Fig. 1. Optical imaging principle and system setup of FPM. (a) Imaging principle of FPM^[26]; (b) system setup of FPM^[37]; (c) constraints in the Fourier domain; (d) illumination of oblique LEDs; (e) LEDs are sequentially turned up during FPM image acquisition

Fig. 2. Schematic diagrams of the range of diffracted light collected by the objective lens at different illumination light angles^[41]. (a) Under normal incident illumination; (b) under oblique incident illumination

Fig. 3. Constraints and algorithm flow of alternating projection^[21]. (a) Constraint with support domain; (b) constraint with amplitude; (c) algorithm flow of alternating projection

Fig. 4. Phase recovery flow chart of Fourier ptychography microscopy^[26]

Fig. 5. Iterative update flow chart of FPM imaging^[44]. (a) Sequential update; (b) batch update

Fig. 6. LED array positional misalignment correction method based on hardware^[56]. (a) System setup; (b1)(b2) criterion for roll angle adjustment of the LED array; (c1)~(c4) criterion for adjustment of central LED position; (d) a mosaic image combining raw images of aperture diaphragm containing bright-field components

Fig. 7. LED array positional misalignment model of FPM setup^[57]. (a) Diagram of a misaligned FPM setup; (b) enlargement of the central windowed part in Fig. 7(a)

Fig. 8. Results and system setup of efficient illumination angle self-calibration FPM^[58]. (a) Brightfield images contain overlapping circles in their Fourier spectra, while darkfield images do not; (b) system setup with quasi-dome type illuminator

Fig. 9. Illumination patterns with planar and dome type LED arrays^[60]. (a) Planar type LED array; (b) dome type LED array

Fig. 10. Dome type LED illumination. (a)(b) Designed by Phillips et al^[60]; (c)--(e) designed by Pan et al^[62]

Fig. 11. Schematic diagram of intensity distribution along different angles of LED^[41]

Fig. 12. Algorithm flow of full-field FPM algorithm based on gradient descent method^[65]

Fig. 13. Reconstruction results from uncorrected FPM and EPRY-FPM algorithms ^[64]. (a1)(a2) Reconstructed sample amplitude and phase using uncorrected FPM algorithm; (b1)(b2) reconstructed sample amplitude and phase using EPRY-FPM algorithm; (c1)(c2) reconstructed pupil function modulus and phase using EPRY-FPM algorithm

Fig. 14. Reconstruction results based on full-field FPM ^[65]. (a) Reconstructed full-FOV amplitude with full-field FPM; (b1)(c1)(d1) reconstructed amplitudes corresponding to three positions; (b2)(c2)(d2) reconstructed phase corresponding to three positions; (b3)(c3)(d3) raw images corresponding to three positions

Fig. 15. FPM system constructions with laser illumination. (a)(b) FPM imaging setup and the Fourier spectrum distribution based on 2D Galvo mirror system proposed by Chung et al^[67]; (c) FPM imaging setup based on DMD proposed by Kuang et al^[68]; (d)(e) FPM imaging setup based on LCD proposed by Guo et al^[69]

Fig. 16. Schematic diagrams of light collection angle range of objective lens^[38]. (a) Critical angles of bright-field and dark-field images; (b) illumination angle that corresponds to boundary of bright-field and dark-field images; (c) diagram of bright-field and dark-field images

Fig. 17. FPM reconstructed results with different denoising methods^[72]. (a)(b) Reconstructed result with no denoising and part of the enlarged image; (c)(d) reconstructed result with conventional thresholding denoising and part of the enlarged image; (e)(f) reconstructed result with adaptive denoising and part of the enlarged image

Fig. 18. Iteration equations weighting factor and penalty function factor corresponding to varied probe amplitude for different algorithms^[78]. (a) Iteration equations weighting factor; (b) penalty function factor

Fig. 19. Typical probe patterns in PIE/FPM^[44]. (a) Convergent beam probe commonly used in X ray; (b) random speckle probe commonly used in visible light imaging; (c) ideal circular lowpass filter probe used in FPM

Fig. 20. Results comparison of adaptive step-size, fixed step-size and some other FPM algorithms^[47]. (a) Comparison of convergence speed and reconstruction accuracy of the adaptive step-size approach and four state-of-the-art FPM reconstruction algorithms under 50% Poisson noise; (b) comparison of reconstruction results and runtime of the adaptive step-size approach and four state-of-the-art FPM reconstruction algorithms

Fig. 21. Comparison of experimental results between AcFPM and FPM^[83]. (a1)(a2) Schematic of traditional FPM and AcFPM; (b1)--(e1) reconstructed amplitude and the corresponding enlarged results of traditional FPM; (b2)--(e2) reconstructed amplitude and the corresponding enlarged results of AcFPM

Fig. 22. Diagram of neural network structure used to replace the traditional FPM phase recovery algorithm^[87]

Fig. 23. Physical-guided neural network of FPM^[86]

Fig. 24. Different sample models for FPM. (a) 2D thin sample; (b) 3D thin sample; (c) 3D thick sample

Fig. 25. Digital refocusing results of FPM introduced with phase propagator and convergence index^[95]. (a1)--(a3) FPM reconstructions without digital refocusing; (b1)--(b3) FPM reconstructions with digital refocusing; (c) FPM convergence index as a function of defocused distances

Fig. 26. Algorithm flow of digital refocusing FPM inserted phase propagator in the reconstruction iteration^[26]

Fig. 27. USAF chart reconstruction results of digital refocusing FPM based on geometric characteristic of the imaging system^[97]

Fig. 28. Paramecium sample reconstruction results of digital refocusing FPM based on geometric characteristic of the imaging system^[97]. (a1)--(a4) Reconstructed amplitude and phase of conventional and digital refocusing FPM with the defocus distance of 79 μm; (b1)--(b4) reconstructed amplitude and phase of conventional and digital refocusing FPM with the defocus distance of 80 μm; (c1)--(c4) reconstructed amplitude and phase of conventional and digital refocusing FPM with the defocus distance of 60 μm

Fig. 29. 3D FPM system setup and reconstruction results based on Multi-Slice model^[99]. (a) FPM system setup; (b) 3D reconstruction results

Fig. 30. Diagrams of sub-spectrum distributions in 2D and 3D FPM. (a) FPM system setup^[102]; (b) 2D spectrum andthe corresponding sub-spectrum distribution^[102]; (c) 3D spectrum and the corresponding sub-spectrum distribution^[101]

Fig. 31. Algorithm flow of 3D FPM and results of tomography based on phase recovery and sub-spectrum splicing in the 3D Fourier domain^[101]. (a) Algorithm flow of 3D FPM; (b) reconstructed results of tomography 3D FPM

Fig. 32. HDR FPM reconstructed results^[104]. (a) Raw image of a blood smear sample; (b) reconstructed result without HDR combination process; (c) reconstructed result with HDR combination process; (d) reconstructed result with sparsely sampled FPM

Fig. 33. Principle and reconstructed results of adaptive HDR FPM with RGB camera^[105]. (a)(b) Imaging principle of Bayer filter; (c) system setup of adaptive HDR FPM; (d) large FOV LR images acquired by this system; (e1)--(e3) reconstructed results with single channel of RGB; (f) reconstructed result with adaptive HDR FPM

Fig. 34. Diagrams of different illumination patterns^[106]. (a) FPM with single LED pattern; (b) FPM with LED positions multiplexing pattern; (c) FPM with wavelengths multiplexing pattern

Fig. 35. FPM algorithm flow of spectral multiplexing^[106]. (a) LED positions multiplexing; (b) LED wavelengths multiplexing

Fig. 36. FPM reconstructed results of coded illumination by multiple LEDs^[73]. (a) Low resolution image acquired by camera; (b) a zoom-in on the smallest features; (c) reconstruction result from traditional FPM; (d) multiplexing FPM with 293 images; (e) multiplexing FPM with 74 images

Fig. 37. FPM reconstructed results corresponding to different overlapping of sub-spectrum^[104]。 (a1)(a2) Input intensity and phase with high resolution in the simulation; (b)--(d) FPM reconstructions with different spectrum overlapping percentages in the Fourier domain

Fig. 38. Updated LED illumination arrays of FPM^[108]. (a) System setup of FPM; (b)(c) different perspectives of illuminators

Fig. 39. Updated FPM system setups. (a)(b) FPM system setup based on a cellphone lens^[112]; (c) FPM system setup based on Raspberry Pi and the corresponding open sources^[113]; (d) FPM system setup based on multi-aperture camera array^[114]; (e) FPM system setup based on industrial camera and telecentric objective^[56]

Fig. 40. Imaging principle of lens-less FPM imaging. (a) System setup of lens-less FPM proposed by Luo et al^[116]; (b) system setup of lens-less FPM proposed by Zhang et al^[117]

Fig. 41. System setup of FPM with reflection and epi-illumination sources. (a) System setup of FPM based on epi-illumination^[121]; (b) system setup of FPM based on reflection illumination designed by Guo et al^[119]; (c) system setup of FPM based on a parabolic mirror^[123]

Fig. 42. Principle and system setup of macroscopic FP imaging. (a) Reflection geometry FP imaging^[125]; (b)(c) aperture-scanning FP imaging^[126]

Table 1. Object update functions for PIE, ePIE and rPIE algorithms