CT fan-beam parametrizations leading to shift-invariant filtering

Abstract

The problem of two-dimensional tomographic image reconstruction from fan-beam projections via shift-invariant filtering (convolution) followed by backprojection has a solution for two well known fan-beam parametrization classes. These parametrizations are associated either with equidistant collinear detector cells or with equi-angular fan rays. In this paper, the problem of finding all such fan-beam parametrizations is solved. Two new parametrization classes are found, which define new CT reconstruction algorithms. Two interpretations are given for the new parametrizations. First, the associated loci of equidistant detector cells are found by solving numerically the differential equations that characterize them. Application of the numerical approach also unveils a new detector geometry associated with one of the two previously known parametrization classes. Secondly, by mapping the parametrizations onto the familiar third-generation CT geometry, variable resolution samplings of the scan field of view are found. Also, the problem of CT image reconstruction on a fourth-generation scanner is addressed. Although it is found that the corresponding parametrization does not belong to the convolution solution set, a 'natural' shift-invariant filter decomposition is described that provides a close approximation to the reconstruction filter.

Generalized‐equiangular geometry CT: Concept and shift‐invariant FBP algorithms

“…Shift invariant property tells us that the filtering operation on projection $$p(x)$$ has to be written in such a form below $$$$\begin{equation} p(x_0) = \int p(x)h(x_0 - x) \mathrm{d} x. \end{equation}$$$$So, in order to keep the shift‐invariant property of the filtering step in Equation (), one needs to find a weighted convolution decomposition that satisfies 34 $$$$\begin{equation} K(\gamma _0, \gamma ) = A(\gamma )B(\gamma _0-\gamma )C(\gamma _0). \end{equation}$$$$In Appendix A, we prove that $$K(\gamma _0, \gamma )$$ cannot be exactly decomposed as $$A(\gamma )B(\gamma _0-\gamma )C(\gamma _0)$$ except for $$k=0$$ or $$k=1$$.…”

“…In the literature, numerous studies have been conducted to investigate the shift‐invariant property of FBP‐type algorithms. For example, Besson studied the general parametrizations of fan‐beam CT that could have a direct FBP method without rebinning 34 . Kachelriess also proposed several interesting detector shapes as well for compact CT systems, which could perform FBP in the native geometry without rebinning 35 .…”

Interesting detector shapes for third generation CT scanners

“…Our analysis is inspired by but does not follow Ref. 1. Compared thereto our derivation is more simple.…”

“…However, at least since 1996 it is known that detector shapes other than circular or linear exist that also allow for filtered backprojection in the native geometry. 1 Our aim is to analyze these shapes with respect to their practicability for third generation clinical CT scanners with a particular focus on minimizing detector costs for compact scanners where the radius R M of the field of measurement (FOM) is close to the radius R F of the focal spot trajectory, e.g., for scanners with ρ = R M /R F = 80%. Given the fact that some manufacturers use image reconstruction algorithms that perform a rebinning to fan-parallel domain, we also aim at finding an even better design that is not restricted to allowing for filtered backprojection property in the native geometry.…”

Comparison of analytical and algebraic 2D tomographic reconstruction approaches for irregularly sampled microCT data

“…Besson demonstrated that the fourth-generation CT geometry does not belong to the few ray sampling configurations which allow exact reconstruction via shift-invariant filtering and he proposed an approximate filter [11].…”