Hindawi Publishing CorporationEURASIP Journal on Advances in Signal ProcessingVolume 2010, Article ID 262869, 12 pagesdoi:10.1155/2010/262869
Review Article
Fluctuation Analyses for Pattern Classification inNondestructive Materials Inspection
A. P. Vieira,1 E. P. de Moura,2 and L. L. Goncalves2
1 Instituto de Fısica Universidade de Sao Paulo, 05508-090 Sao Paulo, SP, Brazil2 Departamento de Engenharia Metalurgica e de Materiais, Universidade Federal do Ceara, 60455-760 Fortaleza, CE, Brazil
Correspondence should be addressed to L. L. Goncalves, [email protected]
We review recent work on the application of fluctuation analyses of time series for pattern classification in nondestructive materialsinspection. These analyses are based on the evaluation of time-series fluctuations across time intervals of increasing size, and wereoriginally introduced in the study of fractals. A number of examples indicate that this approach yields relevant features allowing thesuccessful classification of patterns such as (i) microstructure signatures in cast irons, as probed by backscattered ultrasonic signals;(ii) welding defects in metals, as probed by TOFD ultrasonic signals; (iii) gear faults, based on vibration signals; (iv) weld-transfermodes, as probed by voltage and current time series; (v) microstructural composition in stainless steel, as probed by magneticBarkhausen noise and magnetic flux signals.
1. Introduction
Many nondestructive materials-inspection tools provideinformation about material structure in the form of timeseries. This is true for ultrasonic probes, acoustic emission,magnetic Barkhausen noise, among others. Ideally, signa-tures of material structure are contained in any of thosetime series, and extracting that information is crucial forbuilding a reliable automated classification system, which isas independent as possible from the operator’s expertise.
As in any pattern classification task, finding a set ofrelevant features is a key step. Common in the literatureare attempts to classify patterns from time series by directlyfeeding the time series into neural networks, by measuringstatistical moments, or by employing Fourier or wavelettransforms. These last two approaches are hindered by thepresence of noise, and by the nonstationary character ofmany time series. Sometimes, however, relevant informationis hidden in the “noise” itself, as this can reflect mem-ory effects characteristic of underlying physical processes.Analysis of the statistical properties of the time series canreveal such effects, although global calculations of statisticalmoments miss important local details. Here, we showthat properly defined local fluctuation measures of time
series can yield relevant features for pattern classification.Such fluctuation measures, which are sometimes referredto as “fractal analyses”, were introduced in the study ofmathematical fractals, objects having the property of scaleinvariance. It turns out that they can also be quite usefulin the study of general time series. Early applications [1–3]of fluctuation analyses to defect or microstructure recogni-tion relied on extracting exponents and scaling amplitudesexpected to characterize memory effects on various systems.The approach reviewed here, on the other hand, is based onmore general properties of the fluctuation measures.
The remaining of this paper is organized as follows.In Section 2, we define mathematically the fluctuation (orfractal) analyses used to extract relevant features from thevarious time series. In Section 3, we review the tools usedin the proper pattern-classification step, illustrated by severalapplications in Section 4. We close the paper by presentingour conclusions in Section 5.
2. Fluctuation Analyses
All techniques of fluctuation analysis employed here startby dividing the signal into time intervals containing τ
2 EURASIP Journal on Advances in Signal Processing
points. Each technique then involves the calculation of theaverage of some fluctuation measure Q(τ) over all intervals,for different values of τ, thus gathering local informationacross different time scales. For a signal with genuine fractalfeatures, Q(τ) should scale as a power of τ,
Q(τ) ∼ τη, (1)
at least in an intermediate range of values of τ, correspondingto 1 � τ � L, L being the signal length.
In general, the exponent η is related to the so-calledHurst exponent H of the time series [4, 5]. This exponentis expected to gauge memory effects which somehow reflectthe underlying physical processes influencing the signal. Asimple example is provided by fractional Brownian motion[5–7], in which correlated noise is postulated, leading topersistent or antipersistent memory, and to a standarddeviation σ(t) following:
σ(t) =(
2Kf t)H
, (2)
where t is the time elapsed since the motion started, and Kf
is a generalized diffusion coefficient. A Hurst exponent equalto 1/2 corresponds to regular Brownian motion, while valuesof H different from 1/2 indicate the presence of long-rangememory mechanisms affecting the motion; H > (1/2) (H <1/2) corresponds to persistent (antipersistent) behavior ofthe time series.
Real-world time series, however, originate from a muchmore complex interplay of processes, acting at differentcharacteristic time scales, and which, therefore, competeto induce memory effects whose nature may change as afunction of time. As the series is probed at time intervalsof increasing size, the effective Hurst exponent can vary. Inthat case, any other exponent η related to H would likewisevary. This variation of η with the size τ of the time interval isprecisely what the present approach exploits.
Once the relevant features are obtained from the vari-ation of η with τ, the different patterns can be classifiedwith the help of statistical tools available in the pattern-recognition literature. Here, as discussed in Section 3, wemake use of principal component analysis (PCA) andKarhunen-Loeve transformations. (See, e.g., [8] for a thor-ough account of statistical pattern classification).
2.1. Hurst (R/S) Analysis. The rescaled-range (R/S) analysiswas introduced by Hurst [4] as a tool for evaluating thepersistency or antipersistency of a time series. The methodworks by calculating, inside each time interval, the averageratio of the range (the difference between the maximum andminimum values of the accumulated series) to the standarddeviation. The size of each interval is then varied.
Mathematically, the R/S analysis is defined in the follow-ing way. Given an interval Ik of size τ, we calculate 〈z〉τ,k , theaverage of the series zi inside that interval,
〈z〉τ,k =1τ
∑
i∈Ikzi. (3)
We then define an accumulated deviation from the mean as
Zi,k =i∑
j=�k
(zj − 〈z〉τ,k
), (4)
(�k labelling the left end of Ik), and from this accumulateddeviation we extract the range
Rτ,k = maxi∈Ik
Zi,k −mini∈Ik
Zi,k, (5)
while the standard deviation is calculated from the seriesitself,
Sτ,k =√√√√1τ
∑
i∈Ik
(zi − 〈z〉τ,k
)2. (6)
Finally, we calculate the rescaled range Rτ,k/Sτ,k , and take itsaverage over all nonoverlapping intervals, obtaining
ρ(τ) ≡ 1nτ
∑
k
Rτ,k
Sτ,k, (7)
in which nτ = �L/τ is the (integer) number of nonoverlap-ping intervals of size τ than can be fit onto a time series oflength L.
For a purely stochastic curve, with no underlying trends,the rescaled range should satisfy the scaling form
ρ(τ) ∼ τH , (8)
where H is the Hurst exponent.
2.2. Detrended-Fluctuation Analysis. The detrended-fluctuation analysis (DFA) [9] aims at improving theevaluation of correlations in a time series by eliminatingtrends in the data. In particular, when a global trend issuperimposed on a noisy signal, DFA is expected to providea more precise estimate of the Hurst exponent than R/Sanalysis.
The method consists initially in obtaining a new inte-grated series Zi,
Zi =i∑
j=1
(zj − 〈z〉
), (9)
the average 〈z〉 being taken over all points,
〈z〉 = 1L
L∑
i=1
zi. (10)
After dividing the series into intervals, the points inside agiven interval Ik are fitted by a polynomial curve of degree l.One usually considers l = 1 or l = 2, corresponding to first-and second-order fits. Then, a detrended variation functionΔi,k is obtained by subtracting from the integrated data thelocal trend as given by the fit. Explicitly, we define
EURASIP Journal on Advances in Signal Processing 3
where Zfi,k is the value associated with point i according to
the fit inside Ik. Finally, we calculate the root-mean-squarefluctuation Fτ,k inside an interval as
Fτ,k =√√√√1τ
∑
i∈IkΔ2i,k, (12)
and average over all intervals, obtaining
F(τ) = 1nτ
∑
k
Fτ,k. (13)
For a true fractal curve, F(τ) should behave as
F(τ) ∼ τα, (14)
where α is a scaling exponent. If the trend is correctlyidentified, one should expect α to be a good approximationto the Hurst exponent H of the underlying correlated noise.
2.3. Box-Counting Analysis. This is a well-known method ofestimating the fractal dimension of a point set [7], and itworks by counting the minimum number N(τ) of boxes oflinear dimension τ needed to cover all points in the set. For areal fractal, N(τ) should follow a power law whose exponentis the box-counting dimension DB,
N(τ) ∼ τ−DB . (15)
For stochastic Gaussian processes, the box-counting andthe Hurst exponents are related by
DB = 2−H. (16)
2.4. Minimal-Cover Analysis. This recently introducedmethod [10] relies on the calculation of the minimal areanecessary to cover a given plane curve at a specified scalegiven by the window size τ.
After dividing the series, we can associate with eachinterval Ik a rectangle of height Hk, defined as the differencebetween the maximum and minimum values of the series ziinside the interval,
Hk = maxi0�≤i≤�i0+τ−1
zi − mini0�≤i≤�i0+τ−1
zi, (17)
in which i0 corresponds to the left end of the interval. Theminimal area is then given by
A(τ) = τ∑
k
Hk , (18)
the summation running over all cells.Ideally, in the scaling region, A(τ) should behave as
A(τ) ∼ τ2−Dμ , (19)
where Dμ is the minimal cover dimension, which is equal to1 when the signal presents no fractality. For genuine fractalcurves, it can be shown that, in the limit of infinitely manypoints, the box-counting and minimal-cover dimensionscoincide [10].
2.5. Detrended Cross-Correlation Analysis. This is a recentlyintroduced [11] extension of DFA, based on detrendedcovariance calculations, and is designed to investigate power-law correlations between different simultaneously recordedtime series {xi} and {yi}.
The first step of the method involves building theintegrated time series
Xj =j∑
i=1
xi, Yj =j∑
i=1
yi. (20)
Both series are then divided into N − (τ − 1) overlappingintervals of size τ, and, inside each interval Ik, local trendsXfj,k and Y
fj,k are evaluated by least-square linear fits. The
detrended cross-correlation Cτ,k is defined as the covarianceof the residuals in interval Ik,
Cτ,k = 1τ
∑
j∈Ik
(Xj − X f
j,k
)(Yj − Y f
j,k
), (21)
which is then averaged to yield a detrended cross-correlationfunction
C(τ) = 1N − τ + 1
∑
k
Cτ,k. (22)
3. Pattern-Classification Tools
Having obtained curves of different fluctuation estimatesQ(τ) as functions of the time interval size τ, we make useof standard pattern-recognition tools in order to group thesignals according to relevant classes. The first step towardsclassification is to build feature vectors from one or morefluctuation analyses of a given signal. In the simplest case,a set of d fixed interval sizes {τj} is selected, and the valuesof the corresponding functions Q(τj) at each τj , as calculatedfor the ith signal, define the feature (column) vector xi of thatsignal,
xi =
⎛⎜⎜⎜⎜⎝
Q(τ1)Q(τ2)
...Q(τd)
⎞⎟⎟⎟⎟⎠. (23)
In our studies, unless stated otherwise, we select as intervalsizes the nearest integers obtained from powers of 21/4,starting with τ1 = 4 and ending with τd equal to the lengthof the shortest series available.
It is also possible to concatenate vectors obtained frommore than one fluctuation analysis to obtain feature vectorsof larger dimension. This usually leads to better classifiers.
The following subsections discuss different methodsdesigned to group feature vectors into relevant classes. Allmethods initially select a subset of the available vectorsas a training group in order to build the classifier, whosegeneralizability is then tested with the remaining vectors.This procedure has to be repeated for many distinct choicesof training and testing vectors, as a way to evaluate theaverage efficiency of the classifier. One can then study theresulting confusion matrices, which report the percentage ofvectors of a given class assigned to each of the possible classes.
4 EURASIP Journal on Advances in Signal Processing
3.1. Principal-Component Analysis. Given a set of N featurevectors {xi}, principal-component analysis (PCA) is basedon the projection of those vectors onto the directions definedby the eigenvectors of the covariance matrix
S = 1N
N∑
i=1
(xi −m)(xi −m)T , (24)
in which m is the average vector,
m = 1N
N∑
i=1
xi, (25)
and T denotes the vector transpose. If the eigenvalues of S arearranged in decreasing order, the projections along the firsteigenvector, corresponding to the largest eigenvalue, definethe first principal component, and account for the largestvariation of any linear function of the original variables.In general, the nth principal component is defined by theprojections of the original vectors along the direction ofthe nth eigenvector. Therefore, the principal components areordered in terms of the (decreasing) amount of variation ofthe original data for which they account.
Thus, PCA amounts to a rotation of the coordinatesystem to a new set of orthogonal axes, yielding a new setof uncorrelated variables, and a reduction on the numberof relevant dimensions, if one chooses to ignore principalcomponents whose corresponding eigenvalues lie below acertain limit.
A classifier based on PCA can be built by using the firstfew principal components to define modified vectors, whoseclass averages are determined from the vectors in the traininggroup. Then, a testing vector x is assigned to the class whoseaverage vector lies closer to x within the transformed space.This is known as the nearest-class-mean rule, and would beoptimal if the vectors in different classes followed normaldistributions.
3.2. Karhunen-Loeve Transformation. Although very helpfulin visualizing the clustering of vectors, PCA ignores anyavailable class information. The Karhunen-Loeve (KL) trans-formation, in its general form, although similar in spirit toPCA, does take class information into account. The versionof the transformation employed here [8, 12] relies on thecompression of discriminatory information contained in theclass means.
The KL transformation consists in first projecting thetraining vectors along the eigenvectors of the within-classcovariance matrix SW , defined by
SW = 1N
NC∑
k=1
Nk∑
i=1
yik(xi −mk)(xi −mk)T , (26)
whereNC is the number of different classes,Nk is the numberof vectors in class k, and mk is the average vector of classk. The element yik is equal to one if xi belongs to class k,and zero otherwise. We also rescale the resulting vectors by
a diagonal matrix built from the eigenvalues λj of SW . Inmatrix notation, this operation can be written as
X′ = Λ−1/2UTX, (27)
in which X is the matrix whose columns are the trainingvectors xi, Λ = diag(λ1, λ2, . . .), and U is the matrixwhose columns are the eigenvectors of SW . This choice ofcoordinates makes sure that the transformed within-classcovariance matrix corresponds to the unit matrix. Finally,in order to compress the class information, we project theresulting vectors onto the eigenvectors of the between-classcovariance matrix SB,
SB =NC∑
k=1
Nk
N(mk −m)(mk −m)T , (28)
where m is the overall average vector. The full transformationcan be written as
X′′ = VTΛ−1/2UTX, (29)
V being the matrix whose columns are the eigenvectors of SB(calculated from X′).
With NC possible classes, the fully-transformed vectorshave at most NC − 1 relevant components. We then classify atesting vector xi using the nearest-class-mean rule.
4. Applications
4.1. Cast-Iron Microstructure from Ultrasonic BackscatteredSignals. An early application of the ideas described in thisreview aimed at distinguishing microstructure in graphitecast iron through Hurst and detrended-fluctuation analysesof backscattered ultrasonic signals.
As detailed in [2], backscattered ultrasonic signals werecaptured with a 5 MHz transducer, at a sampling rateof 40 MHz, from samples of vermicular, lamellar, andspheroidal graphite cast iron. Double-logarithmic plots ofthe resulting R/S and DFA calculations, shown in Figure 1,reveal that in all cases two regimes can be identified,reflecting short- and long-time structure of the signals,respectively. From the discussion in Sections 2.1 and 2.2, thisimplies that one can define two sets of exponents, related tothe short- and long-time fractal dimensions of the signals,as estimated from the corresponding values of the Hurstexponent H and the DFA exponent α. See (16).
Lamellar cast iron is readily identified as having smallershort- than long-time fractal dimension, contrary to bothvermicular and spheroidal cast irons. These latter types, inturn, can be identified on the basis of the relative values of Hand α on the different regimes.
As discussed in the following subsections, this fortunateclear distinction on the basis of a very small set of exponentsis not possible in more general applications. Nevertheless, aset of relevant features can still be extracted from fluctuationor fractal analyses by using tools from the pattern recognitionliterature.
EURASIP Journal on Advances in Signal Processing 5
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(c)
Figure 1: Double-logarithmic plots of the curves obtained from Hurst (R/S) and detrended-fluctuation (DF) analyses of backscatteredultrasonic signals propagating in lamellar (a), vermicular (b), and spheroidal (c) cast iron. The values of 〈α〉 and 〈H〉 are obtained byaveraging the slopes of all curves in the corresponding intervals, as shown by the solid lines.
4.2. Welding Defects in Metals from TOFD Ultrasonic Inspec-tion. The TOFD (time-of-flight diffraction) technique aimsat estimating the size of a discontinuity in a material bymeasuring the difference in time between ultrasonic signalsscattering off the opposite tips of the discontinuity. Forwelding-joint inspection, the conventional setup consists ofone emitter and one receiver transducer, aligned on eitherside of the weld bead. (Longitudinal rather than transversewaves are used, for a number of reasons, among which ishigher propagation speed.)
In the case studied in [13], 240 signals of ultrasoundamplitude versus time were captured, with a TOFD setup,from twelve test samples of steel plate AISI 1020, weldedby the shielded process. (Details on materials and methodscan be found in [14].) The signals used in the studywere extracted from sections with no visible defects in thewelding, and from sections exhibiting lack of penetration,
lack of fusion, and porosities. Each of the four classes wasrepresented by 60 signals, each one containing 512 datapoints, with 8-bit resolution. Examples of signals from eachclass are shown in Figure 2.
By combining curves obtained from Hurst, lineardetrended-fluctuation, minimal cover, and box-countinganalyses into single vectors representing each ultrasonic sig-nal, a very efficient classifier is built using features extractedfrom a Karhunen-Loeve transformation and the nearest-class-mean rule. The average confusion matrix obtainedfrom 500 sets of 48 testing vectors is shown in Table 1. A max-imum error of about 27% is obtained, corresponding to themisclassification of porosities. A slightly poorer performanceis obtained by first building feature vectors from each of thefour fluctuation analyses, performing provisional classifica-tions, and then deciding on the final classification by meansof a majority vote (with ties randomly resolved). In this
6 EURASIP Journal on Advances in Signal Processing
50040030020010000
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100
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(a)
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(b)
50040030020010000
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100
150
200
250
300
(c)
50040030020010000
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100
150
200
250
300
(d)
Figure 2: Typical examples of signals obtained from samples with (a) lack-of-fusion defects, (b) lack-of-penetration defects, (c) porosities,and (d) no defects. The horizontal axes correspond to the time direction, in units of the inverse sample rate of the equipment.
Table 1: Average percentage confusion matrix for testing vectorsbuilt from a combination of fluctuation analyses. The possibleclasses are lack of fusion (LF), lack of penetration (LP), porosity(PO), and no defects (ND). Figures in parenthesis indicate thestandard deviations, calculated over 500 sets. (Notice that in [13]these figures were erroneously reported.) The value in row i, columnj indicates the percentage of vectors belonging to class i which wereassociated with class j.
case, as shown in Table 2, the overall error rate is somewhatincreased, although the classification error of samples associ-ated with lack of penetration decreases. In any case, both ofthese approaches yield considerably better performance thanclassifiers based on either correlograms or Fourier spectra ofthe signals, and at a smaller computational cost.
4.3. Gear Faults from Vibration Signals. As detailed in [15],vibration signals were captured by an accelerometer attachedto the upper side of a gearbox containing four gears, one ofwhich was sometimes replaced by a gear either containinga severe scratch over 10 consecutive teeth, or missing onetooth.
Several working conditions were studied, consisting ofdifferent choices of rotation frequency (from 400 rpm to1400 rpm) and to the presence or absence of a fixed externalload. For each working condition, 54 signals containing 2048points were captured (with a sampling rate of 512 Hz), 18signals corresponding to each of the three possible classes ofgear (normal, scratched, or toothless). Linear DFA was thenperformed on the signals, and feature vectors were built fromcurves corresponding to 13 interval sizes τ ranging from 4to 32. Figure 3 shows representative signals obtained underload, at a rotation frequency of 1400 rpm, along with thecorresponding DFA curves.
Principal-component analysis was applied to the result-ing vectors, and a nearest-class-mean classifier was builtfrom the first three principal components of 36 randomlychosen training vectors. With averages taken over 100
choices of training and testing vectors, the classifier wasalways capable of correctly identifying scratched gears, whilethe classification error of testing vectors corresponding tonormal or toothless gears, although unacceptably high fortwo working conditions in the absence of load, lay below 6%for most conditions under load. See Tables 3 and 4. Althougha similar classifier based on Fourier spectra yields superiorperformance, this comes at a much higher computationalcost, since feature vectors now have 1024 points [15].
4.4. Weld-Transfer Mode from Current and Voltage TimeSeries. As detailed in [16], voltage and current data werecaptured during Metal Inert/Active Gas welding of steelworkpieces, with a simultaneous high-speed video footage,allowing identification of the instantaneous metal-transfermode. The sampling rate was 10 kHz, and a collection of ninevoltage and current time series was built, with three seriescorresponding to each of three metal-transfer modes (dip,globular, and spray). The typical duration of each series was4.5 seconds, and examples are shown in Figure 4.
A systematic classification study was performed byfirst dividing each time series into smaller series con-taining L points (L being 512, 1024, 2048, or 4096).These smaller series were then processed with Hurst, lin-ear detrended-fluctuation, and detrended-cross-correlationanalyses. Figure 5 shows example curves. Selecting 80% ofthe obtained feature vectors for training (with averages over100 random choices of training and testing sets), classi-fiers were built from voltage or current signals separatelyprocessed with Hurst or detrended-fluctuation analyses,as well as from voltage and current signals simultane-ously processed with detrended-cross-correlation analysis.A Karhunen-Loeve transformation was finally employedalong with the nearest-class-mean rule. In the poorestperformance, obtained from signals with L = 512 pointssubject to Hurst analysis, the maximum classification errorwas 27% for signals corresponding to spray transfer mode,with 100% correctness achieved for globular transfer mode.
Table 5 shows the average classification error of eachclassifier, for different series length L. The overall perfor-mance of classifiers with L = 1024 and L = 2048 is better
than with the other two lengths. This can be traced to thefact that, as illustrated by Figure 5, distinguishing features(such as average slopes and discontinuities) between curvescorresponding to different transfer modes tend to happenat intermediate time scales. For a given length, detrended-cross-correlation analysis of voltage and current signalsyields an intermediate classification efficiency as comparedto either voltage or current signals analyzed separately.The best classifier is obtained with the Hurst analysis ofsignals containing L = 2048 points, yielding a negligibleclassification error of 0.1%.
In contrast, as shown in the bottom two rows of Table 5,similar classifiers in which feature vectors are defined by thefull Fourier spectra of the various signals yield much largerclassification errors, and at a much higher computationalcost (since the size of feature vectors scales as L, whereas forfluctuation analyses it scales as logL).
4.5. Stainless Steel Microstructure from Magnetic Measure-ments. Barkhausen noise is a magnetic phenomenon pro-duced when a variable magnetic field induces magneticdomain wall movements in ferromagnetic materials. Thesemovements are discrete rather than continuous, and arecaused by defects in the material microstructure, generatingmagnetic pulses that can be measured by a coil placed on thematerial surface.
Magnetic Barkhausen noise (BN) and magnetic flux(MF) measurements were performed on samples of stainless-steel steam-pressure vessels, as detailed in [17]. Thesepresented coarse ferritic-pearlitic phases (named stage “A”)before degradation. Owing to temperature effects, two dif-ferent microstructures were obtained from pearlite that haspartially (stage “BC”) or completely (stage “D”) transformedto spheroidite. Measurements were performed by using asinusoidal magnetic wave of frequency 10 Hz, each signalconsisting of 40 000 points, with a sampling rate of 200 kHz.A total of 144 signals were captured, 40 signals correspondingto stage A, 88 to stage BC, and 16 to stage D. Typical signalsare shown in Figure 6. Notice that, as regards the magneticflux, the difference between signals from the various stagesseems to lie on the intensity of the peaks and troughs,
8 EURASIP Journal on Advances in Signal Processing
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Figure 3: Representative signals and DFA curves obtained from the three types of gear, working under load at a rotation frequency of1400 rpm. In the signal plots, time is measured in units of the inverse sampling rate.
Table 4: The same as in Table 3, but now for gears working under load.
EURASIP Journal on Advances in Signal Processing 9
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Figure 4: Examples of voltage (left) and current (right) time series obtained during the welding process under dip (top), globular (center),and spray (bottom) metal-transfer modes.
10 EURASIP Journal on Advances in Signal Processing
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Figure 5: Examples of curves obtained from Hurst (top), detrended-fluctuation (center), and detrended-cross-correlation (bottom) analysesto current (I) and voltage (V) sample signals obtained under dip (top), globular (center), and spray (bottom) metal-transfer modes.Logarithms are in base 10, and the time window size is measured in tenths of a millisecond.
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Figure 6: Typical signals of (a) magnetic flux and (b) Barkhausen noise obtained from stainless-steel samples at different stages ofmicrostructural degradation. Plots in (b) have been vertically shifted for clarity.
Table 5: Average percentage classification errors of testing voltage (V) and current (I) signals containing L points, produced by classifiersbased on Hurst, detrended-fluctuation (DF), or detrended-cross-correlation (DCC) analyses. Also shown are results for classifiers based onFourier spectra.
L 512 1024 2048 4096
DF, V 3.1± 0.2 2.2± 0.4 3.6± 0.7 5.3± 1.3
Hurst, V 6.5± 0.4 3.1± 0.5 0.1± 0.1 0.7± 0.7
DF, I 2.1± 0.2 0.6± 0.2 0.5± 0.3 1.6± 1.1
Hurst, I 14.5± 0.5 5.4± 0.6 4.0± 0.9 2.7± 1.3
DCC, V + I 3.2± 0.3 1.5± 0.3 2.4± 0.7 7.7± 1.3
Fourier, V 23.6± 0.9 21.8± 0.8 18.7± 1.2 36.7± 2.3
Fourier, I 22.7± 2.5 27.5± 1.9 8.7± 0.9 14.5± 1.9
Table 6: Average percentage of correctly classified testing signals coming from stainless-steel samples in different degradation stages.Classifiers employed detrended-fluctuation (DFA), Hurst (RS), or Fourier spectral (FS) analyses on either Barkhausen noise (BN) ormagnetic flux (MF).
although there is also a fine structure in the curves which isnot visible at the scale of the figure.
Results from classifiers based on detrended-fluctuationand Hurst analyses, with a KL transformation as the finalstep, are shown in Table 6, for both BN and MF signals,with averages over 100 sets of training and testing vectors.Also shown for comparison are results from classifiers based
on Fourier spectral analysis (making use of magnetic-fluxsignals with 512 points extracted from the original signalsby selecting every 78th point, in order to build featurevectors with a manageable number of dimensions). Theperformance of classifiers based on Barkhausen noise ismuch inferior to that of classifiers based on magnetic fluxsignals, which is now discussed.
12 EURASIP Journal on Advances in Signal Processing
The best performance is obtained by the Hurst classifier,with maximum error of about 10%, followed by the DFAclassifier, with a maximum error around 17%. Somewhatsurprisingly, in view of the long-time regularity of themagnetic flux signals evident in Figure 6, the Fourier-spectral classifier shows the worst performance, with anaverage classification error of 25%.
5. Conclusions
We have reviewed and supplemented recent work on applica-tion of fluctuation analysis as a pattern-classification tool innondestructive materials inspection. This approach has beenshown to lead to very efficient classifiers, with a performancecomparable, and usually quite superior, to more traditionalapproaches based, for instance, on Fourier transforms. Thepresent approach also requires less computational effort toachieve a given efficiency, which would be an important issuewhen building automated inspection systems for field work.
An extension of the present approach to defect recogni-tion from radiographic or ultrasonic images can be achievedbased on generalizations of the fluctuation analyses to mea-sure surface roughness [18, 19]. Given any two-dimensionalimage, a corresponding surface can be built by a color-to-height conversion procedure, and mathematical analyses canthen be performed.
Acknowledgments
The authors acknowledge financial support from the Brazil-ian agencies FUNCAP, CNPq, CAPES, FINEP (CT-Petro),and Petrobras (Brazilian oil company).
References
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The 2011 European Signal Processing Conference (EUSIPCO 2011) is thenineteenth in a series of conferences promoted by the European Association forSignal Processing (EURASIP, www.eurasip.org). This year edition will take placein Barcelona, capital city of Catalonia (Spain), and will be jointly organized by theCentre Tecnològic de Telecomunicacions de Catalunya (CTTC) and theUniversitat Politècnica de Catalunya (UPC).EUSIPCO 2011 will focus on key aspects of signal processing theory and
li ti li t d b l A t f b i i ill b b d lit
Organizing Committee
Honorary ChairMiguel A. Lagunas (CTTC)
General ChairAna I. Pérez Neira (UPC)
General Vice ChairCarles Antón Haro (CTTC)
Technical Program ChairXavier Mestre (CTTC)
Technical Program Co Chairsapplications as listed below. Acceptance of submissions will be based on quality,relevance and originality. Accepted papers will be published in the EUSIPCOproceedings and presented during the conference. Paper submissions, proposalsfor tutorials and proposals for special sessions are invited in, but not limited to,the following areas of interest.
Areas of Interest
• Audio and electro acoustics.• Design, implementation, and applications of signal processing systems.
l d l d d
Technical Program Co ChairsJavier Hernando (UPC)Montserrat Pardàs (UPC)
Plenary TalksFerran Marqués (UPC)Yonina Eldar (Technion)
Special SessionsIgnacio Santamaría (Unversidadde Cantabria)Mats Bengtsson (KTH)
FinancesMontserrat Nájar (UPC)• Multimedia signal processing and coding.
• Image and multidimensional signal processing.• Signal detection and estimation.• Sensor array and multi channel signal processing.• Sensor fusion in networked systems.• Signal processing for communications.• Medical imaging and image analysis.• Non stationary, non linear and non Gaussian signal processing.
Submissions
Montserrat Nájar (UPC)
TutorialsDaniel P. Palomar(Hong Kong UST)Beatrice Pesquet Popescu (ENST)
Procedures to submit a paper and proposals for special sessions and tutorials willbe detailed at www.eusipco2011.org. Submitted papers must be camera ready, nomore than 5 pages long, and conforming to the standard specified on theEUSIPCO 2011 web site. First authors who are registered students can participatein the best student paper competition.
Important Deadlines:
P l f i l i 15 D 2010
Industrial Liaison & ExhibitsAngeliki Alexiou(University of Piraeus)Albert Sitjà (CTTC)
International LiaisonJu Liu (Shandong University China)Jinhong Yuan (UNSW Australia)Tamas Sziranyi (SZTAKI Hungary)Rich Stern (CMU USA)Ricardo L. de Queiroz (UNB Brazil)
Webpage: www.eusipco2011.org
Proposals for special sessions 15 Dec 2010Proposals for tutorials 18 Feb 2011Electronic submission of full papers 21 Feb 2011Notification of acceptance 23 May 2011Submission of camera ready papers 6 Jun 2011
