NDT&E International 43 (2010) 661–666

Contents lists available at ScienceDirect

NDT&E International

0963-86

doi:10.1

n Corr

E-m

IbarraC@

miguel.

(X. Mald

journal homepage: www.elsevier.com/locate/ndteint

Infrared thermography processing based on higher-order statistics

Francisco J. Madruga a,n, Clemente Ibarra-Castanedo b, Olga M. Conde a,Jose M. Lopez-Higuera a, Xavier Maldague b

a Photonic Engineering Group, University of Cantabria, E.T.S.I.I.T. Av. De Los Castros s/n 39005 Santander, Cantabria, Spainb Computer Vision and Systems Laboratory, Department of Electrical and Computer Engineering, 1065 Av. de la Medecine, Laval University, Quebec City, Canada G1V 0A6

a r t i c l e i n f o

Article history:

Received 7 August 2008

Received in revised form

1 July 2010

Accepted 9 July 2010Available online 24 July 2010

Keywords:

Thermography

TNDT

Statistic method

Skewness

Kurtosis

Higher-order central moment

95/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ndteint.2010.07.002

esponding author.

ail addresses: francisco.madruga@unican.es (F

gel.ulaval.ca (C. Ibarra-Castanedo), olga.con

lopezhiguera@unican.es (J.M. Lopez-Higuera)

ague).

a b s t r a c t

Active thermography has reached a high status as a non-destructive evaluation method due to both

ease and speed of inspection. Nevertheless, automatic processing of an infrared (IR) sequence is

essential in order to reduce human intervention. Unfortunately, this target is difficult to achieve given

the amount of data recorded by the IR camera during a typical inspection process and human

participation is absolutely necessary. In this paper, higher-order statistics (HOS) analysis is employed to

process IR sequences and to compress the most useful information into a unique image for each

inspection. Pulsed infrared thermographic temporal response is well-known with a statistical

behaviour. This statistical behaviour is analyzed and the results of its application to carbon fibres

reinforced plastic (CFRP) samples are reported.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Inspection techniques for the non-destructive testing andevaluation (NDT&E) of materials and manufactured componentshave been used for years. However, these techniques haveexperienced a huge development in last decades due to theirimportance and efficiency as a tool in quality control processes.Thermal inspection is nowadays recognized as one of the mostvaluable techniques for the detection and characterization ofdefects. Particularly, thermography based on infrared radiancesupplies fast, contactless and safe measurements. It transformsthe thermal energy, emitted by the surface of an object in theinfrared band of the electromagnetic spectrum, into a visibleimage.

Active thermography allows the inspection of large structuresby heating (or sometimes cooling) the surface of a sample with ashort or long heat pulse or with a continuous periodic heat source,while an IR camera records the thermal changes in the samplesurface. Raw thermograms from typical active thermography testsare rarely suitable for quantitative analysis as acquired. Proces-sing is a mandatory step prior to defect detection and/orcharacterization. A wide variety of methods from the field of

ll rights reserved.

.J. Madruga),

de@unican.es (O.M. Conde),

, MaldagX@gel.ulaval.ca

machine vision and signal processing have been specificallyadapted for thermal non-destructive testing (TNDT) [1]. Theproblems associated with the application of pulsed IR thermo-graphy in TNDT can be summarized as follows:

1.

The exponential rate of attenuation of defect mark with depth(thermal diffusion) considerably limits the depth and thespatial resolution of TNDT.

2.

A TNDT experiment requires high capacity storage equipmentin order to collect raw data.

3.

Raw thermograms from typical active thermography tests arerarely adequate as acquired for quantitative analysis given thatnoise of many forms contaminates the data and advancedsignal processing techniques are necessary.

However, to develop an automatic procedure that eliminatesor at least reduces the human subjectivity of the process is not aneasy task and not all the available techniques can be used.

A wide variety of techniques can be applied to thermographicpulsed data to analyze the information and to avoid theaforementioned problems. Signal transformation using trans-forms such as Fourier [2], Hough [3] and Radon [4], among others[5] can be used with the purpose of finding an alternative domainwhere data analysis is more straightforward. Other attractivetechniques based on the assumption that temperature profiles fornon-defective pixels should follow the decay curve given by theone-dimensional solution of the Fourier equation have beenproposed in order to increase defect contrast [6] or to reduce the

F.J. Madruga et al. / NDT&E International 43 (2010) 661–666662

amount of stored data. Thermal signal reconstruction (TSR) storesthe n+1 coefficients of an n degree polynomial, which fits thedecay curve of each pixel in a double logarithmic space [7].Similarly, an exponential curve fitted for the surface temperaturedata of each pixel that allows to store only two coefficients in adimensionless heat transfer model has been recently presented[8]. Techniques based on matrix factorization [9] or statisticparameters such as principal component thermography (PCT)[10], and more recently the skewness parameter [11], have beenreported and tested for solving TNDT limitations with success. Onthe other hand, the uses of the Skewness and the kurtosisparameters on thermographic images have been proposed insome studies. For instance, used in biological sciences for theanalysis of breast thermograms with morphological imagesegmentation [12], and for the non-destructive detection andquantification of corrosion damage on stonework corrosionevaluation[13]. Nevertheless, in these two applications a singleimage (thermogram) is of interest contrary to pulsed thermo-graphy where the information about subsurface defects iscontained in the whole thermogram sequence. Analysis of asingle image is rarely appropriate in this case.

The purpose of this paper is to present the performance ofhigher-order statistic (HOS) parameter analysis applied to pulsedthermographic inspection. Third, fourth and fifth order statisticparameters have been evaluated showing high defect contrastlevels relative to unprocessed data. Furthermore, HOS analysiscompresses the information of the whole data sequence into asingle and meaningful image. Application to experimental data isconsidered as well. A CFRP composite sample containing Teflons

inserts was inspected using pulsed thermography and the largeresponse data sequence was compressed and analysed using theproposed statistic parameter method.

2. Statistic parameters applied to pulsed thermography

A typical TNDT based on pulsed thermography consists in an IRcamera that records the transient thermal cooling process on asurface heated typically with one or two heat pulse sources such ashigh power photography flashes (Fig. 1). A data volume of P IRimages (M�N pixel) has to be processed in order to extract therequired information. The analysis of thermal evolution in time isoften simplified to a 1-D solution in TNDT. The temperature valuesobtained from the thermographic sequence for a chosen temporalwindow can be represented in a histogram of temperature values asdepicted in Fig. 2, for three pixels: Pt1 with dash–dot line is theshallowest defect, Pt2 with dashed line is deeper defect and Pt3 withsolid line is point corresponding to a sound area (free of defects).

The temperature response of a thick solid sample to aninstantaneous uniform heating is described by the 1-D Fourier’sdiffusion equation

@2T

@2z¼

1

a@T

@tð1Þ

where a is the thermal diffusivity. The surface temperature for aslab of thickness L is obtained calculating the solution of Eq. (1)with the following boundary conditions:

@T

@z¼ 0, z¼ L, tZ0

�k@T

@z¼QdðtÞ, tZ0

ð2Þ

resulting in

Tð0,tÞ ¼Q

rCL1þ2

X1n ¼ 1

exp�an2p2t

L2

� �" #ð3Þ

where C is the specific heat capacity, r the density of the sampleand Q the input energy flux. We observe that the surfacetemperature scores obtained from Eq. (3) are not a normaldistribution. They can be approximated by a standard Weibulldistribution where the HOS parameters reach significant values.The shape of distribution and the statistic parameter will changein the presence of subsurface defects. We can thus obtain a HOSmapping of the surface temperature evolution that will bemodified by the presence or absence of defects. And is possibleto generate a unique image with these values in order to detectand quantify the subsurface defects of the sample.

The most commonly employed statistic parameters aremeasures of central tendency and variability, with the meanand the variance being the most representative parameters.Theoretically, only the first four statistic parameters have aphysical definition in the mathematical study of distribution.These are the mean, variance, skewness and kurtosis, correspond-ing to the first, second, third and fourth statistical moments,respectively. The mean m (also referred to as the arithmetic mean)is the average score in a distribution

m¼ E X½ � ¼1

P

XP

n ¼ 1

Xn ð4Þ

The variance s2 is the second central moment of a distribution.It is a measure of statistical dispersions about the mean of thedistribution [14]

s2 ¼ E½ðX�E½X�Þ2� ð5Þ

The standardized central moments MI, where the subscript I

indicates the moment order, can be defined as

MI ¼E½ðX�E½X�ÞI�

sIð6Þ

Skewness is the third standardized central moment (I¼3); itrepresents a measure of symmetry, or more precisely, the lack ofsymmetry of a distribution. Kurtosis is the fourth moment (I¼4)and it characterizes the relative flatness of a distribution inrelation to the shape of a normal distribution. Standardizedcentral moments of higher order present large values due to thehigh power terms involved in their calculations and it cannotoften be defined physically. They are associated with the presenceof outliers in the distribution.

In order to evaluate the defect detection capability and toquantify the pixel contrast between defect and sound area,a thermographic sequence has been simulated with theThermocals software. The simulated sample contains two defectsdefined with different depths and the same thermal diffusivity.The histograms of surface temperature temporal evolution ofdefect and sound area pixels are presented in Fig. 2. They arestrongly non-symmetrical and right-skewed, that is their skew-ness values are positive. However, sound area pixels will present ahigher skewness value than defect pixels and the minimumskewness value will be obtained from the shallowest defect.

The kurtosis measurement reflects the degree to which thedistribution is peaked, that is it provides information regardingthe height of a distribution relative to the value of standarddeviations. The histograms of the pixels presented in Fig. 2 arecharacterised by a high degree of peakedness (leptokurtic)especially the sound area pixel, so its kurtosis value will behigher than defect pixels.

The fifth order central moment (FCM) value is not reflected onthe histogram characteristics and it is related to the outliers of thedistribution, which do not exist in simulated thermographicsequence. For example, in real thermographic sequence, the FCMvalue will bring out the fibres of composite specimens as CRPF orglass fibre pieces.

Fig. 1. Setup of data acquisition system in pulsed thermography and an example of three temperature profiles (defects: Pt1 and Pt2; sound area: Pt3).

Fig. 2. Histograms of three points: Pt1(defect depth¼1 mm), Pt2 (defect depth¼1.5 mm), Pt3 (sound area) obtained from thermographic sequence of 1000 images.

F.J. Madruga et al. / NDT&E International 43 (2010) 661–666 663

3. Experimental results and discussion

In order to demonstrate the above, a CFRP specimen(300�300�2 mm3, 10 plies) was tested in reflection mode usingtwo high power photographic flashes (6.4 kJ for 5 ms) as shown inFig. 1. The specimen contains a total of 25Teflons squareinclusions having 5 lateral sizes (3, 5, 7, 10 and 15 mm) and5 depths (0.2, 0.4, 0.6, 0.8 and 1 mm), as depicted in Fig. 3(a).

The inspection parameters were 157 Hz frame rate and 1896frames were recorded after the flash. The analysed sequence wastruncated to 1800 frames of cooling process, for establishing sameinitial conditions in the processing.

Fig. 3(b) shows the raw thermogram at 1.27 s, where most ofthe subsurface defects can be detected. Fig. 3(c–e) shows imagesobtained from the skewness, kurtosis and 5th standardizedcentral moment for each pixel, respectively. They show that

50 100 150 200

20406080

100120140160180200220 27.5

28

28.5

29

29.5

30

30.5

31

50 100 150 200

20406080

100120140160180200220 5.2

5.45.65.866.26.46.66.87

50 100 150 200

20406080

100120140160180200220 40

45

50

55

60

65

50 100 150 200

20406080

100120140160180200220 400

450500550600650700750

Fig. 3. CFRP specimen with Teflon inclusions (a) geometry, (b) raw thermogram at t¼1272 ms obtained with two flash excitation, (c) 3rd order moment (skewness) value

image, (d) 4th order moment (kurtosis) value image and (e) 5th order moment value image.

F.J. Madruga et al. / NDT&E International 43 (2010) 661–666664

23, 23 and 22 over 25 defects have been detected qualitatively inone unique image. Only the deepest and smallest defects were notdetected. The anisotropic nature of CFRP, presenting the highestthermal conductivity in the longitudinal direction of fibre ishighlighted mainly in the 5th central moment image (Fig. 3e)displaying the net structure of carbon fibres.

To facilitate comparison of contrast for each statistic image,the signal-to-noise ratio (SNR) has been calculated. Several SNRdefinitions can be found in the literature. The noise is typicallycharacterised by its standard deviation s. However, the char-acterisation of the signal can differ. For instance, the SNR for each

defect can be calculated by taking the average value of the pixelsover the defects minus the average of the background pixels anddividing by the standard deviation of the background pixels [1,15]

SNR1 ¼ 20log10

absðSareaaverage�Nareaaverage Þ

s

� �ðdBÞ ð7Þ

Therefore, an area of 30�40 pixels (A1) over several defects isdefined as depicted in Fig. 4. The areas over the selected defectshave the same size 7�7 mm2 but they are at different depths. Aninternal area of 7�8 pixels (Sarea) is also defined for containing

F.J. Madruga et al. / NDT&E International 43 (2010) 661–666 665

the defective area. This is the internal area representing the signaland the rest of area A1 (Narea) represents the noise.

However, if the signal is known to lie between two boundaries,sminososmax, then the SNR can be defined as [16]

SNR2 ¼ 20log10maxðA1Þ�minðA1Þ

s

� �ðdBÞ ð8Þ

The results for these two SNR definitions are shown in Table 1.The values for SNR2 are approximately 50% higher than for SNR1,but the trends are the same for both. SNR is degraded by thedefect depth, although for the worse case (defect depth¼1 mm)the SNR value is 3.27 dB, which is enough to automatically detectthe defect. The best SNR values for different depth of defects inthis case correspond to the skewness parameter. CFRP iscomposed of fibre with high thermal conductivity in the

Fig. 4. Skewness value image for CFRP specimen with Teflons i

Table 1Summary of SNR values (dB) for each defect and statistic parameter.

Defects size: 7�7 mm2

Depth: 0.2 mm 0.4 mm 0

SNR1 SNR2 SNR1 SNR2 S

Skewness 20.67 28.42 17.76 26.10 1

Kurtosis 21.60 28.85 16.64 25.09 1

5th order 21.51 28.35 15.61 24.07 1

longitudinal direction contributing to signal degradation in thethermographic sequence. Higher statistic parameters have arelation to thermal conductivity in the longitudinal direction ofCFRP samples; the higher the statistic parameter, the moresensible it becomes. The best (highest) SNR value is processedby the 4th standardized central moment or kurtosis for theshallowest defect. The shallowest defects modify the distributionshape of the surface temperature of the sample, that is, thedistribution is less peaked because higher temperature valuespresent more frequency in the histogram. However, if the defectmaterial has a higher diffusivity value than the host material, thedistribution shape will be more peaked and the kurtosisparameter will present a higher SNR than the skewnessparameter. The distribution shape characterised by kurtosis isless affected by the deeper defect. The skewness of the

nclusions. Window used as signal and background (noise).

.6 mm 0.8 mm 1 mm

NR1 SNR2 SNR1 SNR2 SNR1 SNR2

3.48 20.86 8.62 15.82 5.56 10.88

2.13 19.33 6.58 13.42 4.10 9.86

1.24 18.21 5.43 11.94 3.27 9.52

F.J. Madruga et al. / NDT&E International 43 (2010) 661–666666

distribution is the most sensible parameter but it is lessdependent on the shallowest defects than kurtosis parameter.

4. Conclusions

The HOS parameters (third, fourth and fifth order) applied toTNDT have been shown to be a powerful tool for detectingsubsurface defect. Additionally, they show high compressioncapabilities reducing the whole thermographic sequence to aunique image, which contains complete information of its defects.Application to experimental data in CFRP specimens calculatingthe SNR for several defects provides evidence of the effectivenessof proposed techniques and its viability as automatic detectionprocessing of subsurface defect by pulsed thermography.

Acknowledgments

This work was supported by the Spanish Science andTechnology Minister under project TEC2007-67987-C02-01 andJose Castillejo program, the Chaire de recherche du Canada(MiViM) and the Minist�ere du developpement economique,innovation et exportation du Quebec.

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