Fig. 1. Sketch for VEM-FPM. (a) Optical layout of a Fourier ptychographic microscopy system. (b) Sketch for the variation EM algorithm. The VEM alternatively finds the estimation of expectation denoted by the green parabola (E-step) and maximizes the likelihood by locating the lateral coordinate of the vertex of the parabola (M-step). (c) Flowchart of VEM-FPM. The latent variable Z is estimated in the E-step through the observed data and predicted data generated by the forward model. In the M-step, the sample wavefront Φ and the pupil function P are obtained through blind deconvolution by solving the quadratic loss function with the plug-and-play prior for in-loop image denoising.

Fig. 2. FPM reconstruction using VEM-FPM. (a) Full-field reconstruction results and the first 9 images for the red channel. (b) Quantitative intensity profile along the white lines in (c) and (d). (c), (d) Zoomed-in image for the yellow boxes in (a). Scale bar: 500 μm for (a); 100 μm for (c) and (d). Full-resolution image is available on Gigapan [42].

Fig. 3. Large FOV reconstruction using VEM-FPM. (a) VEM-FPM result. (b), (c) Zoomed-in image for yellow boxes in (a). (b1) and (b2) are raw images. (c1) and (c2) are from VEM-FPM. Scale bar: 1 mm for (a); 100 μm for (b) and (c).

Fig. 4. Quantitative phase reconstruction using VEM-FPM. (a) Raw image. (b) Reconstructed phase pattern. (c) Zoomed-in image denoted by the yellow box in (b). (d) Evolution of the loss function w.r.t. epoch. (e) Evolution of the phase pattern w.r.t. epoch. (f) and (g) Quantitative phase profile along the white lines in (b) and (c), respectively.

Fig. 5. Validation of VEM-FPM in recovery of aberrations. Simulation studies are performed with 500 groups of testing (a) PSNR and (b) SSIM evaluations for reconstruction quality. (c) Experimental study of VEM-FPM in the blind recovery of the defocused sample. The defocusing distance is known as the ground truth but is not given in the reconstruction. The initial guess of the pupil function is a binary circular mask. (c1) Raw image illuminated by the central LED. (c2) Reconstructed phase pattern. (d) Zernike fitting results for the reconstructed pupil function. The pupil function is smoothed by a Gaussian kernel during the reconstruction.