We are happy that this paper has been accepted and published on Signal Processing

Simulation experiments are available in folder toy-experiment .
Experiment demo is available in folder usaf_demo

We present a simple but efficient and robust reconstruction algorithm for Fourier ptychographic microscopy, termed error-laxity Fourier ptychographic iterative engine (Elfpie), that is simultaneously robust to

(1) noise signal (including Gaussian, Poisson, and salt & pepper noises)
(2) problematic background illumination
(3) vignetting effects
(4) misaligning of LED positions
(5) without the need of calibrating or recovering these system errors

You don't have to worry about your raw data quality and system calibrations ! (^_^)

In Elfpie, we embed the inverse problem of FPM under the framework of feature extraction/recovering and propose a new image gradient-based data fidelity cost function regularized by the global second-order total-variation regularization. The closed-form complex gradient for the cost function is derived and is back-propagated using the AdaBelief optimizer with an adaptive learning rate.

The cost function contains two parts: the fidelity term and penalty term.

We use the gamma corrected intensity measurement in image gradient domian to be the data fidelity term. Since image gradient is sparse, we further use the $\mathrm{L} _{1}\mathrm{-norm}$ to achieve sparsity promotion.

$\mathbf{I}_{n}$ denotes the intensity measurement of n-th low-resolution image.

$\mathbf{F} ^{\dagger}$ denotes the inverse Fourier transfrom.

$\mathbf{P}$ denotes the pupil function of the image system.

$\mathbf{M}_{n}$ denotes the selection mask for n-th LED illumination.

When $\gamma = 1$ the fidelity term uses intensity measurement, while for $\gamma = 0.5$ the fidelity uses amplitude measurement.
Other fractional $\gamma$ can be tested and generates different reconstruction results.

We use second-order total-variation (TV)-regularization imposed on both amplitude and phase of the recontructed high-resolution image as the penalty term to suppress the noise signal

Where

The optimizer used in ELFPIE is a combination of AdaBelif and AdaDelta. The learning-rate $\boldsymbol{\delta}$ is adaptively adjusted according to historical gradient.
We choose $r_1 = 0.9$, $r_2 = 0.999$, and $\boldsymbol{\delta}^0 = 100$ for learning the parameters.
At the t-th iteration, the model parameters are updated according to

We test the ELFPIE on the USAF resolution target using our homemade FPM platform.
The raw image suffers from severe noises and vignetting effect. LED array is not well-calibrated also.
All dark-field images were collected using the same camera setting (exposure time) as that of bright field images.

The iteration convergent with few loops (20 loops).
The Fourier spectrum of the sample inside and outside the Synthetic Aperture are reconstructed.