Fig. 1. ODT. (a) Experimental setup of conventional ODT; (b) demonstration of the Fourier diffraction theorem; (c) illumination angle scanning to collect different spatial frequency components of the sample

Fig. 2. Generalized imaging system of non-interferometric IDT. (a) Schematic of the imaging system; (b) examples of coherent and partially coherent illumination (intensity distribution at the front focal plane of the condenser lens); (c) different illumination angle scanning devices; (d) 2D (in-focus) intensity measurement or 3D (defocus) intensity measurement; (e) various 3D defocus intensity measurement devices

Fig. 3. TIE-based IDT^[72]. (a) Schematic of the experimental setup; (b) flowchart of the 3D RI reconstruction process; (c)‒(f) 3D RI reconstruction of polystyrene microsphere, lung cancer cell, HeLa cell, and pandorina morum

Fig. 4. IDT based on 2D Kramers-Kronig relations^[79]. (a) Schematic of the experimental setup; (b) the phase (imaginary part of a complex function) can be reconstructed from the intensity (real part of a complex function) based on the 2D Kramers-Kronig relations; (c)(d) 3D RI reconstruction of polystyrene microsphere and A549 cell

Fig. 5. Phase optical transfer functions under different illumination conditions and corresponding intensity spectrum^[92]. (a) 2D case; (b) 3D case

Fig. 6. IDT based on 3D Kramers-Kronig relations. (a) Flowchart of the 3D RI reconstruction process^[92]; (b) (c) 3D RI reconstruction of polystyrene microsphere and C. elegans^[92]; (d) partially coherent incident light satisfying the extreme matched illumination condition and its spatial frequency coverage^[96]

Fig. 7. FP-based IDT. (a) Flowchart of the 3D RI reconstruction process^[77]; (b) 3D RI reconstruction of HeLa cell^[77]; (c) accelerated optimization strategy based on sparse bright field illumination and multiplexed dark field illumination^[99]

Fig. 8. Absolute value distribution of the POTFs under different illumination conditions and at different focal planes^[100]

Fig. 9. TIE- and FP-based IDT^[100]. (a) Flowchart of the 3D RI reconstruction process; (b)‒(d) 3D RI reconstruction of polystyrene microsphere, C2C12 cell, and HeLa cell

Fig. 10. Coherent optical transfer function (C-OTF)-based IDT. (a) Flowchart of the 3D RI reconstruction process^[73]; (b) LED illumination satisfying the extreme matched illumination condition^[74]; (c) a condenser lens is added into Fig.10(b) to increase the illumination NA^[75]; (d) acceleration strategy using a sparse multiplexed LED illumination^[76]; (e) RI reconstruction of C. elegans at x-y plane^[74]; (f) 3D time-lapse imaging of C. elegans^[74]

Fig. 11. Partially coherent optical transfer function (PC-OTF)-based IDT. (a) Schematic of the experimental setup; (b) 2D plots of 3D POTF section in an axial plane for four different illumination apertures^[70]; (c)‒(e) 3D RI reconstruction of HeLa cell^[104], polystyrene microsphere^[101], and cymbella subturgidula diatom^[101]

Fig. 12. Multi-slice forward propagation model-based IDT^[78]. (a) Schematic of the multi-slice forward propagation model; (b)‒(e) 3D RI reconstruction of polystyrene microsphere, a C. elegans embryo cluster, 3T3 cell, and whole C. elegans worm

Fig. 13. Comparison of the first-order Born, first-order Rytov, and MSBP models by predicting the scattering of light for a dielectric sphere. (a) SSIM between the ground truth intensity of the wavefield at the y-z plane and those predicted by different forward physical imaging models versus the radius of the dielectric sphere and the RI contrast; (b) scatter-plots and box-plots of the SSIM values in Fig.13(a); (c) representative examples of the intensity of the wavefield at the y-z plane

Fig. 14. Neural network-based IDT. (a) Data-driven approach to establish a forward physical imaging model; (b) physical information neural network is used to perform unsupervised learning to build a forward physical imaging model^[127]; (c) automatic differentiation library is used to perform gradient descent algorithm^[129]; (d) re-parameterization of 3D RI reconstruction using neural network^[130-131]; (e) neural network is used to improve the 3D RI reconstruction performance of traditional physical model-based methods^[133-134]