–An efficient way to perform Radon backprojection is to do it in two steps and each step is a series of 2D backprojections.
–For the ray transform data, we require that Orlov’s data sufficiency condition be satisfied. The ray directions trace a trajectory on the unit sphere. If every great circle intersects this trajectory, Orlov’s condition is met.
–The image reconstruction algorithm for the ray transform depends on the ray direction trajectory. Due to data redundancy, the reconstruction algorithm is not unique. One can either do filtering first or backprojection first.
–Feldkamp et al. developed a simple and robust FBP algorithm for cone-beam circular orbit imaging. Even though this algorithm is a modification of a fan-beam’s FBP algorithm and is not exact, it has wide applications in many fields. The reconstruction errors are not significant if the cone angle is small enough.
–One can use Tuy’s condition to verify if the cone-beam imaging geometry is able to provide sufficient projections. Nonplanar orbits, such as the helix orbit or the circle-and-line orbit, are required to satisfy Tuy’s condition. Tuy developed a relationship between the cone-beam data and the original image; he also developed a cone-beam inversion formula, but it is difficult to use.
–Grangeat’s relationship is that the angular derivative of the cone-beam weighted planar integral equals to the derivative of the Radon planar integral. In Grangeat’s cone-beam image reconstruction algorithm, the image is reconstructed using the Radon inversion formula. A drawback of Grangeat’s cone-beam reconstruction method is the rebinning from the cone-beam data to Radon data. The rebinning step can cause large errors.
–Katsevich’s cone-beam image reconstruction algorithm is truly an FBP algorithm with shift-invariant filtering and cone-beam backprojection. One drawback of Katsevich’s algorithm is its difficulty in selection of filtering directions. Another drawback is that cone-beam projection data are not fully used.
–The readers are expected to understand the Radon inversion formula and Feldkamp’s cone-beam image reconstruction algorithm in this chapter.
Problems
| Problem 5.1 | Calculate the 3D parallel line integrals p(u, v,  ) and parallel plane integrals p(s, ) of a uniform ball, in which the line density and area density are both 1. The center of the ball is at the origin of the coordinate system, and the radius of the uniform ball is R. |
| Problem 5.2 | A cone-beam focal-point orbit is a circle with two lines as shown. The radius of the circular orbit is R. The object to be imaged is a ball of radius r. Determine the length of the linear orbits so that Tuy’s condition can be satisfied. |
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