<title>New extended fan-beam reconstruction formula</title>

Abstract

Because of its high imaging efficiency, fan-beam reconstruction approach has been very popular in recent years. However, fan-beam reconstruction formula had been restricted to a circular scanning locus until an extended fan-beam reconstruction formula was presented by D. B. Smith. Actually, non-circular scanning loci are not only academically interesting, but also practically significant. Smith's extended fan-beam reconstruction formula requires a scanning locus satisfying some nontrivial conditions, contains the derivative of the scanning locus, and in general a spatially variant convolution. In addition, Smith's formula must be specifically discretized for each kind of scanning loci. In this paper, we propose a new extended fan-beam reconstruction formula, which is obtained based on geometrical intuition and is then validated by a strict mathematical proof. The new fan-beam reconstruction formula is the same as the conventional equispatial one, except that the source-to-origin distance depends on the rotation angle, and thus gets rid of the derivative of a locus and always requires a spatially invariant convolution. Furthermore, due to its simple form, the new formula naturally results in a unified discrete version. The new formula requires that the source-to-origin distance be differentiable almost everywhere with respect to the rotation angle and be symmetric with respect to the origin of the reconstruction coordinate system. The numerical simulation results of the new fan-beam formula are also presented.