A uniqueness result for propagation-based phase contrast imaging from a single measurement

Abstract: Phase contrast imaging seeks to reconstruct the complex refractive index of an unknown sample from scattering intensities, measured for example under illumination with coherent X-rays. By incorporating refraction, this method yields improved contrast compared to purely absorption-based radiography but involves a phase retrieval problem which, in general, allows for ambiguous reconstructions. In this paper, we show uniqueness of propagation-based phase contrast imaging for compactly supported objects in the near field regime, based on a description by the projection- and paraxial approximations. In this setting, propagation is governed by the Fresnel propagator and the unscattered part of the illumination function provides a known reference wave at the detector which facilitates phase reconstruction. The uniqueness theorem is derived using the theory of entire functions. Unlike previous results based on exact solution formulae, it is valid for arbitrary complex objects and requires intensity measurements only at a single detector distance and illumination wave length. We also deduce a uniqueness criterion for phase contrast tomography, which may be applied to resolve the three-dimensional structure of micro- and nano-scale samples. Moreover, our results may have some significance to electronic imaging methods due to the equivalence of paraxial wave propagation and Schr\"odinger's equation.