Invertibility of the Fourier Diffraction Relation in Raster Scan Diffraction Tomography

TL;DR

This paper investigates the invertibility of the Fourier diffraction relation in raster scan diffraction tomography, showing that in higher dimensions all Fourier coefficients are uniquely recoverable, while in 2D only partial recovery is possible.

Contribution

It proves the generic uniqueness of Fourier coefficient recovery in higher dimensions and characterizes the limitations in the 2D case for raster scan diffraction tomography.

Findings

In higher dimensions, all Fourier coefficients are generically uniquely determined.

In 2D, only a subset of Fourier coefficients can be uniquely recovered.

The Fourier diffraction relation leads to a linear system for the Fourier coefficients.

Abstract

Diffraction tomography aims to recover an object's scattering potential from measured wave fields. In the classical setting, the object is illuminated by plane waves from many directions, and the Fourier diffraction theorem provides a direct relation between the Fourier transform of the object's scattering potential and the Fourier transform of the measurements. In many practical imaging systems, however, focused beams are used instead of plane waves. These beams are then translated across the object to bring different regions of interest into focus. This article discusses what information about the scattering potential can be extracted from such measurements. As in the classical case, the analysis is based on a recently derived Fourier diffraction relation that relates the measurements to the Fourier coefficients of the scattering potential. However, this relation does not immediately provide an explicit reconstruction formula, but instead leads to a linear equation system for the Fourier coefficients. We therefore prove in this work that all Fourier coefficients appearing in these relations are in dimensions higher than two generically uniquely determined. In the two-dimensional case, on the other hand, only a particular subset of the Fourier coverage is uniquely recoverable, while on the remaining region distinct coefficients may produce identical data.