X-Ray Cone Beam Tomography with Two Tilted Circular Trajectories

Abstract

Recently 3-D cone-beam tomography has become of interest for the nondestructive evaluation of advanced materials. The main field of application in nondestructive testing is the evaluation of structural ceramics. Study of such materials implies high density resolution and high sensitivity to cracks. In fact, with a single circular source trajectory, when the conebeam aperture increases, density is underestimated and cone shaped artifacts may appear at interfaces in the sampie even at relatively small aperture [1][2][3]. These artifacts limit the thiekness we can ex amine with a planar source trajectory. To maintain optimal reconstruction accuracy with a circular source trajectory, the angular aperture must remain within ±10 0 • However Kudo and Saito [4] showed that this limit can be slightly overcome by using a special interpolation of the shadow area. But to examine greater thicknesses and to maintain resolution, we must widen the cone-beam aperture thereby decreasing accuracy. To overcome these aperture limitations, Tuy [5] introduced the double circular source trajectory idea.Until now, most of the experiments presented in the literature were performed with a planar source trajectory [1,2,3,4,6]. Recently, a new method presented by Smith [7,8] has been applied to nonplanar source trajectories by Kudo and Saito [9]. The inversion presented by Kudo uses the Hilbert transform of the first derivative of the Radon transform. This inversion has been experienced with real data on two circular trajectories at 90° with intersection of the two axis of the trajectories (orthogonal scan).We have presented [10] reconstructions on simulated with double circular source trajectories with an exact reconstruction method using the inversion of the first derivative of the 3-D Radon transform. These simulations were performed with an angle smaller than 90° between the two trajectories. Here we quantify the advantages of this method and we present experimental reconstruction in this implementation. We show also that according to the Radon space sampling, the axes of the two trajectories do not have to intersect, but they must be close enough. MATHEMATICALBACKGROUNDGrangeat [11,12] showed that we can link exactly the X-ray transform of an object and the first derivative of its 3D Radon transform. Given an object function f(M) where M is a given point of the space, let us define the X-ray transform Xf(S,A), as the radiographie reading at point A corresponding to a source position S: +00Xf(S,A)= J f (S + a.u~)da with u~ = a = 0

51464 Development of an advanced 3D cone beam tomographic system

“…This is an empirical extrapolation from wh at is done in the elassical 2D CT algorithm. It produces good results when the cone aperture is lower than ±5 degrees, but cannot be used for larger apertures [5,7].…”

“…Incomplete data lead to artifacts in reconstructed object volumes. These artifacts decrease the contrast and spatial resolution when the apert ure is larger then ±5 degrees [5,7]. The only way to avoid this "shadow area" is to use a better suited trajectory.…”

“…These studies led to an accurate original algorithm [1,3,4] based on a correct mathematical framework. One of the main features of this algorithm is the ability to deal with complex trajectories of the source in order to acquire complete data [5]. In collaboration with its industrial partner INTERCONTROLE, an NDT company of the CEA industry group, LETI initiated a European BRITE project to make use of this new technique on materials research.…”

“…This is because the critical defect sizes could be smaller than 100 microns. The mechanical subsystem allows acquisitions of complete data by providing complex object motions such as a double tilted circular trajectory [5].…”

Development of an Advanced 3D Cone Beam Tomographic System

“…The first one, called EVEGEN, uses the classic approach proposed by Feldkamp [6] whereas the second one, RADON, computes Grangeat's algorithm [3,4] backprojection [5]. The RADON code consists in three main steps [1,3,4].…”