Statistical detection of defects in radiographic images in ... - CiteSeerX

Signal Processing 82 (2002) 791 – 801 www.elsevier.com/locate/sigpro

Statistical detection of defects in radiographic images in nondestructive testing I.G. Kazantseva;∗ , I. Lemahieub , G.I. Salova , R. Denysc a Institute

of Computational Mathematics and Mathematical Geophysics (Computing Center), 630090, Novosibirsk, Russia 41, Electronics and Information Systems Department, Ghent University, B-9000, Ghent, Belgium c Sint-Pietersnieuwstraat 41, Department of Mechanical Construction and Production, Ghent University, B-9000, Ghent, Belgium b Sint-Pietersnieuwstraat

Received 11 August 2000; received in revised form 21 December 2001

Abstract In this paper, we investigate applicability of statistical techniques for defect detection in radiographic images of welds. The defect detection procedure consists in a statistical hypothesis testing using several nonparametric tests. A comparison of rules derived for image thresholding for a given level of false alarm is presented. In this work we consider circular defects such as cavities and voids. Numerical experiments with real data are performed. ? 2002 Elsevier Science B.V. All rights reserved. Keywords: Defect detection; Radiographic 5lm; Nondestructive testing; Nonparametric tests; Thresholding

1. Introduction Film radiography is a traditional imaging inspection technique for nondestructive examination of industrial equipment components in order to locate any cavities, inclusions, lack of fusion and so on that may have been formed during the manufacturing or operation process [2]. Radiographic testing with 5lm is an expensive and time-consuming technique (exposure time and development of the 5lm). Some attempts have been made to speed up and automate di:erent stages of the radiographic inspection cycle. Digital radioscopy, using X-ray detectors coupled to an image acquisition and processing system, permits real-time inspection. ∗ Corresponding author. Current address: Medical Image Processing Group, Department of Radiology, University of Pennsylvania, Blockley Hall, Fourth Floor, 423 Guardian Drive, Philadelphia, PA 19104-6021, USA. E-mail address: [email protected] (I.G. Kazantsev).

However, as the resolution of 5lm is higher than that of digital radiographs, 5lms are still considered as a reference for all the imaging systems, especially in cases of detecting very small defects. To process the 5lm by computer, one needs to digitize the 5lm by a scanner. After that, digital image processing methods are used for both techniques to help a human operator in the interpretation of visual data. This makes the inspection system more reliable, but in return requires the development of high-level image processing methods to replace the expert’s knowledge. Sophisticated image analysis of digitized 5lms and digital radiographs is a widely studied research 5eld, with much recent approaches using adaptive thresholding [8,9], a wavelet-based multi-resolution image representation [24], variational methods [14], a model-based statistical segmentation [19], mathematical morphology [10,11,22], pattern recognition and neural techniques [13,23] and data fusion [4,6]. We refer to the works [1,3,8,19] where reviews with

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I.G. Kazantsev et al. / Signal Processing 82 (2002) 791 – 801

a comparative study of the methods can be found. The general failings of the majority of published techniques can be attributed to four areas: (1) unacceptable false alarm rates due to component structure and noise, (2) inability to detect defects of all orientations and types, (3) inability to detect defects across di:erent applications, (4) nonrealistic computation times. For these reasons, e:orts to solve at least part of the problems, are continuing. Matching the aforementioned classi5cation, we restrict ourselves to the objective of this work as follows: (1) given a false alarm level, (2) detect blob-like defects (as cavities or voids) in (3) austenitic welds in (4) realistic computer time. We try to investigate the possibilities of automatic image processing of weld defects with the help of statistical hypothesis testing using nonparametric statistical tests. We consider several tests which theoretically, for a given level of false alarm, provide us with a threshold resulting in a map of possible defects. Practical problems of traditional image enhancement—5eld Kattening and noise reduction are discussed as well as the task of automatic thresholding. The software is tested on images of welds of austenitic pipes acquired by radiography. 2. Acquisition of radiographic images Original X-ray 5lms were of about 6 cm wide and about 335 cm long. They were laid on a 30:5 mm thickness pipe in a circumferential way to embrace the weld area. Image acquisition was carried out by radiography. Exposed and developed X-ray 5lms were digitized by portions of about 10 cm long using the Umax Powerlook 3000 digitizer. Refer to the web site http://www.umax.com for a detailed description of the digitizer. The 5lms were digitized at 600 dpi (about 40 m) resolutions and each pixel has a depth of 16 bits with 14 bits of them as meaningful. This results in large X-ray images that must be processed to 5nd anomalies in welds. The images contain reference marks to identify positions in the welds, identi5cation letters, etc. Since only the items within a weld are of interest for the purposes of image processing, we extracted the weld areas free of letters and reference objects interactively. A typical image is shown in Fig. 1

(top). An overview about the applicability of existing 5lm digitization systems to nondestructive testing can be found in [26]. 3. Image analysis and eld attening A radiograph is a photographic record produced by the passage of X-rays through a steel pipe onto a 5lm. After developing the 5lm, the darker regions represent the more penetrable parts of the object, and the lighter regions, those more opaque [2]. The gray-levels of the digitized 5lms are inverted in a way that darker areas on the digitized version of the 5lm correspond to the darker regions in the original X-ray 5lm. In this case, brighter gray-level pixels store information about more dense (than the base metal) areas (for instance, dense metal inclusions). Porosities are seen as darker blobs with a round shape. Pixels from darker areas of digital images have less values than pixels from brighter areas. Another agreement we use is that the weld seam is located horizontally in our illustrations so that columns are vertical sections of the weld picture (Fig. 1). We call them as pro5les as well. There are two main areas in the weld image: the base (parent) metal area and the weld area. The weld area is more bright than the parent metal area, due to the normal overthickness of the weld. The weld area consists of a middle area and two side areas, clearly distinguishable. X-ray imaging is inherently noisy because of the quantum nature of radiation; there may be only a few photons per pixel per exposure time. In addition there is a noise contribution from the digitizing procedure. The image su:ers severely from both types of noise and a median 5lter is applied for its suppression. Watching the pro5les of the 5ltered image columns we observe that the pro5les roughly resemble a nonsymmetric Gaussian (Fig. 2, top left). Segmentation techniques [1,11,14] that are widely used in extraction of objects from background use, as a rule, an assumption that the image under investigation is a composition of piecewise constant functions. Therefore, we need to Katten the pro5les of the weld images. Methods of 5eld Kattening [2,10] mostly perform a subtraction of a smooth component from the original image. This procedure is nonstable and results in an image with high gradients that needs additional smoothing.


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Fig. 1. Top: Radiographic image of 30:5 mm wall thickness austenitic pipe obtained using the conventional X-ray system. The visualized part of the pipe has horizontal and vertical sizes of about 92 and 15 mm, respectively. The image is a result of median 5ltering. Its size is 1143 × 179 (after zooming the original digital version with factor 2) with pixel resolution 80 m. Middle: The image is a result of subtraction from each pro5le its closest parabola. Bottom: The image is a result of subtraction from each pro5le its smoothed version.

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Fig. 2. Upper left: Plot of w(x)—the 570th pro5le of the weld. Upper right: Plot of p(x)—the parabolic background (dashed) and s(x)—window-smoothed version of the pro5le (solid). Lower left: Plot of the weld’s pro5le after parabolic Kattening. Lower right: Plot of the weld’s pro5le after smoothing-based Kattening.

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A window for smoothing should be of the same (or larger) size as the expected Kaw, in order to avoid redundant borders inside the Kaw area. In [2], the authors write: “: : : the 5eld Kattening is an absolutely crucial phase of computer-based analysis”. In this paper we use two approaches to the problem. As a 5rst rough approximation, we assume that each weld’s pro5le w(x); x ∈ [x0 ; x1 ], where [x0 ; x1 ] is the support of w(x), is nearly a parabola p(x)=ax2 +bx+c with coeQcients which can be found from boundary conditions f(x0 ) = p(x0 ); f(x1 ) = p(x1 ) by least-squares 5tting. Subtracting parabola p(x) from the weld’s pro5le w(x), we obtain the “softly” Kattened version of the original image (Fig. 1, middle). We refer to this type of Kattening as a parabolic 5tting. The second method of Kattening we explore is subtraction from each pro5le w(x) its smoothed version s(x), where smoothing is ful5lled within i=m a window of 2m + 1 pixels: s(x) = 1=(2m + 1) i=−m w(x + i). Fig. 1 (bottom) shows the resulting image with smoothing parameter m = 20. We can see that subtraction from the pro5le its smoothed version gives us more severe Kattening than that of the parabolic background. Fig. 2 shows us results of Kattening of a single pro5le. Local smoothing (averaging by 3–5 pixels window) is additionally applied to the results of Kattening. The radiograph is a kind of shadow picture, its shadow being more determined by geometrical optics principles than by wave scattering in comparison with ultrasonics. The size and form of the shadow, for example cast by voids, cracks or inclusions present in the test object, depend on the distance and angle between the radiation source and the defect and the distance and angle between the defect and the 5lm. In this work, we do not use physical parameters of the radiographic data acquisition such as size and energy level of the source, the 5lm grain size and characteristic curve, the exposure factor and others. However, we exploit a kind of geometrical model in the form of the parabolic weld pro5le. Due to asymmetry in the weld’s pro5les, this approximation looks more realistic than the methodology based on the assumption that the intensity plot of a weld is Gaussian [15]. Generally, the problem of 5eld Kattening is still open. After the Kattening procedure, the next task is a detection of low-contrast spot-like isolated objects on inhomogeneous and noisy background.

4. Detection of an object in a random background We consider the detection problem as a problem of hypothesis testing [16], in the most important practical formulation—the absence of a priori information about brightness distribution in the points of object and background. We assume that an image under investigation is inhomogeneous and anisotropic and all observed variables have continuous probability distribution functions. In case the image does not contain an object, we suggest that the observations are statistically independent. 4.1. General scheme of detection Let us de5ne a shape of an investigated object by the form of the window (square, circular, etc.) which moves along the image. For each possible discrete location, size and orientation of the window we investigate a problem of defect detection by testing the null hypothesis H0 :

absence of a defect

against the alternative H1 :

presence of a defect:

Rejection or acceptance of H0 is based on values of some test statistics that are functions of the observed data. Hypothesis H0 is rejected when values of the statistics achieve a critical level (threshold). Denote Z as the entire number of all tests of hypothesis H0 . Generally, some of the Z decisions can be false. If the threshold is chosen in a way that the probability of rejection of H0 when it is true is , then under H0 , for large Z stochastically, Z decisions can be false approximately. For each current pixel we choose two circular concentric windows W1 and W2 centered at the pixel with radii R1 and R2 , respectively, R1 ¡ R2 (Fig. 3). It is suitable to operate with circular windows and describe their sizes by a single parameter—by radius, although the technique suggested is easily generalized for windows designed in accordance with di:erent shapes of the detected objects. Let W1∗ = W2 − W1 denote the set of all those points W2 which do not belong to W1 . We will associate the domain W1 with a suspect defect area and domain W1∗ will often be called as a surrounding background. The value of the radius R2


detection and localization from limited tomographic data can be found in [5]. The tomographic theory can be applied to defect detection in a single radiograph only in rare cases when a geometry of the specimen and a forward model of image formation are fully determined [12].

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4.2. Sign test One of the simplest nonparametric tests is a sign test [18]. Put M = N and assume that the pixels from the object are stochastically more brighter (larger) than those from the background (one-sided alternative). The Sign Criterion for testing the null hypothesis H0 is based on the following statistic: =

Fig. 3. Scheme of scanning.

is 5xed by the user. Changing values of the radius R1 from Rmin up to R2 − 1 and testing hypothesis H0 , we decide whether to reject or accept the hypothesis and to mark or not the pixel position under investigation as a center of the defect zone with radius R1 . If the number of tested central pixels is denoted as Np , then the entire number of all tests can be expressed as follows: Z = Np × (R2 − Rmin ): Let x1 ; x2 ; : : : ; xN be arbitrary pixel values from W1 and y1 ; y2 ; : : : ; yM be arbitrary pixels from W1∗ . Denote x = {xi }; i = 1; : : : ; N ; y = {yj }; j = 1; : : : ; M . In this work we choose samples located on circles with radii R1 and R2 (Fig. 3). For detecting the object we test the null hypothesis H0 :

values xi and yj are stochastically equal

N

I {xi − yi ¿ 0};

xi are stochastically larger (or less) than yj :

For testing, we need appropriate statistics. We are going to test the hypothesis H0 with the use of statistics which do not depend on the form of the image brightness distribution functions unknown to the observer. Such statistics and the corresponding tests are often called nonparametric. A statistical approach using a parameterized hypothesis testing method for anomaly

(1)

i=1

that is the number of positive signs among x1 − y1 ; : : : ; x n − yn . Here and in what follows, we de5ne I {A} = 1 if event A occurs and I {A} = 0 if it does not. The sign test rejects the null hypothesis H0 when ¿ , where the threshold = () is determined by the acceptable level and equals the smallest integer such that N i 2−N 6 : (2) N i=

4.3. Rosenbaum test As a second example, consider the Rosenbaum statistic [20]. Denote A1 = the number of x larger than ymax = max{yj }: (3)

against the two-sided alternative hypothesis H1 :

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Given the integer r, it can be shown (see [21]) that the probability of event {A1 ¿ r} under the null hypothesis H0 has the form Pr(A1 ¿ r|H0 ) =

(N + M − r)!N ! : (M + N )!(N − r)!

(4)

The one-sided Rosenbaum test is as follows. Reject H0 only if A1 ¿ , where the threshold is the smallest

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integer such that (N + M − )!N ! 6 : (M + N )!(N − )!

is to reject H0 if (5)

A2 = the number of y larger than xmax = max{xi }: (6) The two-sided Rosenbaum test for H0 against the two-sided alternative H1 is as follows. Reject H0 only if T1 ≡ A1 − A2 ¿ C1 or T1 6 − C2 . Critical levels C1 and C2 depend upon the entire number Z of tests. In case of a single test, the probability of false alarm Pr(A1 ¿ C1 | H0 )+Pr(A2 ¿ C2 | H0 ) is expressed as a ratio (N + M − C2 )!M ! (N + M − C1 )!N ! + ; = (M + N )!(N − C1 )! (M + N )!(M − C2 )! (7) where C1 and C2 should be chosen so that the fractions in (7) are approximately the same and they are the smallest integers for which the relationship (8)

is hold. If M = N , then C1 = C2 and calculations are essentially simpler. In this case, given the probability of false alarm , we reject the hypothesis H0 when

where the threshold C1 is chosen as the smallest integer such that (2N − C1 )!N ! 6 : (2N )!(N − C1 )!

4.5. Wilcoxon–Mann–Whitney test The following nonparametric test for H0 was originally proposed by Wilcoxon [25], Mann and Whitney [17]. The Mann–Whitney test is based on the statistics U+ = U

−

N M

I {xi − yj ¿ 0};

i=1 j=1 N M

=

(11) I {xi − yj ¡ 0};

i=1 j=1

(U + = number of pairs {xi ; yj } with xi ¿ yj ). Values U + and U − are integers from 0 up to NM and distributed symmetrically near the point NM=2 when H0 is true. The Wilcoxon test is based on the statistics W = U + + 12 N (N + 1):

(12)

Let the integer u ¿ 0 be 5xed, then the probability P(U + = u|H0 ) (P(U − = u|H0 )) can be represented in the form M !N ! P(U + = u|H0 ) = 1; (13) (M + N )! p(M )

|T1 | ¿ C1 ;

2

(10)

Critical levels C() and C(=2) can be taken from tables [7].

Let

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T2 = |A1 + B1 − A2 − B2 | ¿ C(=2):

(9)

4.4. Haga test The Rosenbaum test can be modi5ed. Let B1 = the number of y smaller than xmin ; B2 = the number of x smaller than ymin : Then the one-sided Haga test for H0 against the one-sided alternative H1 is as follows. Reject H0 only if A1 + B1 ¿ C(). The two-sided Haga test for H0

where the summation is performed over all partitions p(M ) = {m0 ; m1 ; : : : ; mN } of the number M into N + 1 integer nonnegative addendas M = m 0 + m1 + · · · + mN satisfying the equality N

imi = u:

i=1

The two-sided Mann–Whitney test for H0 against the two-sided alternative H1 is as follows. Reject H0 only if U + ¿ CU (=2) or U − ¿ CU (=2). The one-sided Mann–Whitney test is to reject H0 if U + ¿ CU (). The one-sided Wilcoxon test is to reject H0 if W ¿ CW (). Critical values can be taken from tables in [17,18,25]. In this work we use the


two-sided Mann–Whitney test U = max{U + ; U − }:

(14)

5. Experimental results obtained on radiographs Defect zones are revealed with actual measurements on fracture face using destructive testing. There are three defect clusters in the weld (Fig. 4): defect A—lack of fusion, defect B—Cu-inclusions and wetting, defect C—porosity and slag inclusions. Radiographic visual inspection overestimates longitudinal size of zone A in 6 mm (zone A , Fig. 5), does not reveal area B at all and underestimates longitudinal size of the defect zone C in 15 mm (zone C , Fig. 5). Zone B has very low contrast and is dif5cult to detect without the image enhancement procedure. A methodology of the numerical experiment is as follows. We explore three images: original nonKattend version of the weld (Fig. 1, top), the parabolic 5tting— based Kattened image (Fig. 1, middle) and the local means—based Kattened weld (Fig. 1, bottom). We use two ways of thresholding: (1) Blind, or maximal thresholding, when only maximally possible values for test statistics are used. It means that we visualize only the objects that theoretically correspond to a minimal level of false alarm, i.e., extreme suspects on defectness. (2) For a given level of false alarm we compute thresholds for di:erent tests with 5xed values of such processing parameters as M , N , Rmin , R2 and Z. The sign test, Rosenbaum test, Haga test and Mann– Whitney test—, T1 ; T2 and U are used. In case of rejection of H0 in favour of H1 , we visualize a circle (of maximal screen brightness) with radius R1 , (Rmin 6 R1 ¡ R2 ) corresponding to the biggest value

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of the test statistics computed R2 − Rmin times at the current pixel. Peak performance occurs when the inner window W1 maximally matches the object, leaving the bordering window W2 in a background region. Test statistics are computed within the sliding window W2 of radius R2 . At each pixel we store the maximal value of test statistics computed for concentric windows W1 with radius R1 changing from Rmin to R2 − 1. 5.1. Results of detection with maximal threshold (with minimal ) We visualize circles centered in the points at which the test statistics take their maximally possible values, i.e., with the lowest possibility of false alarm of the used criterion 0 6 6 N; 0 6 |T1 | 6 N; 0 6 T2 6 2N; 0 6 U 6 N 2: Results with sign test are shown in Fig. 6. Here and what follows, the results of detection are positioned within illustrations from top to bottom as it is: original image, image Kattened with parabolic 5tting and image Kattened with local smoothing. The experiments with Haga test T2 thresholded with maximal level C = 40 and with Mann–Whitney test U thresholded with maximal level C = 400 result in the images that are very similar to that of Fig. 7, where the Rosenbaum test T1 is presented. It con5rms the theoretical fact about the equivalence of the Rosenbaum, Haga and Mann–Whitney tests when maximal values of test statistics are chosen as a threshold. In this case the sum in Formula 13 is equal to 1 and this formula becomes identical to Formula 4.

Fig. 4. Zones A; B and C are defect areas (projections of the defect areas onto the 5lm) revealed by actual measurements on fracture face (destructive testing).

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Fig. 5. Zones A and C are defect areas revealed by radiographic inspection (non-destructive testing).

Fig. 6. Results of experiments on detection of dark spots with sign test , maximal threshold C = 20 is chosen ( = 10−6 ), M = 20, N = 20, Rmin = 2, R2 = 15.

Fig. 7. Results of experiments with Rosenbaum test T1 thresholded with maximal level C = 20 and parameters M = 20, N = 20, Rmin = 2, R2 = 15.

The experiments show that unlike the visual radiographic inspection (Fig. 5), the test statistics successfully detect several defect objects from the area B. After a number of numerical experiments we conclude that the user’s experience and expectations are very important in choosing the parameters R2 and Rmin for each detection procedure. The entire set of possible defects should be divided into several virtual groups, for instance of small, middle and large objects. Given

a priori knowledge that sizes of the defects from a certain group are limited by s and S, the user chooses the parameter Rmin slightly less than s and the radius R2 slightly larger than S. Detecting small defects encounters the diQculty of di:erentiation between true defect pixels and noisy impulses. If parameter Rmin is about a few pixels, we expect more false small defects. Therefore, investigation of small defects should be performed separately from the large ones.


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Fig. 8. Results of experiments with Mann–Whitney test statistic values—thresholded with level C = 750 and parameters M = N = 30, Rmin = 5; R2 = 10.

Fig. 9. Image with Rosenbaum test T1 values of nonKattened weld’s image calculated for M = N = 20, Rmin = 2; R2 = 15.

5.2. Results with chosen signi:cance level It is revealed that the images received in experiments with the same parameters M = 20, N = 20, Rmin = 2, R2 = 15 and with thresholds C() theoretically derived from a false alarm level = 0:001, are severely overcrowded by redundantly detected circles for all the investigated tests. It turns out that predicted threshold C = 13 is too low for the sign test (as well as threshold C = 10 is too low for the Rosenbaum test and so on) to provide the user with a minimal number of false defects. Besides of noise, a lack of precision in recording the data can be a reason of such relationship between threshold and false objects. Although there is a zero probability of obtaining a sample observation exactly equal to a certain quantity (mean, max, min, etc.) when the population is continuous, nevertheless in practice a sample value equal to the quantity will often occur due to digitization e:ects. The experiments show that empirically chosen thresholds often are very close to the upper limit of the test statistic values. Varying values of radii Rmin and R2 , we can observe di:erent impacts on thresholds. Samples from W1 can be statistically dependent in case of small values of Rmin . The larger Rmin , the

closer the empirical threshold is to its theoretically predicted level. Small values of Rmin can also lead to detecting the noise points as a center of a spot-like area. To illustrate this point we show the result of detection by the Mann–Whitney test U thresholded by C =750 with parameters Rmin =5; R2 =10, M =N =30 in Fig. 8. The method suggested can be improved by taking sample values not only on circles with radius R1 and R2 but also inside regions W1 and W1∗ . Random choice of samples also can improve performance of the algorithm. Choosing a value of the signi5cance level , the user should take into account that this choice relates to the expectation of about Z false defects. For instance, if the user intends to escape noisy false defects (but with a risk to miss a true defect), small values of have to be chosen. In addition, we show a map of Rosenbaum test T1 values for original nonKattened weld’s image in Fig. 9 to illustrate a alternative to the aforementioned ways of thresholding. We see that points suspect on defects have essentially larger values (more bright pixels) and can be segmented interactively. The maps for the two Kattened versions are not shown here because of their similar (but more noisy) character. As it is seen from the experiments,

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the Kattening procedure improves image visually but introduce impulse noise, so that some false peaks can be detected as a signal. For unsupervised detection it is better to use “soft” versions of the weld’s Kattening techniques. We can derive from the experiments that the smaller the expected size of defect, the less Kattening is needed. The problem of regularization of the Kattening procedure has a strong task-dependent character and is considered to be closely related with the size of searched defect. In practice, we would recommend to combine a model-based techniques with averaging within large windows in order to achieve a trade-o: between global and local properties of estimated background. 6. Remarks and conclusions The results presented here demonstrate that the methodology of hypothesis testing based on nonparametric statistics can be applied to problems of defect detection with some hope of success. The bene5ts in detection of sizes of the defects remain somewhat limited, although we do see images with reliably detected defective zones. Even unsupervised thresholding of the map of the investigated test statistics discloses the defects that are not indicated by visual radiographic inspection. The method proposed in this work for detecting defect indications in radiographic images has a distinction from other well-known procedures. One of the most important features is the distribution-free character of the presented statistical approach. Being based on nonparametric test statistics it has some advantages to other techniques in Kexibility and ability of clear interpretation of the results. For example, it is possible to consider the map of test statistics as a probability or evidence of defect presence and therefore it can serve as an input to a data fusion module [6], which often is not possible for existing segmentation methods. The research presented here may impact other areas in signal detection. For example, a similar technique can be applied to feature detection in 3D ultrasonic nondestructive data and to the microcalci5cations detection problem encountered in mammography. The authors consider these topics as subject for future work.

Acknowledgements The authors are grateful to Dr. Antoon Lefevre for supplying us with a description of the defects and to all members of the MEDISIP Group of the ELIS Department of Ghent University for their help and support. The authors wish to express their appreciation to the referees for their helpful suggestions which greatly improved the presentation of the paper. References [1] H. Boerner, H. Strecker, Automated X-Ray inspection of aluminium castings, IEEE Trans. Pattern Anal. Mach. Intell. 10 (1) (January 1988) 79–91. [2] L.E. Bryant, P. McIntire, R.C. McMaster (Eds.), Nondestructive Testing Handbook, 2nd Edition, Radiography and Radiation Testing, Vol. 3, ASNT, 1985, ISBN 0-931403-00-6. [3] C.H. Chen, Pattern recognition in nondestructive evaluation of materials, in: C.H. Chen, L.F. Pau, P.S.P. Wang (Eds.), Handbook of Pattern Recognition and Computer Vision, World Scienti5c Publishing Company, Singapore, 1998, pp. 455–471. [4] A. Dromigny, Y.M. Zhu, Improving the dynamic range of real-time X-ray imaging systems via Bayesian fusion, J. Nondestructive Eval. 16 (3) (1997) 147–160. [5] A.B. Frakt, W.C. Karl, A.S. Willsky, A multiscale hypothesis testing approach to anomaly detection and localization from noisy tomographic data, IEEE Trans. Image Process. 7 (1998) 825–837. [6] X.E. Gros, NDT Data Fusion, Arnold, London, 1997. [7] T. Haga, A two-sample rank test on location, Ann. Inst. Math. Stat. 11 (1959=1960) 211–219. [8] R. Halmshaw, N.J. Ridyard, A review of digital radiological methods, Br. J. NDT 32 (1) (1990) 17–26. [9] A.K. Jain, M.P. Dubuisson, Segmentaion of X-ray and C-scan images of 5ber reinforced composite materials, Pattern Recognition 25 (3) (1992) 257–270. [10] V. Kaftandjian, A. Joly, T. Odievre, M. Courbiere, C. Hantrais, Automatic detection and characterization of aluminium weld defects: comparison between radiography, radioscopy and human interpretation, Proceedings of the Seventh European Conference on Nondestructive Testing, Copenhagen, Denmark, May 1998, pp. 1179 –1186. [11] V. Kaftandjian, Y.M. Zhu, G. Peix, D. Babot, Automatic recognition of defects inside aluminium ingots by X-ray imaging, Insight 38 (9) (September 1996) 618–625. [12] I.G. Kazantsev, I. Lemahieu, Reconstruction of elongated structures using ridge functions and natural pixels, Inverse Problems 16 (2) (April 2000) 505–517. [13] A. Kehoe, G.A. Parker, An intelligent knowledge based approach for the automated radiographic inspection of castings, NDT & E Int. 25 (1) (1992) 23–36.

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