Fig. 1. In-line holography with multiple scattering. (a) A plane-wave is incident on a 3-D object containing distributed scatterers. The field undergoes multiple scattering within the volume and then propagates to the image plane. A hologram is recorded, which is then used to estimate the unknown scatterers’ distribution. (b) An inline holography setup is used that consists of a collimated laser for illumination and a 4F system for magnification. (c) The raw data are a single hologram. (d) The reconstruction implements a nonlinear inverse multiple scattering algorithm.⁴⁰ (e) The output estimates the 3-D distribution of the scatterers.

Fig. 2. Illustration of the 3-D internal scattered field operator G in Eq. (6). (a) Each object slice f is first voxelwise multiplied by the lower order scattered field uk−1; it is then propagated to every other slice within the volume. (b) This computed scattered-field uks is added to the incident-field uin to obtain the next higher-order Born-field uk. This process is recursively applied to compute the multiply scattered field within the volume.

Fig. 3. Small-scale multiple-scattering inversion. (a) An accurate 3-D forward model is used to simulate the hologram. (b) Multislice 3-D reconstruction is performed from a single simulated measurement using our method. The number of slices in the inverse reconstruction can be flexibly chosen. (c) Full 3-D inversion is performed by reconstructing all axial slices in the original object using our method. The multiple-scattering method outperforms the single-scattering method by providing both more accurate permittivity contrast estimation and improved optical sectioning. (d) Our multislice approach enables 3-D reconstruction using a much reduced number of slices while still maintaining the benefit of incorporating multiple scattering. Reconstruction using only three slices is compared to demonstrate the improved localization capability by our method.

Fig. 4. Effect of particle density on the scattered intensity term |E|2 contribution in the hologram. (a) Contribution is negligible compared to the hologram for low particle densities and becomes gradually important as the particle density increases. (b) The ratio between the total intensity of the hologram and the |E|2 terms for all values of Rg tested in the simulation. For Rg≤0.1, the total intensity of the hologram is at least an order of magnitude larger than the |E|2 term.

Fig. 5. Validation of our multiple-scattering method on large-scale simulation. (a) Convergence properties of the forward model are studied under varying particle densities. Higher-order scattering is generally required for convergence when the object is strongly scattering. In most cases studied, two orders of scattered field sufficiently capture the majority of the contribution. (b) For higher refractive index contrast (δn=0.19), multiple-scattering performs similarly to single-scattering for low concentration (Rg≤0.02), and better than single-scattering for 0.02<Rg≤0.1. Reconstruction fails for very high concentration (Rg>0.1), i.e., when the SNR drops below an empirically chosen value of 1 dB. The error in the predicted versus the ground truth particle concentrations also shows a similar trend. (c) For lower contrast (δn=0.01), multiple scattering contributions are negligible and both methods give similar performance. (d) A 3-D rendering depicting localized particles is shown for δn=0.19 and Rg=0.1. Both methods have similar performance for slices close to the image plane, but our multiple-scattering model performs better at increased depths.

Fig. 6. Reconstruction performance as a function of depth. (a) Segmentation maps of reconstructed slices (zoomed-in 51μm×51μm regions) at different depths (true positive, white; true negative, black; false positive, green; false negative, pink). For object slices close to the hologram, both multiple and single scattering methods provide high accuracy. At larger depths, the accuracy deteriorates for both methods. Our multiple-scattering method performs notably better at larger depths for higher particle densities. (b) The slicewise Dice coefficient plotted as a function of slice depth also indicates that the multiple-scattering model provides improved segmentation accuracy, especially at greater depth. (c) The particle localization accuracy is quantified using the ROC curve. The curves corresponding to the multiple-scattering solutions consistently have larger areas underneath, indicating better localization accuracy as compared to the single-scattering method in all cases studied.

Fig. 7. Experimental validation of our method in large-scale. (a) The multiple-scattering model converges to a lower cost than the single-scattering model for all concentrations indicating better fit to the cost function. (b) The reconstructed particle density follows a trend similar to the simulation where multiple-scattering performs better than the single-scattering method for Rg≤0.1; both methods fail for Rg>0.1. (c) As Rg increases, the hologram gradually resembles speckle patterns, as quantified by the CR.

Fig. 8. A 3-D visualization of the localized particles under different concentrations from our experiment and their 200×200 lateral cross sections at different depths. For low density, both multiple- and single-scattering methods perform similarly. For high density, the underestimation of particles from the single-scattering method is clearly visible, especially at increased depth. Our multiple-scattering model mitigates the underestimation as it accounts for the intercoupling between particles whose strength increases as the depth. The traditional BPM is effective for low density but completely fails for high density and the reconstruction resembles speckles throughout the volume.