Inverse Fourier Transform in the Gamma Coordinate System

Figure 1

3D reconstruction problem in parallel-beam geometry and the cylindrical coordinate system. (a) A parallel-beam scans an object along a semicircle locus on the unit sphere; (b) In the frequency domain, when 𝜙₀ increases from 0 to 𝜋, the frequency plane 𝜙=𝜙₀ scans every point exactly once except for the points on the 𝑘₃ axis, and the end point of the normal vector 𝑂𝑆 moves along the equator from (0,−1,0) to (0,1,0); (c) For simplicity, we suppose that the detector panel passes through the origin.

(a)

(b)

(c)