Recovering low spatial frequencies in wavefront sensing based on intensity measurements

Abstract

The transport of intensity equation (TIE) offers a convenient method to retrieve the phase of a wave function from maps of the irradiance (images) recorded at different planes along the optic axis of an optical system. However, being a second-order partial differential equation, even for noise-free data a unique solution of the TIE requires boundary conditions to be specified which are generally not accessible experimentally, jeopardizing retrieval of the low-frequency information in particular. Here we introduce an iterative algorithm which forgoes the need for explicit boundary conditions and combines the well-known reciprocal space solution of the TIE with the charge-flipping algorithm that has originally been developed to solve the crystallographic phase problem in X-ray diffraction. Application of this algorithm to experimental data and comparison with conventionally used algorithms demonstrates an improved retrieval of the low spatial frequencies of the phase.

Automatic software correction of residual aberrations in reconstructed HRTEM exit waves of crystalline samples

“…Focal series reconstruction has a well-known limitation that it cannot accurately reconstruct lower spatial frequencies [12, 16]. This leads to exit wave phase images in the reconstructions that are somewhat flatter (lower peak-to-peak range) than the true exit wave phases.…”

“…One possible solution is to reconstruct the complex electron wavefunction via inline holography, by taking a defocus series and employing an exit wave reconstruction (EWR) algorithm such as Gerchberg-Saxton or the Transport of Intensity Equation [10–16]. Alternatively, an exit wave can be reconstructed by interferometric methods, i.e.…”

Multi-resolution convolutional neural networks for inverse problems

“…In conventional phase retrieval schemes, a trade-off between simple invertibility and accuracy has to be made. When approximating the imaging process by the transport of intensity equation (TIE) 29 , the relationship between the wavefront and a gradient measurement in the intensity domain can be approximated by a second-order partial differential equation, the solution of which can be found by assuming either periodic 33 or other, often more appropriate boundary conditions 34 . To obtain a reliable gradient estimation, multiple intensities have to be measured at focal planes below and above the plane of focus, in pairs symmetrically to the in-focus plane 31,32 .…”

Gradient-flipping-assisted Flux-preserving Wave Reconstruction for Improved Lowspatial- Frequency Recovery and High-contrast Imaging Using in-line Holography

“…Here, we introduce an in-line electron holography reconstruction algorithm that allows a wide range of spatial frequencies to be recovered by combining gradient ipping 19,20 and phase prediction with incoherent background subtraction and a ux-preserving nonlinear imaging model 21,22 .…”