  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 problem, (3) vignetting effects, and (4) misaligning of
LED positions, without the need of calibrating or recovering of these system
errors. In Elfpie, we embedded the inverse problem of FPM under the framework
of feature extraction/recovering and proposed a new data fidelity cost function
regularized by the global second-order total-variation regularization (Hessian
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 to update the entire Fourier spectrum of the sample and system
pupil function. The Elfpie is tested on both simulation data and experimental
data and is compared against the state-of-the-art (SOTA) algorithm. Results
show the superiority of the Elfpie among other SOTA methods, in both
reconstruction quality under different degeneration issues, and implementation
efficiency. In general, compared against SOTA methods, the Elfpie is robust to
Gaussian noise with 100 times larger noise strength, salt & pepper noise with
1000 times larger noise strength, and Poisson noise with 10 times noise
strength. The Elfpie is able to reconstruct high-fidelity sample field under
large LED position misalignments up to 2 mm. It can also bypass the vignetting
effect in which all SOTA methods fail to reconstruct the sample pattern. With
parallel computation, the Elfpie is able to be K times faster than traditional
FPM, where K is the number of used LEDs.
