In [1] an algorithm was proposed, inverse dynamical photon scattering (IDPS), which uses as a forward model the propagation of the optical wave through the sample and the objective lens with the multislice method. By recasting the forward model as an artificial neural network (ANN), an error metric can be chosen, and the derivatives of this metric with respect to the unknown values of the discretized object become available at the low computational cost of one extra pass through the ANN. These gradients are deployed in a derivative-based optimization scheme to retrieve a three-dimensional reconstruction of the specimen; Polak-Ribière conjugate gradients (PRCG) and alternate directions augmented Lagrangian (ADAL) are opted for. IDPS is implemented on the graphics processing unit.

IDPS is verified using the open source data from [2,3], where two stacked 1995 US Air Force resolution test charts, 110 mm apart, were adopted as specimen. This dataset is acquired with a microscope with a numerical aperture (NA) of 0.1. An LED array is placed sufficiently far away to consider each individual LED to illuminate the specimen with a spatially coherent plane wave. Nine bright-field and 284 dark-field images are recorded with LEDs up to 0.44 illumination NA, providing an effective NA of 0.54. Since the central wavelength of the LEDs is 643 nm, the 0.54 NA corresponds to a lateral resolution of 1.20 µm. The resolution of the reconstructions is evaluated as the center-to-center distance of the bars in the smallest element in the resolution target that is still discernible: 2.76 µm for element 4 of group 8, and 1.38 µm for element 4 of group 9.

Given that results are obtained from an initial guess of zero, no preprocessing steps such as light field refocusing are necessary. The free choice of error metric is important as the standard choice of sum of squared differences (SSD) leads to a high amount of resolution-limiting noise, see Fig. 1, and to erratic behavior in the R-factor. The alternative sum of normalized absolute differences (SNAD) yields a better-behaved and lower R-factor, as well as a better resolution (see Fig. 2). This poses a problem for Gerchberg-Saxton-type algorithms since they can be thought of as implicitly minimizing SSD [1].

Estimation of certain nuisance parameters—the focal value and illumination angle in this case—together with the specimen itself, improves the result by decreasing the R-factor and diminishing cross-talk between layers. Combined with a total variation regularization of the object, achieved by the ADAL approach, a reconstruction with the same resolution and considerably smoother and strongly diminished spurious oscillations is obtained, as is shown in Fig. 3. [4]

[1] X. Jiang et al., “Inverse dynamical photon scattering (IDPS): an artificial neural network based algorithm for three-dimensional quantitative imaging in optical microscopy,” Optics Express (2016), accepted.

[2] L. Tian, “3D FPM on LED array microscope,” (2015).

[3] L. Tian and L.Waller, “3D intensity and phase imaging from light field measurements in an LED array microscope,” Optica 2, 104–111 (2015).

[4] The Carl Zeiss Foundation is gratefully acknowledged by all authors. C.T. Koch also acknowledges the DFG (KO 2911/7-1). The authors acknowledge the helpful discussion and the open source data provided by L. Waller and L. Tian, University of California, Berkeley.

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