Abstract
The fundamental principle of fan-beam CT is to reconstruct the object from its known projection by using two basic types of analytical reconstruction formulas filtered back-projection (FBP) and differentiated back-projection (DBP). Both approaches are used for various applications, and each type has its own advantage properties. Discussing their theory analysis and application introduction appeared in several papers. However, there is no previous discussion about the internal mathematical relation between them. In this paper we present a derivation of the fan-beam reconstruction formula from the starting point of parallel-beam projection by using the equality between the Hilbert transform of object`s 2D Radon transform and the Hilbert transform of fan-beam projection. Because the Ramp filter can be divided into a single Hilbert filtering step and a derivatives step, we rewrite the derived reconstruction formula by exchanging the calculation order of the Hilbert filter step and the derivatives step. As a result, not only the mathematical relation between FBP and DBP algorithm is clarified, but also present six formulas that include all type of analytical fan-beam reconstruction algorithms.
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