Gradient flipping algorithm: introducing non-convex constraints in wavefront reconstructions with the transport of intensity equation

Parvizi, Amin; Broek, Wouter Van den; Koch, Christoph T.

doi:10.1364/oe.24.008344

Cited by 6 publications

(1 citation statement)

References 15 publications

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“…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 .…”

Section: Introductionmentioning

confidence: 99%

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

Keskinbora

Broek

Boothroyd

et al. 2022

Preprint

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In-line holography in the transmission electron microscope is a versatile technique, which provides real-space phase information that can be used for the correction of imaging aberrations, as well as for measuring electric and magnetic fields and strain distributions. It is able to recover high-spatial-frequency contributions to the phase effectively but suffers from the weak transfer of low-spatial-frequency information, as well as from incoherent scattering. Here, we combine gradient flipping and phase prediction in an iterative flux-preserving focal series reconstruction algorithm with incoherent background subtraction that gives extensive access to the missing low spatial frequencies. A procedure for optimizing the reconstruction parameters is presented, and results from MgO cubes, and Fe-filled C nanospheres are compared with phase images obtained using off-axis holography.

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Section: Introductionmentioning

confidence: 99%

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

Keskinbora

Broek

Boothroyd

et al. 2022

Preprint

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Wave front reconstruction vi the transport of intensity equation: Introduction of non‐convex constraints

Parvizi

Broek

Blessing

2016

European Microscopy Congress 2016: Proceedings

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The transport of intensity equation (TIE) is an elliptical, second order partial differential equation which relates the intensity variation along the optical axis to Laplacian‐like expression involving of the unknown phase. Due to its simple mathematical formulation and the straight forward procedure to acquire the experimental data, the TIE has attracted enormous attentions from various research communities such as transmission electron microscopy, X‐ray microscopy, neutron imaging, etc.. The FFT approach to the TIE due to its deterministic nature and computational speed is widely used. However, periodic boundary condition inherent to the FFT approach results in low frequency artifacts in case of non‐periodic objects. Alternatively, one can impose an additional constraint on the solution in order to overcome the problem of only weakly encoded low spatial frequency phase information and unknown boundary conditions. For example, by total variation minimization approach employs gradient based optimization techniques to minimize the convex l 1 ‐norm of the derivative of the phase is minimized, leading a solution which is preferentially flat. However, the TV‐minimization approach should only be considered for piece‐wise constant objects. Here, we report on an iterative algorithm namely, the gradient flipping algorithm (GFA) [1,2], which imposes non‐convex constraint on phase. The GFA assumes that the wavefront to be recovered is sparse in its gradient basis and therefore, combines the reciprocal space solution of the TIE with the principle of the charge flipping algorithm [3] by flipping the sign of phase gradients below a determined threshold. This leads to the sparsest solution in the gradient domain. In an iterative manner, the boundary conditions are updated in such a way that consistency of the recovered wavefront with the experimental data is assured and at the same time the solution is sparse. The algorithm iterates until the convergence criterion is fulfilled. Figure 1(a) depicts the variation of the image intensity along the optical axis of images recorded of HeLa cells which was estimated from a focal series comprising 20 images with defocus step of 1 µm and laser illumination at a wavelength of λ = 520 nm. Figures 1(b), 1(c) and 1(d) show the phase reconstructed by a Tikhonov‐regularized FFT‐based solution of the TIE (q ‐2 ‐> (q 2 +α) ‐1 , with q being the reciprocal space coordinate) for α =0.001 µm ‐2 ,α=0.01 µm ‐2 and α=0.1 µm ‐2 where α is the regularization parameter. The phase retrieved by employing the TVAL3 package [4], (convex TV‐minimization) is shown in Fig. 1(e). Finally, Fig. 1(f) presents the phase map reconstructed by the proposed approach. This GFA‐reconstruction provides a physically very reasonable solution while that of TV‐minimization suffers from missing low frequency information, and the Tikhonov regularized FFT‐based reconstruction suffers from low frequency artifacts when ααα is small and is not capable of recovering low frequency information when the regularization parameter is increased.

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Quantifying the data quality of focal series for inline electron holography

2021

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Gradient flipping algorithm: introducing non-convex constraints in wavefront reconstructions with the transport of intensity equation

Cited by 6 publications

References 15 publications

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

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

Wave front reconstruction vi the transport of intensity equation: Introduction of non‐convex constraints

Quantifying the data quality of focal series for inline electron holography

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