A standard and linear fan-beam Fourier backprojection theorem

We propose a theoretical formulation for the tomographic fan-beam backprojection in standard and linear geometries. The proposed formula is obtained from a recent backprojection formulation for the parallel case. Such formula is written as a Bessel-Neumann series representation in the frequency domain of the target space in polar coordinates. A mathematical proof is provided together with numerical simulations compared with conventional fan-beam backprojection representations to validate our formulation

Tomographic inspection system using X-rays

Tomography refers to the cross-sectional imaging of an object from either transmission or reflection data collected by illuminating the object from many different directions. This technique revolutionized diagnostic medicine since it enabled doctors to view the slices of internal organs of the patient using x-rays. For the same reason, the method is being used in industry for applicationa such as inspection of turbine blades, rocket motors, ceramics, electronic components, castings, etc. The mathematical basis of CT was established by J.Radon in 1917 when he showed that it is possible to determine the value of a function over a region of space if the set of line integrals is known for all ray paths through the region. In the case of CT, the line integrals are derived from the x-ray intensities sensed by the detectors, and the function to be determined is the distribution of the x-ray attenuation coefficient over the object. However, the large number of calculations needed to accomplish the reconstruction ruled out any practical application to x-ray data until the availability of relatively rapid computers. Hounsfiled and Cornack first received a nobel prize in 1979 in medicine for their x-ray brain scanner with reconstruction time of 2 days. Since then several advances have been made resulting in fast reconstruction algorithms. Fourier weighted backprojection developed by Ramchandran and Laxminarayan is one of the most commonly used algorithm. This algorithm bring out the splendor and power of mathematical formulation of a problem. With very few assumptions, cross-sectional view of an object can be obtained with unprecedented accuracy. The amount of computation involved is still complex enough to demand considerable computing power

Automatic alignment for three-dimensional tomographic reconstruction

In tomographic reconstruction, the goal is to reconstruct an unknown object from a collection of line integrals. Given a complete sampling of such line integrals for various angles and directions, explicit inverse formulas exist to reconstruct the object. Given noisy and incomplete measurements, the inverse problem is typically solved through a regularized least-squares approach. A challenge for both approaches is that in practice the exact directions and offsets of the x-rays are only known approximately due to, e.g. calibration errors. Such errors lead to artifacts in the reconstructed image. In the case of sufficient sampling and geometrically simple misalignment, the measurements can be corrected by exploiting so-called consistency conditions. In other cases, such conditions may not apply and we have to solve an additional inverse problem to retrieve the angles and shifts. In this paper we propose a general algorithmic framework for retrieving these parameters in conjunction with an algebraic reconstruction technique. The proposed approach is illustrated by numerical examples for both simulated data and an electron tomography dataset