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Tomographical Methods of Crack Detection for Components of Nuclear Power Industry U. Ewert, B. Redmer, Y. Onel BAM, Berlin, Germany V. A. Baranov Institute of Introscopy, Tomsk, Russia Contact

A new approach for space filtering of highly degraded images is presented, which is based on the principle of "local symmetry" (i.e. symmetry of image structural elements). The local symmetry (LS) is defined by its symmetry group and statistically estimated in moving window to form brightness characteristics of the resulting image. Some examples of images processing by the proposed method are considered and its prospects of application in NDT are discussed.

Introduction

The development of digital methods in radiology imposes serious demands on X-ray image de- tection means. It is especially true in regard to tomographical techniques, which are assumed to be based on exact measurement of beam-sums. Normally, X-ray images in NDT are degraded by noise of different nature and far of this optimum. The situation can be overcome partly by elaboration of highly specialized measurement subsystems, however this way is economically unthinkable for large-scale use in NDT.

The alternative is to create special image processing techniques, which enable to manage suc- cessfully with erratic image information. An essential incompleteness of this information leads to reconsideration of the image processing concept. First, it is necessary to go away from rep- resentations of all details of the raw image, but to focus on retrieval of some substructures of interest. Second, in these conditions the need of a priory knowledge for a selection, classification and restoration of substructures is of fundamental nature, therefore, an explicit and systematic way has to be developed to introduce it into retrieval procedures.

Principles

With these assumptions, a structure-oriented approach to image processing was proposed [1, 2, 3, 4, 5, 6]. In this concept the recognition of structural elements is performed statistically and usually is brought to the form of a classical space filtering procedure (i.e. in moving window mode).

An initial 2-dimensional image I(x; y) is supposed to be representative in additive form as

I(x; y) = U(x; y) + B(x; y) + n(x; y)

(1)

where B(x, y) describes a background (not obligatory low-frequency one), n(x, y) is a high- frequency noise and U(x; y) corresponds to a "useful signal". The function I(x, y) can be rarely used for direct statistical evaluation of U(x, y) because in general case the terms in the right side of equation (1) are not statistically independent. To release interdependence between U(x, y) and B(x, y) a integro-differential operator D (generally non-linear one) is constructed, so that V (x, y) = D U(x, y) and the evaluation of U(x, y) is based on function V (x, y). In a simple case the operator D acts as differentiation in one or two directions and smoothing in an subset of the moving window.

The images in (1) are represented as vectors of a functional space L. Introduce a projectional operator W, which transforms U into some vector U_w of a space L_w , (L_w Ì L) as

W U(x, y) = Uw(x, y), " U Î L, Uw Î Lw Ì L

(2)

In the framework of this approach image structural elements are to be estimated by their projections (2), more exactly by functions WDI (x; y), inside some "environment". In such a manner the function U(x; y) is evaluated indirectly and represented by its integral characteristics.

The choice of a particular projectional operator (2) is decisive to consider different aspects of the problem and to create specific versions of the image processing algorithm, both in space and frequency domain. In one practically important case the projector (2) is specified as a cut-out operator W = Wr (X, Y, a, b) by a rectangular moving window (where (X, Y ) are coordinates of the window center and (a, b) are its half-width and half-height), so that W_r U(x, y) is equal to U(x, y) inside the window and 0 outside it. Because in this case an "environment" is depending explicitly on the window parameters and an elaboration of straightforward procedures in a traditional form of classical space filtering is possible.

Fundamentally, a priory knowledge for the isolation of the substructure and representation of its elements is specified by a set of constrains H which is regarded as a statistical hypothesis. The remark is necessary; that generally the operator D transforms this set (H (r) HD by action of D). To estimate the hypothesis H_D a statistic S is introduced, which can be represented as a functional of W_r DI (x, y) (W_r DI 2 L_wr Ì L) and is depending of some parameters

S = S (Wr DI; HD; X, Y, a, b)

(3)

Further, the estimation (3) is used in different ways in specific procedures:

1. as a variable in Boolean expressions,
2. in suppressive factor functions or
3. directly as a brightness of resulting image (maybe, with some look-up-table transform).

In a similar manner, a symmetry of image structural elements (or "local symmetry") is considered as a statistical hypothesis and estimated by (3).

Consider a group G of isomorphisms g in L, so that gA(x, y) Î L if g Î G and A(x; y) Î L. According to the basic assumptions of group theory G is a symmetry group for the image U(x, y) if U(x, y) is invariant with respect to group element operation i.e. g U(x, y) = U(x, y), " g Î G. Similar to this the local symmetry (LS) is also described by the group G of isomorphisms in L, however LS is attributed to projections (2), so the classical definition is to be modified as

W (g U(x, y)) = W U(x, y), " g Î G

(4)

In a special case W º e, where e is an identity operator, (4) turns into the classical definition and describes the symmetry of image U(x, y) as a whole. It should be noted that the group G is not intended to describe all the possible elements of LS (in particular, such elements which act in a null-space of projector W) but is considered as a statistical hypothesis to be verified. In this context an incompleteness of symmetry (in other words inaccuracy of (4) is of basic importance and a relevant estimation statistic S for LS can be interpreted as a measure of non- total symmetry (NTS). Because a finite group G generates N images g U(x, y) this measure of NTS can be regarded as multiple similarity measure of these N images (4).

In the case W = W_r the group element operators g_w are explicitly depending on parameters of the moving window and by this means located. The group G and the operator W_r generate a system of microimages U_wr (d x, d y) = g_w (X, Y, a, b) U(x, y)), " _gw Î G_w where dx = x - X,d y = y - Y. A hypothesis H_LS,wr = H_LS,wr (X, Y, a, b) about LS is verified. Because the operator D changes the relevant group of symmetry (G_w (r) G_wD and HLS,wr (r) H_LS,wrD by action of D), indeed, the hypothesis H_LS,wrD concerning identity of N microimages

qwr (d x; d y) = gwd (X, Y, a, b) Wr D I(x; y)); " gwd Î GwD

(5)

should be estimated by (3). In this case (3) is constructed like a multiple measure of similarity of N microimages (5) inside the moving window.

Examples of image processing

To gain a better insight into this approach consider two specific examples of its application.

Fig 1: Comparision of measured and processed projections of a steel reinforced concrete girder in a building. (a) measured projection; (b) processed projection to visualise the complete structure.

1. A symmetry-based filtering algorithm was used for a preprocessing of projections for ferroconcrete walls. To obtain radiographs a Co⁶⁰-source and imaging plates with the biomedical system BAS2000 (Fuji Film) were used. The wall thickness was 400 mm. The geometry of data acquisition was coplanar (like in classical tomosynthesis). The distance from imaging plate to source plane was equal to 1000 mm. The source positions have been arranged along a straight line on the source plane (there were 7 positions within the step equal to 200 mm) perpendicular to steel rods in concrete.

   In this case a statistical hypothesis concerning isotropy of microimages (5) was verified, so that non-isotropism was identified with the presence of structural elements (steel reinforcement in concrete). The structural elements are located, so that the cut-out projector Wr is suitable to describe "environment" as a moving window.

   The verification of isotropy was performed by comparison of beam-sums for the beams, which are incidental to the center of microimage, on the basis of one-factor analysis of variance and a Fisher's F-ratio was used as the estimator (3). In these calculations a group of rotations was approximated by its finite subgroup with only four possible directions. The parameters a and b matched to the typical thickness of steel rodes in reinforcement. Further, some simplified versions of this algorithms have been elaborated as well.

   One can see some results of filtering by this algorithms on Fig. 1. The preprocessed projections were used for 3D tomographical reconstruction of the ferro-concrete wall.

2. The second example is connected with crack detection in nuclear power industry components (water filled pipes) [5, 6]. In this case due to a special orientation (dangerous cracks are oriented along the welding) the problem was reduced to a one-dimensional one. The crack profile is assumed to be a d-like distribution with finite width and intensity which is related to a typical size of a moving interval, non-negative, not equal to zero only in the interval, has a maximum in the middle of the interval and is symmetric.

   Because the group of symmetry generates only two microimages in this instance, it is natural to take a covariance of these microimages as an estimation (3). The advantage of the symmetric covariance is that it is not only a measure of symmetry but a measure of energy for a signal in the interval as well. However, a correlation between the crack profile and background is significant, so to release this interdependence the operator D as differentiation in x-direction was applied to the initial signal. The operator D changes the parity of functions and transforms crack profiles into antisymmetric functions. Therefore, (3) can be represent as a symmetric covariance in form

   S (X) = -cov(DI(- d x + X), DI(+ d d + X))

   (6)

   where I(x) is an initial signal and X is the middle of the interval. The algorithm has been additionally improved by non-linear suppressive factors in regard to the symmetry of the energy characteristics in the left and right parts of the interval. Some results of filtering are represented in Fig.2. Three filtered projections were used further for tomographical reconstruction by tomosynthesis.

Fig 2: Extraction of crack indication from three radiometric scans with different radiation angle by co-variance analysis.

Conclusion

The proposed method were successfully applied to different NDT objects. The results are good even in difficult situations, when other methods were inappropriate.

The intimate relationship between symmetry and statistics enables to combine them naturally in the framework of this structural approach to form a new mode of thought and to use an extremely flexible and powerful formalism of both combined concepts to create new image processing algorithms. Information, which is revealed by group-theoretical analysis is focussed statistically. Because of this image processing procedures are equally flexible, sensitive and almost noise-immune.

Although LS is regarded as a special case among possible constraints HD. It is the most important case, because usually after some transformations geometrical and brightness constrains can be refined in terms of symmetry. The method has a convenient means for change-over among possible substructures. A modification of HD by content enables to "illuminate" another substructure, if exist.

By its very nature a recognition of a structure element in this method is "situational", although brightness and energetical characteristics of an image are taken into account as well. The recognition is "complementary" i.e. with partial retrieval of information which was lost on raw image.

At the present time the proposed method is unique and quite adequate for processing of highly degraded images, which is the case in NDT. Due to abovementioned advantages the method is applicable to the majority of NDT objects.

References

1. V. Baranov, U. Ewert, Industrial application of digital tomosynthesis and special types of non-linear backprojection algorithms, Proc. of International Symposium "Computer methods and Inverse Problems in Non-Destructive Testing and Diagnostics (CM NDT - 95) ", Minsk, 1995, p. 82-86.
2. U. Ewert, V. Baranov, K.Borchardt, New non-linear backprojection algorithms as a tool for cross-sectional imaging of building constructions using imaging plates, Proc. of International Symposium "Non-Destructive Testing in Civil Engineering (NDT-CE)", Berlin, 1995, p. 1267- 1274.
3. U. Ewert, A. Schumm, C. Nockemann, V. Baranov, Fortschritte auf dem Gebiet der digitalen Laminographie, in Jahrestagung 1995, Zerstoerungsfreie Materialpruefung, 100 Jahre Roentgenstrahlen und die heutige Vielfalt industrieller ZfP-Praxis, Aachen, 1995, P6.
4. U. Ewert, V. Baranov, K.Borchardt, Cross-sectional imaging of building elements by new non-linear tomosynthesis techniques using imaging plates and Co-60 radiation, NDT & E International, 1997, Vol. 30, No. 4, pp. 243-248.

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