features in the signal that carry
discriminatory information are
identified and extracted. These
features serve as a compact signature
of the signal.
- The feature vector is classified by using
a standard pattern classification
technique such as a clustering
algorithm or a neural network.
2.1 EXTRAÇÃO DE CARACTERÍSTICA
Feature Extraction
Feature extraction serves two major
functions, namely data compression and
invariance with respect to parameters
such as frequency and gain. Features are
data attributes that capture similarities in
signals from the same class as well as
dissimilarities in signals from different
classes. In addition to containing
discriminatory information, the feature
vector is typically of lower dimension
than the signal, resulting in data
compression. Features can be physical or
structural (peak value, rise time,
peak-to-peak separation and others) or
transform based (fast fourier transform,
discrete cosine transform and others).
Transform based features are more easily
implemented by using numerical
algorithms. Examples of commonly used
transform based features are discrete
fourier transform coefficients, (R01) discrete
cosine transform coefficients (R08) and scale
based features such as discrete wavelet
transform coefficients, (R09) principal
components (R10) and linear predictive
coding coefficients. (R11) These feature
extraction schemes are described below.
2.1.1 TRANSFORMADA DE FOURIER DISCRETA
Discrete Fourier Transform
One of the earliest and simplest
techniques used for feature extraction is
the discrete fourier transform. The discrete
fourier transform of a signal x(n) can be
expressed as the weighted sum of complex
exponential basis functions. For a series
x(n) with N samples, the discrete fourier
transform is expressed as:
onde k = 0,1, ..., N-1
For smooth signals, the magnitude of
the coefficients can be shown to decay at
the rate of n~ with the result that the
energy can be compacted in very few
discrete fourier transform coefficients.
Magnitudes of the discrete fourier
transform coefficients are the simplest
and most commonly used feature vector
for representing signals for classification.
2.1.2 TRANSFORMADA DISCRETA DE COSSENO
Discrete Cosine Transform
The discrete cosine transform is a special
case of the discrete fourier transform
where the basis functions consist of
cosines instead of complex exponentials.
Repeating Eq. 10, the discrete cosine
transform of an N-dimensional discrete
time signal x is given by:
where B,; denotes the N transformed
values and where the coefficients for c(k)
are given by:
e:
para 1 < k < N-1
This technique results in a feature
vector composed of a smaller number of
coefficients B, that can represent the
signal.
2.1.3 TRANSFORMADA WAVELET DISCRETA
Discrete Wavelet Transform
The discrete wavelet transform (R09) of a signal
x(n) is a joint time scale transform that
provides both time and frequency
localization of a signal. The discrete
wavelet transform can be expressed as the
weighted sum of basis functions:
where realizations of the wavelet basis
function w,,;(n) are derived froma single
function y(n), referred to as the mother
wavelet, by dilations v and translations k
according to:
The discrete wavelet transform
coefficients C,, are determined by
projecting the signal x(n) onto the wavelet
basis set yy,,(7). It is usually implemented
as a series of subband filters. The most
common version is the two-band discrete
wavelet transform, which uses two finite
impulse response filters — a low pass filter
and a high pass filter.
The computation of the discrete
wavelet transform coefficients for a data
vector x of length n (where n indicates the
number of points in the signal) is
indicated in Fig. 8, which presents a fast
implementation of the discrete wavelet
transform usinga filter bank approach.
The output of each filter is downsampled
by a factor of 2 by discarding every other
sample. The output of the high pass filter
represents the discrete wavelet transform
coefficients at the first resolution level.
The output of the low pass filter is then
applied to the same set of filters and
sampled again. The output of the high
pass filter is retained as the discrete
wavelet transform coefficients at the
second resolution level. This process is
repeated until the number of samples is
reduced to 1. The number of possible
resolution levels is given by a:
Legenda:
DWT= discrete wavelet transform
g = high pass function
h = low pass function
k
= iteration number
n = number of points in signal
x = data vector
Ficure 8. Filter bank approach for discrete wavelet transform computation.
(R09)
Because the discontinuity related
information is typically present in the
discontinuity scale subspace, an
appropriate set of coefficients in the
discontinuity subspace can be used as
features.
2.1.4 ANÁLISE DO PRINCIPAL COMPONENTE
Principal Component Analysis
Principal component analysis is a
statistical technique that linearly
transforms a time series sequence into a
substantially smaller set of uncorrelated
variables that contains most of the
information in the original data set.!° The
overall goal of principal component
analysis is to reduce the dimensionality of
the original data set. Principal component
analysis allows the reconstruction of the
original pattern from linear projections
required to have sequentially maximal
variances. The basis vectors of the
representation are constrained to be
mutually orthonormal. If X is ann xn
data matrix of measurement vectors with
mean M, and covariance matrix 2, (where
subscript x represents a datum) an
orthogonal set of eigenvectors may be
found that diagonalizes the covariance
matrix. By arranging the eigenvectors in a
matrix in accordance with decreasing
eigenvalues (largest first), an ordered
orthogonal basis may be created that has
the greatest degree of variability of the
data along the first eigenvector. Retaining
only p largest eigenvalues provides a
feature extraction operator ®, a p x p
matrix of p eigenvectors. Using this
transformation matrix, a data set X may
be transformed to matrix Y:
making Y an orthonormal projection of X
onto the columns of the transformation
matrix. The inverse transformation may
be used to reconstruct the original data
set X by:
where ®! represents the transpose of
matrix ®. The matrix Y represents X in
Ne domain spanned by the vectors @, ..
These columns ofY are referred to as
aprincipal components of the data set X
and are of a lower dimension than the
original data vectors.
2.1.5 COEFICIENTES DA CODIFICAÇÃO PREDITIVA LINEAR
Linear Predictive Coding
Coefficients
Linear predictive modeling (R11) is commonly
used in the processing of speech signals.
Linear predictive coding coefficients are
known to accurately represent speech
signals with a small set of parameters. The
approach can be used also for extracting
features from test signals.
In linear predictive coding analysis, it
is assumed that the present value of the
sample s(n) can be represented as a
weighted sum of the past samples. The
linear predictive coding coefficients are
estimated by minimizing the mean
squared error between the predicted value
and true value. The error e(n) is given by:
where 0; represents the estimates of the
linear predictive coding coefficients.
Setting the partial derivatives of the mean
squared error with respect to oj to zero for
j=1, 2, ..., p gives:
for i= 1, 2, ..., p. Equation 31 can be
rearranged:
for i= 1, 2, ..., p, where ©, is the
autocorrelation function:
The linear predictive coding coefficients
in Eq. 33 can be solved recursively by
using Durbin’s algorithm. (R11)
Nondestructive test signals can be
represented by a small set of linear
predictive coding coefficients, thereby
achieving data reduction and compaction.
The coefficients represent the signal and
serve as a reduced dimensional feature
vector.
2.2 AVALIAÇÃO DE CARACTERÍSTICA
Feature Evaluation
Once the features are computed, a feature
evaluation and selection step may be used
to eliminate redundancy in the
representation and to evaluate the
features on the basis of the discriminatory
information. More importantly, the
selection step offers an opportunity to
choose features invariant either to
changes in test conditions or to some
selected aspect of the test object
properties. Because nondestructive test
signals are acquired under varying test
conditions, the results are sensitive to
instrument drift and to variations in
probe characteristics, scanning speeds,
gain settings, operating frequencies and
test object conductivity and permeability.
A major challenge lies in the development
of signal processing schemes to
compensate the signal for variations in
experimental test parameters. Such
schemes are crucial for rendering the
overall signal classification performance
insensitive to the environment in which
the signal was acquired.
A number of procedures capable of
selecting features on the basis of
discriminatory information in them have
been proposed. The process begins with
the selection of candidate features.!%13
Each candidate is then evaluated and
either accepted or rejected on the basis of
the amount of discriminatory information
contained in it. In the second step, the
goal is to identify features that contain
the greatest amount of discriminatory
information. One popular technique for
feature reduction is called fisher linear
discrimination.'4 The technique uses a
statistical weight function for each feature
to determine the optimum feature set
with the greatest amount of
discriminatory information to be selected
for signal classification. Fisher linear
discrimination quantifies the
discriminatory content of the different
features.
A typical fisher linear discriminant
implementation is carried out by using
scatter matrix analysis. The within-class
and between-class scatter matrices are
computed as follows:
Let 2; be the scatter matrix of the
sample vector x in class C; around their
respective mean m;:
Then Sw can be defined:
and:
where S,, is the within-class scatter matrix
showing the average scatter 2; of the
sample vector x in class C; around their
respective mean m,, where P(C)) is the
prior probability of class C; and where S,
is the between-class scatter matrix,
representing the scatter of the conditional
mean vectors mj; around the overall mean
vector m.
Various measures are available for
quantifying the discriminatory power, the
commonly used one being:
Where W is the optimal discrimination
Here W
projection vector, which can be obtained
by solving the generalized eingenvalue problem:
where A is an eigenvalue.
An example of a feature extraction
procedure that offers dimensionality
reduction as well as invariance properties
involves fourier descriptors.(R16) The
technique has been used for representing
eddy current impedance plane trajectories.
The model not only represents the signal
by a few coefficients, which are invariant
under rotation, translation and scaling of
the eddy current impedance plane
trajectory, but also allows the resynthesis
of the original signal from the stored
coefficients.
2.3 ALGORÍTIMO DE CLASSIFICAÇÃO
Classification Algorithms
The features computed in the previous
step are applied as input to a classification
algorithm for data interpretation. Two of
the most widely used pattern
classification techniques are (1) clustering
algorithms and (2) neural networks. These
techniques are described next.
2.3.1 CLUSTERIZAÇÃO K-MEANS
K Means Clustering
Clustering algorithms treat a feature
vector as a point in the N-dimensional
feature space.!© Feature vectors from a
similar class of signals then form a cluster
in the feature space. The most popular of
the clustering algorithms is the K means
clustering algorithm, which uses an
iterative procedure that classifies each
input signal into one ofK classes.
2.3.2 ALGORÍTIMO K-MEANS
K Means Algorithm
The objective of the K means clustering
algorithm is to partition the feature space
into K mutually exclusive regions. The
partitioning is performed in a way that
minimizes a performance index or cost
function F equal to the sum of the square
of distance between the cluster center and
all points within the cluster.
Let the number of patterns be N..
- Assign anyK (first K, randomly
selected K or user assigned K) patterns
as the K cluster centers z;, where i = 1,
2, .., K.
- Assign each of the remaining N.-K
patterns at the jth iteration to one of
the K clusters whose center is closest
(using the euclidian norm). Equações 39 e 40.
where wis the mth cluster in the jth
iteration and:
para 1 < m, n < K.
3. Update the cluster centers zy, pH, (2;
... K, in a manner that minimizes the
performance index:. Equação 41.
where Fj is the cost function
corresponding to the ith cluster in the
jth iteration and N/ is the number of
patterns in the cth cluster in the jth
iteration. It can be shown that the
centers Zit, i= 1, 2,..., K), which
minimize the above performance
index, are the sample mean of all
points within the cluster:
4. If z/*! = z/ for alli = 1, 2, ..., K, the
algorithm has converged and the
process can be terminated. Otherwise
go to step 2.
The K means algorithm converges if the
classes are linearly separable and the
performance generally is better if the
initial cluster centers are chosen from the
K classes.
2.3.3 REDES NEURAIS
Neural Networks
Neural networks provide an alternate
approach for classification. Interest in this
approach arose froma desire to mimic
biological nervous systems with respect to
architecture as well as information
processing strategies.!? The network
consists of simple processing elements
interconnected by weights. The network is
first trained using an appropriate learning
algorithm for the estimation of
interconnection weights. Once the
network is trained, unknown test signals
can be classified. The class of neural
networks used most often for
classification tasks is the multilayer
perceptron network.
The multilayer perceptron network
(Fig. 9) generally consists of an input layer
of nodes, one or more hidden layers of
nodes and an output layer of nodes.
Nodes within the same layer are not
connected. However, each layer of nodes
is fully interconnected to the nodes in the
next layer. All units within a layer process
data in parallel but the outputs of
different layers are calculated sequentially
starting from the input layer and moving
toward the output layer. Each node
generates an output that is a nonlinear
function of the weighted sum of all its
input signals. This nonlinear function is
primarily used to limit the output of a
node between the values of 0 and 1.
Ficure 9. Architecture of multiplayer perceptron neural network.
The network is trained using the
backward error propagation algorithm!®
where training patterns are sequentially
applied to the network. The overall
algorithm is summarized in Fig. 10. The
algorithm uses a gradient search
technique for minimizing the squared
error between the actual output and the
desired output by adapting the
interconnection weights iteratively. The
algorithm cycles through the training data
repeatedly until the error drops below a
specified threshold value. Neural networks
have been used with success for the
classification of eddy current and
ultrasonic signals. (R19)
Legenda:
d
= output at node j
t = time
w = eighting factor
x = input signal
y = network output at node j
alfaMUDAR = momentum parameter
deltaMUDAR = variable defined by equation
netaMUDAR= learning parameter
tauMUDAR = preset threshold value for error
Ficure 10. Flow chart of backpropagation training algorithm
for multilayer perceptron networks.
3. CARACTERIZAÇÃO DOS SINAIS
Part 3. Signal Characterization
Signal characterization involves a more
complete solution to the inverse problem.
In material science, the inverse problem
involves reasoning from effects (that is,
indications) in order to draw inferences
about test objects. Characterization
techniques use information contained in
the signal to estimate the size, shape and
location of discontinuities. In other
words, characterization procedures
involve the full two-dimensional or
three-dimensional reconstruction of
discontinuity profiles in terms of the
spatial distribution of the material
properties of the test object. In general,
the objective of the signal or
discontinuity characterization procedure
can be described as the identification of a
mapping f such that:
where S$
represents the measurement
vector from a scan in two dimensions M
and Q:
andD represents the discontinuity profile:
The value of dj represents the depth of
the discontinuity at a location (i,j).
Several approaches have been
developed for solving the inverse problem
in nondestructive testing. These solutions
can be categorized as either
phenomenological or
nonphenomenological. Phenomenological
techniques are based on the underlying
physical process of the nondestructive test
technique. Examples of the
phenomenological approach for inversion
are based on analytical solutions of the
underlying governing equation, which is
in general a difficult problem.
Nonphenomenological approaches do not
depend on the physics of the inspection
technique. These approaches model the
nondestructive test system as a black box
or as a linear system and use signal
processing techniques to invert the
measured signal. Typical signal processing
approaches for inversion use neural
networks for solving the discontinuity
characterization problem. An approach
using a radial basis function neural
network for the inversion of magnetic
flux leakage signals is described next. (R20)
3.1 REDES DE FUNÇÃO DE BASE RASIAL
Radial Basis Function
Networks
Radial basis function networks can be
viewed as tools for multivariate
interpolation. (R20) Such networks can be
used for estimating a hypersurface that
provides what can be called the best fit to
the training data. The architecture of the
radial basis function network is in many
respects similar to that of a multilayer
perceptron, defined above. A nonlinear
transformation of the signal is performed
between the input and hidden nodes
followed bya linear transformation
between the hidden and output nodes.
Mathematically, the radial basis function
network computes a multidimensional
function:
where 9; is a set of basis functions, c; are
the basis centers and w; are the weights.
Substituting the values in the training
data {x;, f(x), i = 1, ..., N} in Eq. 46 makes
it possible to derive the matrix equation:
The training of the radial basis function
network consists of estimating the
expansion coefficients, which can be done
by inverting Eq. 47:
Once the weights are estimated by
using the training data, the radial basis
function network can be used to invert a
test signal x according to Eq. 46.
Reconstruction results can be further
improved by using a variation of the
radial basis function network, a
multiresolution approach that uses neural
networks with wavelet basis functions. (R21)
These networks, called wave nets, use a
hierarchical architecture associated with
multiple levels of resolution or scale for
both global and local interpolation as
shown in Fig. 11. The network is trained
hierarchically, first to learn the mapping
between inputs and outputs at the
coarsest resolution and later to augment
the mapping with details at higher
resolutions.
Legenda:
L = scale index (superscript)
x=
input to network
f{(x) = function output
phiMUDAR = scaling function
nãoseiMUDAR = wavelet basis function
Ficure 11. Architecture of wavelet basis
function network.
Mathematically, the approximation of
a scalar function f(é) can be expressed as:
where dj denotes the corresponding
expansion coefficients; fo), the mapping
function to be estimated; sf,
the scaling
coefficients; WjK(x, the wavelet basis
functions generated via translations and
dilations of a mother wavelet (x); and
of, the scaling basis functions at
resolution L.
The first term represents the
approximation of the function at the Lth
resolution and the second term represents
additional information pertaining to the
details at the corresponding resolution.
The accuracy of the discontinuity
reconstruction can then be controlled by
selecting the number of resolution levels
in the network architecture. The training
algorithm for wave nets is similar to that
used for training a radial basis function
network.
Initial results obtained using both
radial basis function networks and
wavelet basis function networks using a
two-dimensional magnetic flux leakage
signal as input are shown in Fig. 12. The
magnetic flux leakage signal is obtained
from a rectangular notch machined on
the pipe wall. The results obtained using
the trained radial basis function network
and the wavelet basis function network
are compared with the true profile. These
results show that such
nonphenomenological techniques for
inversion can be trained to perform well
for measurements as long as the
measurements are similar to those used
for training the network.
   
Ficure 12. Neural network characterization of magnetic flux leakage signal from 75 mm (3 in.) long and 75 mm (3 in.) wide
discontinuity:
(a) magnetic flux leakage signal;
(b) discontinuity profile;
(c) prediction using radial basis function network;
(d) prediction using wavelet basis function network. (R21)
3.2 SUMÁRIO
Summary
Advances in digital processing have made
sophisticated signal and image processing
techniques available for practical
applications in nondestructive testing.
When integrated in software programs for
discontinuity classification, signal
processing algorithms make possible the
automation of diagnostic procedures and
quality assurance protocols.
Autores:
- Lalita Upda, Michigan State University, East Lansing, Michigan
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