# Alternative Fan-beam Backprojection and Adjoint Operators

## TL;DR

This paper introduces new analytic formulas for fan-beam backprojection and its adjoint in tomography, derived from parallel tomography theorems, offering more robust performance in noisy data scenarios.

## Contribution

It provides novel Bessel-Neumann series-based formulations for fan-beam backprojection, improving computational efficiency and noise robustness over traditional methods.

## Key findings

- Achieves $O(N^{2.3729})$ computational complexity.
- Demonstrates increased robustness with noisy data.
- Validates formulations through numerical simulations.

## Abstract

We present in this work alternative analytic formulations for the fan-beam tomographic backprojection operation and its associated adjoint transform in standard (equiangular) and linear (equidistant) detector geometries. The proposed formulations are obtained from a recent backprojection theorem in parallel tomography. Such formulations are written as a Bessel-Neumann series in the frequency domain that can be implemented as an $O(N^{2.3729})$ matrix multiplication. Proofs are provided together with numerical simulations compared with conventional fan-beam $O(N^{3})$ backprojection representations showing more robustness when dealing with highly noisy data.