
XRay tomography

Xray radiation is a unique tool for diagnosis, because diffraction and refraction are minimal in this case, and hence the simple ray optics model can be employed. In terms of this model the inverse problem of tomography can be reduced to solving linear integral equations (the Radon transform). In the case of fullrange data, hundreds of efficient methods can be proposed to address the inverse problems arising in xray tomography. By fullrange data we mean that the object can be illuminated from all directions and the detectors can be placed on all sides around it. Figure 1 shows a real tomogram of a seashell. Cross sections 1 to 6 obtained in Z=const planes are shown in the righthand panel of the figure.


Figure 1 Tomogram of a seashell: (a) Xray projection image; (b) reconstructed crosssections. 
In the case of limitedrange data the solution of inverse problem is generally nonunique. It makes no sense trying to solve it by inventing new algorithms. The only way is to use supplementary information about the object examined. The obvious information that the soughtfor function ρ(r) describing the attenuation is positive is of no particulat help. The situation changes essentially if the supplementary information is the fact that the 3D object examined consists of a set of layers. Figure 2 shows the scheme of a tomographic experiment involving the diagnosis of a fivelayer printed circuit board (PCB)
.


Figure 2 Scheme of the tomography experiment 
Figure 3 shows the reconstructed images of the fivelayer printed circuit board. The board examined has the size of 50×50 cm, the width of conductors is 400 μm, and the interlayer spacing is 300 μm. The fragments shown in the figure are 5 × 5 cm in size.


Figure 3: (a)  Input data (an image registered by the detectors); (b)(f) reconstructed images of the metal layers 
Mathematically the inverse problem in the case of tomographic diagnosis of layered objects reduces to solving an illconditioned set of linear equations with a large number of unknowns. [1]

References

1. Bakushinsky A., Goncharsky A. Illposed problems: Theory and applications. Kluwer Acad Publ., DORDRECHT/Boston/, London, 1994.
