High emperatures ^ High Pressures, 2001, volume 33, pages 127 ^ 134 15 ECP Proceedings pages 791 ^ 798 DOI:10.1068/htwu269 Sequential method for thermal diffusivity discrimination by infrared thermography: application to the use of an uncooled plane-array camera David Mourand, Jean-Christophe Batsale Laboratoire d'energëtique et Phënome nes de ransfert (LEP), Ecole Nationale Supërieure d'arts et Me tiers (ENSAM), Esplanade des Arts et Mëtiers, 33405 alence Cedex, France; fax: +33 5 56 84 54 36; email: mourand@lept-ensam.u-bordeaux.fr Presented at the 15th European Conference on hermophysical Properties, Wu«rzburg, Germany, 5 ^ 9 September 1999 Abstract. A one-dimensional transient flash method based on an infrared thermography device permits retrieval of the thermal diffusivity map. However, each pixel of the temperature image presents a very noisy temporal signal. Moreover, in the case of an uncooled focal-plane-array camera, each plane-array detector presents independent calibration coefficients and independent offset levels. An identification method is presented which is based on the estimation of thermal diffusivity variations around a nominal value by an asymptotic gradient development. his method has the advantage of being an optimal discrimination method in the linear least-squares sense. Moreover, the diffusivity variation parameter is not correlated with the calibration parameter of each detector. hus, the identification algorithm can be used without previous calibration of the camera. Various theoretical results are presented, illustrating the noisy temporal signal propagation in the parameter estimation, in order to show that this method can be used with a cheap thermal imaging camera. Some experimental results obtained with an uncooled thermal imaging camera, the PalmIR 250 (Raytheon) with a plane array of 320 240 uncooled ferroelectric detectors, are presented. 1 Introduction A transient flash experiment based on infrared thermography allows us to obtain a diffusivity variation map, using the asymptotic gradient method presented by Mourand et al (1998). he aim of this work was to design and build a cheap and real-time nondestructive evaluation device (drying process control, line control, etc) for materials of parallelipedic form and with thickness of about 1cm. o do this, we used an asymptotic gradient method with a cheap infrared camera. he measurement noise can be processed by the use of a large number of measurements (Mourand 1998). We have chosen an uncooled thermal imaging camera with 240 320 ferromagnetic plane-array detectors: the PalmIR 250 of Raytheon (141 Spring Street, Lexington, MA 02421, USA). First, we recall the principle of the identification method based on the estimation of thermal diffusivity variations around a nominal value by an asymptotic gradient expansion. hen, we show that the asymptotic gradient method can be used with an uncooled thermal imaging camera such as the PalmIR 250. he propagation of the measurement noise in the parameter estimation is illustrated with some theoretical results. hen, an experiment with the uncooled thermal imaging camera is presented on polyvinyl chloride samples of different thicknesses generated by machining. 2 Estimation method In the ideal case, that is to say with an insulated sample, at an initial time (t ˆ 0), its relative temperature is assumed to be uniform [(x, t ˆ 0) ˆ 0], and it is submitted to a thermal excitation. If the heat transfer is assumed to be one-dimensional (1D), then the temperature response on the rear face (x ˆ L) to an instantaneous and uniform thermal
128 D Mourand, J-C Batsale 15 ECP Proceedings page 792 pulse on the front face (x ˆ 0), is given by (Parker et al 1961): t ˆ lim 1 2 X1 1 n exp n 2 p 2 at ˆ L 2 lim f a, t, (1) n ˆ 1 where a is the thermal diffusivity, a=l 2 is the characteristic frequency, and f is the reduced temperature response. From equation (1), the estimation of a is the same as the estimation of a=l 2 when L is known. Both parameters (a or a=l 2 ) are considered interchangeably. In the case where the average diffusivity, a 0, of the material is known, and where the aim is only to estimate its local variation, Da ˆ a a 0, the temperature at time t i can be written with a first-order development: t i ˆ lim f a 0, t i Da lim. (2a) qa a 0, t i In matrix notation, we have the vectors and f: ˆ t 1, ::::, t N Š, f ˆ f a 0, t 1, ::::, f a 0, t N Š, qa ˆ, ::::, qa a 0 t 1 qa,, a 0, t N and parameters b 1 ˆ lim and b 2 ˆ Da lim. Relationship (2a) becomes the following: ˆ f b1 b1 ˆ X. (2b) qa b 2 b 2 his result lead to an excellent approximation (better than 1%) in a neighbourhood of 10% around a 0 (see figure 1). he simplifying assumption here is that the measurement noise on each component of the experimental temperature vector, ^, has a constant standard deviation and is not correlated. It means that we assume that the noise relative to each detector on the plane array is stable during time, and does not depend on the neighbourhood, even if the standard deviation of the noise relative to each detector differs from one pixel to another. 40 emperature arbitrary units 35 30 25 20 15 10 5 (a 0 Da) (a 0 ) imaging (a 0 ) imaging (a 0 Da) 0 0 10 20 30 40 50 ime s Figure 1. Comparison of temperature response to a rear-face transient method in the classical case (^^^^) and when the temperature is measured with the thermal imaging PalmIR 250 (ö öö).
hermal diffusivity discrimination by infrared thermography 129 15 ECP Proceedings page 793 In the domain of validity of equation (2a), the optimum estimator of the parameter vector, ( ^b 1 ^b2 ) is then obtained by (see Beck and Arnold 1977): ^b1 ˆ X ^b X 1 X ^. (3) 2 As Da ˆ b 2 =b 1, the variation of diffusivity can be written in vector notation: f f qa ^ qa f f ^ D^a ˆ f ^ f. (4) qa qa qa qa ^ he linearisation given by equation (2b) not only allows the estimation of the parameter Da, but also allows the confidence interval of this estimation to be studied. An intermediate stage is the covariance matrix of the estimation of the vector ( ^b 1 ^b2 ), given as a function of the standard deviation of the temperature measurement noise, s : ^b1 cov ˆ X ^b X 1 s 2. (5) 2 he random variable relative to the estimation error on the parameter Da can be expressed as e Da. his variable is linked with random variables relative to the estimation errors on the parameters b 1 and b 2 expressed by e b1 and e b2. Because Da ˆ b 2 =b 1,an asymptotic expansion gives: e Da ˆ b 2 1 eb1 b 2 b 1 1 e b2 ˆ Y eb1 e b2. (6) he variance characterising the confidence interval of the estimation of Da is then given by Beck and Arnold (1971): ^b1 var e Da ˆYcov Y ^b ˆ sy X 2 X 1 Y. (7) 2 Equations (3) and (5) show that the method does not depend on the length of the vector ^. hus, scalar products ( =qa) ^ and f ^ can be incremented when the camera records images. his sequential method considerably eases the problems of storage and image manipulation. It allows above all real-time signal processing, which is necessary for a nondestructive evaluation device. Moreover, the asymptotic gradient method is optimised (in the least-squares sense) in the validity domain of the linear approximation. herefore, it constitutes a simple treatment of discrimination even out of the validity domain of the model described by equation (2a). It is not limited to the rear-face transient method, without heat losses. We can model more complex responses [measured on the front face, with heat losses, etc; see Mourand (1998)] with the vectors f and =qa. 2.1 Application of the asymptotic gradient method We have chosen a thermal imaging camera: the PalmIR 250 of Raytheon. It uses 240 320 ferromagnetic uncooled plane-array detectors. he camera, which only has one video output, was connected to an acquisition board in order to record and process a blast of 8 bytes of coded images in real time. hus, the data coming from the camera constitute a 240 320 matrix with 256 temperature levels. We should specify that the thermal imaging measurement noise is very important, practically of the same order as the signal amplitude. However, as has been shown
130 D Mourand, J-C Batsale 15 ECP Proceedings page 794 1.0 0.9 Sensitivity arbitrary units 0.8 f 0.7 0.6 0.5 f im 0.4 0.3 0.2 qa 0.1 0.0 0 10 20 30 40 50 ime s Figure 2. Sensitivity curves numerically calculated for the steady-state temperature ( f ), the thermal diffusivity (=qa), and the imaging temperature ( f im ). (Mourand 1998; Mourand et al 1998), the asymptotic gradient method is suitable for processing very noisy thermograms. As we can see in equation (4), this method does not need the absolute temperature in order to determine the thermal diffusivity variations. We just need a measurement proportional to the temperature. hus, it is not necessary to calibrate the different detectors of the camera, or to know the emissivity of the material. he main problem of the PalmIR 250 camera is that it continuously modifies the mean level of the image, in order to optimise the image dynamics. hus, if they had no measurement noise, the rear-face transient pulse responses obtained would be such as those presented in figure 1. herefore, we have numerically studied the influence of the mean level variations of the thermal imaging camera. In fact, only the appearance of the temperature response to the excitation changes during the experiment. hus, it is only necessary to replace the steady-state temperature sensitivity, f, in the estimation process [in equations (2) ^ (5)], by what we call the imaging temperature sensitivity, f im (see figure 2). herefore, we have performed a numerical simulation of the parameter estimation of the thermal diffusivity variation, Da, with the imaging camera temperature response depending on the standard deviation of the measurement temperature. 2.2 heoretical study We can calculate the temperature response to an instantaneous thermal stimulus on the rear face by equation (1). We have chosen for this simulation a thermal diffusivity of 10 6 m 2 s 1, thermal conductivity, l ˆ 1 Wm 1 K 1, and a sample thickness, L ˆ 1 cm. his thermogram is made numerically noisy to simulate an experimental thermogram. he nominal value of thermal diffusivity allowing us to calculate the steady-state temperature sensitivity, X lim, and the thermal diffusivity sensitivity, X a (see figure 2), to make use of the estimation method is chosen to be a 0 ˆ 1:2 10 6 m 2 s 1.Here,we have studied the influence of the measurement noise amplitude on the thermal diffusivity variation estimation by a theoretical simulation. In tables 1 and 2, we give the mean value of the estimated parameters from 500 noisy thermograms with a relative standard deviation numerically calculated from these 500 estimated values, and a relative standard deviation analytically calculated with the following expression given by the linear least-squares theorem: cov ^a ˆs X 2 X 1.
hermal diffusivity discrimination by infrared thermography 131 15 ECP Proceedings page 795 able 1. Simulation results for the rear face, two-parameter estimation ( lim and a). Real value: a ˆ 1:1 10 6 m 2 s 1 ; nominal value: a 0 ˆ 10 6 m 2 s 1. (s = max ) % 0.29 0.58 2.88 5.76 28.76 numerical (s a =a) % 0.065 0.13 0.65 1.29 6.58 analytical (s a =a) % 0.071 0.14 0.72 1.43 7.13 10 6^a m 2 s 1 1.0972 1.0971 1.0973 1.0978 1.0957 able 2. Simulation results for the rear face, two-parameter estimation ( imaging value: a ˆ 1:1 10 6 m 2 s 1 ; nominal value: a 0 ˆ 10 6 m 2 s 1. and a). Real (s = max ) % 0.29 0.58 2.88 5.76 28.76 numerical (s a =a) % 0.09 0.18 0.90 1.80 9.02 analytical (s a =a) % 0.098 0.20 0.98 1.96 9.80 10 6^a m 2 s 1 1.0992 1.0991 1.0995 1.0999 1.101 10 6 a m 2 s 1 1.18 1.14 1.10 1.06 classical PalmIR thermogram real value 1.02 0.29 0.58 2.88 5.76 28.76 (s = max ) % Figure 3. Mean values of the thermal diffusivity estimations with their confidence intervals (vertical bars) depending on the standard deviation of the temperature measurement simulating noise (see tables 1 and 2). emperature arbitrary units 60 50 40 30 20 10 0 10 classical response simulation PalmIR 250 measurement simulation 20 0 10 20 30 40 50 ime s Figure 4. Noisy thermogram with s = max ˆ 28:76% in the case of a classical response to a thermal transient pulse, and in the case where the measurement is made with a thermal imaging camera.
132 D Mourand, J-C Batsale 15 ECP Proceedings page 796 Each value of tables 1 and 2 is averaged on 500 estimates taken from 500 classical thermograms (see figure 2) of 250 noisy points with s = max from 0.29% to 28.76%. he estimation is made with sensitivities X imaging and X a presented in figure 2. In order to compare the results of the tables easily, we have presented them in figure 3. One can easily see that the thermal diffusivity estimation is almost the same in the two cases, a bit less biased but also a bit less accurate for an estimation from a thermal imaging camera measurement. Our method is really suitable to a thermal imaging camera such as the PalmIR 250, in spite of variations in the middle level of detectors, and the very important measurement noise (see the magnitude of noise when s = max ˆ 28:76% infigure4). 3 Experimental results o illustrate the theoretical results, an experiment was performed on a polyvinyl chloride slab. Instead of varying the diffusivity, we varied the thickness. Indeed, in all the previous presentations the variations of the parameter a=l 2 may be analysed instead of a. Figure 5 presents a schema of the experimental device. he experimental results were obtained with the infrared camera type PalmIR 250. he maximum frame rate is 30 images s 1, each image is composed of 320 240 pixels. Each experiment is composed of more than 1000 images recorded at regular time intervals. After the experiment, the data taken from the image sequence are directly recorded on the disk, and then processed (on a PC Pentium II 233). Figure 6 presents an experimental thermogram corresponding to a pixel aimed at by the camera, after a flash. he flash heating device is made with a halogen lamp which irradiates the sample during a short time (here only 0.1s is necessary). Figure 6 presents the image obtained by applying expression (4). One observes different zones corresponding to the different thicknesses of the plate. he minimum diffusivities of the image are the darkest. Figure 7 presents the average profile taken from the image in figure 6. he good agreement between the estimated thickness and the real thickness may be noted. Such average thickness profiles are compared with results obtained under the same conditions with a mechanical scanning camera of type AVIO VS 2000 and previously published (Mourand et al 1998). Some discrepancies (two-dimensional effects not observed with AVIO VS 2000) can be attributed to the over-simplified assumption relative to the nondependence of the detectors on the array. Some studies to take into account such effects are being developed. 100 mm infrared camera 4.95 mm 4.9 mm 4.8 mm 4.6 mm 100 mm 25 mm PVC pulse of radiant energy Figure 5. Schema of the rear-face transient method. he values of thickness L are given.
hermal diffusivity discrimination by infrared thermography 133 15 ECP Proceedings page 797 10 20 6.6 Pixel line position 30 40 50 60 10 3 (a=l 2 ) m 2 s 1 70 80 50 100 150 200 250 Pixel column position 0.5 Figure 6. hermal diffusivity map corresponding to the sample described in figure 5 and obtained with the PalmIR 250. 10 7 Apparent thermal diffusivity m 2 s 1 1.6 1.5 1.4 1.3 estimated thickness with the PalmIR 250 estimated thickness with the AVIO VS 2000 real thickness 4.60 4.80 4.90 4.95 hickness mm 0 50 100 150 200 250 Pixel column position Figure 7. Average thickness profiles obtained with the PalmIR 250 camera taken from figure 6 and comparison of results obtained under the same conditions with a camera of type AVIO VS 2000. 4 Conclusion Our estimation method associated with an uncooled thermal imaging camera like the PalmIR 250 of Raytheon constitutes a low-cost nondestructive evaluation device using infrared thermography. his work shows that the poor performances or the particularities of the infrared camera can be compensated for by adapted data treatment.
134 D Mourand, J-C Batsale 15 ECP Proceedings page 798 References Beck J V, Arnold K J, 1977 Parameter Estimation in Engineering and Science (New York: John Wiley) Mourand D, 1998, ``Contribution a mise au point de me thode de controª le non destructif thermiqueötraitement de signaux fortement bruite s'', the se de l'ensam Centre Bordeaux, France Mourand D, Gounot J, Batsale J-C, 1998 Rev. Sci. Instrum. 69 1437 ^ 1440 Parker W J, Jenkins W, Abott J, 1961 J. Appl. Phys. 32 1679 ^ 1684 ß 2001 a Pion publication printed in Great Britain