Schwartz Symmetry Condition for Coherent Diffraction Imaging Patterns

Abstract

We demonstrate a symmetry condition for the mixed partial derivatives of measured data when performing a coherent diffraction imaging (CDI) experiment for differentiable samples under scientific investigation. The proposed condition can be used as a physical restriction to improve real data measurements and has been used within the most celebrated phase-retrieval inversion algorithms as an ad hoc constraint without proof. The symmetry relies on John’s ultrahyperbolic equation for the X-ray transform, which is also demonstrated to be valid in the imaging regime for CDI. The obtained conditions are easy to implement and can be used as a constraint by computational imaging methods.