Computer Methods and Inverse Problems in Nondestructive Testing and Diagnostics

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1 Computer Methods and Inverse Problems in Nondestructive Testing and Diagnostics Minsk, October 20-23, 1998 Finite series expansion method modified for multi-step reconstruction from limited number of projections and views S. Zolotarev, V. Vengrinovich, Institute of Applied Physics, Minsk (Belarus) G.-R. Tillack, Bundesanstalt fur Materialforschung und -prufung (BAM), Berlin (Germany) INTRODUCTION Industrial X-ray Tomography from a limited number of projections and limited views of observations is an inevitable practical demand.. Its aim is to implement possibly perfect restoration of the internal structure of the object under radiative testing in the conditions of fundamental lack of experimental data. The probabilistic Bayesian approach to the data analysis [1] leaves only one way to overcome this obstacle: to introduced the a priori: knowledge in the calculus.having the incomplete data we are constrained " to ask a question by setting a priori probability" for any particular configuration of the attenuation coefficients in the elementary volume cells of the object. The article modifies the finite-series expansion method to provide the 3D image restoration from incomplete data by introducing prior knowledge in the calculus. STATEMENT OF THE PROBLEM Main features and conditions limited data computerized Tomography (CT) are: 1) one-side limited access to the object under testing; 2) extremely limited number of projection (2-10); 3) exposure by divirgent conic beam. These problems result in a fundamental incompleteness of basic data and the require development of specialized algorithms for reconstruction.the main task is to exploit maximum a priori information about admissible solution.the 185

2 introduction of an additional information reduces time of calculations and creates restrictions for stochastic distortions of an information in the calculus. The calculations should be conducted in a few stages, alternating numerical transformations with introduction of a priori constrains, taking into account the sinqularity function of the problem. In work [2] it was offered the concept of multi-step reconstruction, which was evolved in the series of subsequent works [3], [4]. We shall firstly specify a model of reconstruction which we exploit in the present article. We consider, that the object under reconstruction represents a three-dimensional body with a mathematically defined surface. The X-rays absorption function is assumed to be fixed, but quantitavily unknown values everywhere inside an object, excluding local defects (porosity, slag, corrosion), where it is changed arbitrarily. We shall consider the total volume of local defects being essentially smaller then the full volume of the object. In this article we shall consider for the simplicity the coefficient of absorption of the matrix is known material, though in case of the small total volume of local defects we can estimate this value quite unbiasly. We have the goal to restore the configuration of local inclusions and distribution of the coefficient of an absorption inside them. MODIFIED NUMERICAL ALGORITHM The three-dimensional variants of methods the finite-series expansion (ART, SIRT, ILST) do not present rigid specifications to the number of projections and range of varying the angle of observation. Continuing the article [5], where ART was investigated, in this work for the same purposes use SIRT. For the best understanding of the stated approach we shall remind the basic aspects. Methods, based on the expansion of an image in to finite series result in the following discrete problem of reconstruction: using a projective matrix R on a vector of the measured values y to evaluate a vector of an image x. The problem is reduced to a solution of a system of the linear equations y = Rx + e (1) Where the matrix R = ^ (i - \J\j = I, J), - length of intersection of the i-th ray with j-th voxel, x - J - dimensional column vector (x x x 2^xj), y And e -1 - dimensional column vectors. It is necessary to have in mind that of the beginning of iterative process the system (1) appears to be undetermined owing of the lack of information. But losing the below desribed methodologies at the end of iterative process we can come to case, when it will appear to be overdetermineted. As a rule, in this case it appears to be incompatible (i.e. for it there is no solution in a usual meaning x = R~ l (y -e)). A vector of an initial approximation is obtained with the help of discrete back projection procedure.thus, we consider x (1) as a vector obtained from an initial vector x ( 0 ) After single run through all projections. According to the work [4] let's introduce a virtual set 186

3 VDS, true set TDS, and also null defect set NDS, as a voxel set, which belong to the set VDS, but not belong to the set TDS that is: VDS = TDS U NDS. (2) Let's introduce into consideration a new set SVDS (Surface of VDS) as the voxel set, pertaining to the boundary of a set VDS, and also conjugated set SNDS. We will consider, that numerical representation of a virtual set of defects VDS ^ is realized by the set of numbers on 3D lattice { y }, where (aj 9 J3j,yj) are varied inside 3D area of reconstruction. Let's define an a priori estimate of probability that the voxel under consideration (a j,pj,y 7-) on k-th of iteration belongs to virtual defects space VDS: = n^a}*rjt J», J *ti(r,s,t)ea,(a J,fij,rj) ez 3 ] (3) Moreover the aperture A will be considered as symmetrical relatively to the origin of coordinates and the containing it in itself: (r,s,t) ea=> { r s t) EA,(0,0,0) ea. The three-dimensional function F depends on number and density of voxels, which contained inside the aperture A. In the most simple approaches it can realize various kinds of a spatial filtration, though there can be used also various combined estimates (variance, gradient) and so on. After completion of the k-th of iteration we allocate from a set SVDS ^ A set SNDS ( k ) = {xflxf } esvds {k \z {k) < 3mzxzf\3 e(0,l)}. After that make the next approximation of null defect set NDS ( k ) = SNDS ( k ) UNDS ( k ~ l ). (4) Then from the previous approximation VDS ^ get the next approximation VDS = VDS (k) r\nds (k). (5) Evidently, that while increasing the number of iterations NDS ^ NDS and, hence respectively VDS (^ TDS. The sense of this algorithm, based on idea of multi-step reconstruction, is: 1) sweeping of the voxels which belong to shaded artefacts, is carried out starting from the surface of the next approximation of virtual defect set VDS ^, 2) separation of the voxels belonging to shadows from voxels, belonging to the surface set TDS is carried out on the basis of 3D spatial filtering and also on the basis of taking into account other spatial estimates(variance,gradient, etc.). 187

4 The allocation of the set SVDS { > on the set VDS { k ) is carried out with the help of effective superhigh-speed algorithm, which allows to establish the required surface thickness in voxels. Application of this algorithm of reconstruction, results in sequential reduction of shadow artefacts, i.e. the spatial structure of shadows becomes more and more "friable", that allows to effectively separate them from defect space. Moreover, during reconstruction separately located shadow "phantoms", which have sufficiently dense spatial structure are arised. Let's name these formations as the tops of null defect space while the voxel set, which belong to theml let's designate as TOPNDS. With the help of special geometric analysis of the set SVDS 1 J it becomes possible to definitely uniquely extract voxels which belong to the set TOPNDS even for limited number of projections (3-5). Unlike the traditional technique for bringing in the corrections by passing consequently down one ray the new technique was introduced. The idea of the approach assumes taking into account of the fact, that each ray contributes not only in the voxels located on its way, but also in the voxels located rather close to it. Along with the voxels which are located on the ray, determineted by pixel p(ij), the voxels located on the rays, determined by the adjacent pixels were considered: p(i - (M + 1) / 2 + k, j - (M + 1) / 2 + /); k \JA\ I = where M = 2m width of the rectangular window. For voxels, located on the adjacent rays there were introduced the local weighted coefficients = y/(py), where p.. - distance from the center of the voxel, located on the ray neighboring to the central one. Thus, the correction for the central rays voxels was distributed between voxels, located on the central ray according to coefficients r-, while for the voxels, located on the neighborhood rays - relatively to local weight coefficients. From theoretical reasons follows, that the magnitude of local weight coefficients should decrease in inverse proportion to some degree p... In the given work the following empirical formula is used: w^rij l{(pja) a *b + \). (6) Here coefficient a - size of an edge of a cubic voxel, the coefficients b and a define the spatial derivative describung the decreasing of local coefficients depending on a distance from the centre of the considered voxel to the central ray. Actually used values were a - 3, b = 8. From a physical point of view: the magnitude of local coefficients decreases in inverse proportion to a cube of a distance from the central ray, and for voxels concerning central ray the magnitude of effective coefficient of interaction decreases twice. Numerical experiments conducted for a width of the window the M=3 yield noticeable improvement of the quality of reconstruction. Thus, to compensate the lack of particular data we used the step by step compression of a virtual set of defects on the one hand while on the other hand we account the fact that the contribution introduced by an integral of a density in an arbitrary point of restored object represents the function of a distance from the given point to the ray. It is necessary to note also, that using of a pointwise correction though has enlarged time of the calculation, but has allowed considerably to increase quality of reconstruction on a comparison with ART method. 188

5 EXAMPLE OF APPLICATION THE ALGORITHM The proposed technique applied with the SIRT method was checked for the same example as in [4]. Image reconstruction of the seven rings of the various forms located inside three-layers object was implemented. Upper and lower layers made from PVC, and central layer represents an aluminium plate. The first reconstruction was implemented from the four projections which provided all-sided access to the object during exposures and indicated in fig.l, a)-d). The second reconstruction was realized having also four projections, namely fig.l, a), b) and fig. 2, a), b), which provide the access to the object within 135 grad. during exposures. Parallel beam sheme was used. The corresponding restored images of the object are shown in fig. 3, b) and fig. 4, b) respectively, while in fig. 3, a) and fig. 4, a) the corresponding VDS are presented. In fig. 5 and fig. 6 the top views of corresponding restored images but in consideration with the real densities in the voxels (grey images) are shown. The complete number of iterations made during reconstruction was seven. The samples and the exposures were done by Dr. U. Ewert in BAM (Germany) and authors are greatfiil to him for the permission to use the data. DISCUSSION The indicated results concern the extension of the reconstruction method, called 3D SIRT. It has shown much letter outcomes comparing to the method ART, though in general the last one is also quite efficient being applied to the same technique of reconstruction. The further explorations will be directed on the study of the influence of various methods of correction, in particular, MART algorithm, which allows to solve the problem of an entropy maximization. THE LITERATURE 1. Skilling J., Probabilistic data analysis: an introductory guide. Journ. of microscopy, vol Pts 1/2, 1998, pp Vengrinovich V.L., Denkevich Y.B., Tillack G.-R., Heine S. X -ray 3D reconstruction using minimal projections and maximum a priori knowledge. Proc. International Conference "Computer methods and Inverse Problems in nondestructive Testing and Diagnostics", Nov., Minsk, 1995, pp Vengrinovich V.L., Zolotarev S.A., Tillack G.-R., C.Nockemann. X-ray 3D reconstruction of objects with unhomogeneous internal structure using. Proc. International Conference "Computer methods and Inverse Problems in nondestructive Testing and Diagnostics", Nov., Minsk, 1995, pp Vengrinovich V.L., Denkevich Y.B., Tillack G.-R. And C. Nockemann. Multistep 3D X-Ray Tomography for a Limited Number of Projections and Views. Rev. Prog. In QNDE, Vol 16, ed. By D.O. Thompson and D.E. Chimenti, Plenum Press New-York, 1997, pp Zolotarev S.A., Vengrinovich V.L. And Tillack G.-R. 3D Reconstruction of Flaw Images with Inter -Iterational Suppression of Shadow Artefacts. Rev. Prog. In QNDE, Vol 16, ed. By D.O. Thompson and D.E. Chimenti, Plenum Press New-York,

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7 a) b) Fig. 3 VDS (a) and restored binarized image (b) of the object having all sided access during 4 exposures. Fig. 4 VDS (a) and restored binarized image (b) of the object having limited (within 135 grad.) access during 4 exposures. 191

8 o Fig. 5 Restored grey images (in consideration with the densities in voxels) of the object having all-sided access during 4 exposures. mj-fy,^m '] Fig. 6 Restored grey images (in consideration with the densities in voxels) of the object having the limited (within 135 grad.) access during 4 exposures. 192