·Table of Contents
Tomographical Methods of Crack Detection for Components of Nuclear Power IndustryU. Ewert, B. Redmer, Y. Onel
BAM, Berlin, Germany
V. A. Baranov
Institute of Introscopy, Tomsk, Russia
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.
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 Uw of a space Lw , (Lw 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 Wr 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 HD a statistic S is introduced, which can be represented as a functional of Wr DI (x, y) (Wr DI 2 Lwr 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:
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 = Wr the group element operators gw are explicitly depending on parameters of the moving window and by this means located. The group G and the operator Wr generate a system of microimages Uwr (d x, d y) = gw (X, Y, a, b) U(x, y)), " gw Î Gw where dx = x - X,d y = y - Y. A hypothesis HLS,wr = HLS,wr (X, Y, a, b) about LS is verified. Because the operator D changes the relevant group of symmetry (Gw (r) GwD and HLS,wr (r) HLS,wrD by action of D), indeed, the hypothesis HLS,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.
|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.|
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.
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.|
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.
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