Max Leo Leidl (Jülich / DE; Munich / DE), Patricia Römer (Munich / DE), Frank Filbir (Munich / DE), Carsten Sachse (Jülich / DE; Düsseldorf / DE), Knut Müller-Caspary (Jülich / DE; Munich / DE)
Abstract text (incl. figure legends and references)
The Wirtinger Flow (WF) is a variational approach to the phase retrieval problem. It minimizes a loss function which is adapted to a specific noise model. It was studied in the mathematical literature in greater detail [1-3] for a least squared loss function
Ig(ȳ,y) = |ȳ - y|2
that is related to an additive Gaussian noise and to a loss function derived for Poisson noise
Ip(ȳ,y) = y · log(ȳ) - ȳ,
where ȳ is the guess and y the measurement. However, in the situation of low-dose measurements these results do not apply as the gradient has singularities.
The widely used reconstruction algorithms like the extended iterative engine (ePIE) and the Wigner deconvolution can be analysed for a Gaussian noise model. However, for the Poisson noise model the analysis is missing. In contrast, the WF method can be adapted even for lowest-dose measurements. This is of particular importance when dose sensitive samples like viruses have to be imaged. However, the low dose case is not well examined and WF is mostly tested on simulated [1,2] or x-ray data [3].
We compare unblinded (known illumination) WF reconstructions of low-dose scanning transmission electron microscopy (STEM) simulations with a convergence semi-angle of 5 mrad and a dose of 30 e/Ų of apoferritin (PDBe: 7a6a [4]) in vacuum. A Gaussian, Poissonian and an approximation of the Poissonian probability distribution [5] are assumed during the reconstruction. The reconstructions are compared with the object transmission function (ground truth) and an integrated centre of mass (iCOM) reconstruction in real space and by using the Fourier ring correlations (FRC).
Our results show that the differences between the different reconstructions are rather small (Fig. 1). However, all WF reconstructions show a higher dose efficiency for most of the resolution shells and less noise at low frequencies than the iCOM reconstruction. However, the loss function based on Poissonian noise, which is theoretical favourable at low dose, does not perform better than a Gaussian loss function at a dose of 30 e/Ų. A reason is the singularity that is included in the logarithm of the Poissonian loss function. Therefore, it is difficult to control the gradient update for the Poissonian loss function. We will discuss the difficulties of the Poissonian loss function, the possibility to combine the Gaussian and Possonian loss function and demonstrate further dose-dependent WF reconstructions of experimental STEM data of 2D materials for comparison.
Figure 1: (a) iCOM reconstruct. (b) to (d) WF reconstructions with different loss functions that are given in the label. (e) FRC curves that compare the iCOM reconstruction and the phase of the WF reconstructions with the phase of the ground truth.
[1] Candes, et al. IEEE Transactions on Information Theory 61.4 (2015)
[2] Bian, et al. Scientific reports 6.1 (2016)
[3] Thibault, et al. New Journal of Physics 14.6 (2012)
[4] Yip, et al. Nature 587.7832 (2020)
[5] Thibault, et al. New Journal of Physics 14.6 (2012)