Three-Dimensional Optical Diffraction Tomography With Lippmann-Schwinger Model

Abstract

A broad class of imaging modalities involve the resolution of an inverse-scattering problem. Among them, three-dimensional optical diffraction tomography (ODT) comes with its own challenges. These include a limited range of views, a large size of the sample with respect to the illumination wavelength, and optical aberrations that are inherent to the system itself. In this work, we present an accurate and efficient implementation of the forward model. It relies on the exact (nonlinear) Lippmann-Schwinger equation. We address several crucial issues such as the discretization of the Green function, the computation of the far field, and the estimation of the incident field. We then deploy this model in a regularized variational-reconstruction framework and show on both simulated and real data that it leads to substantially better reconstructions than the approximate models that are traditionally used in ODT. IntroductionOptical diffraction tomography (ODT) is a noninvasive quantitative imaging modality [1,2]. This label-free technique allows one to determine a three-dimensional map of the refractive index (RI) of samples, which is of particular interest for applications that range from biology [3] to nanotechnologies [4]. The acquisition setup sequentially illuminates the sample from different angles. For each illumination, the outgoing complex wave field (i.e., the scattered field) is recorded by a digital-holography microscope [5,6]. Then, from this set of measurements, the RI of the sample can be reconstructed by solving an inverse-scattering problem. However, its resolution is very challenging due to the nonlinear nature of the interaction between the light and the sample. Related WorksTo simplify the reconstruction problem, pioneering works focused on linearized models. These include Born [1] and Rytov [7] approximations, which are valid for weakly scattering samples [8]. Although originally used to deploy direct inversion methods, these linearized models have been later combined with iterative regularization techniques to improve their robustness to noise and to alleviate the missing-cone problem [9,10].Nonlinear models that adhere more closely to the physic of the acquisition are needed to recover samples with higher variations of their refractive index. For instance, beam-propagation methods (BPM) [11][12][13][14] rely on a slice-by-slice propagation model that accounts for multiple scatterings within the direction of propagation (no reflection). Other nonlinear models include the contrast source-inversion method [15] or the recursive Born approximation [16]. Although more accurate, all these models come at the price of a large computational cost.The theory of scalar diffraction recognizes the Lippmann-Schwinger (LS) model to be the most faithful. It accounts for multiple scatterings, both in transmission and reflection. Iterative forward models that solve the LS equation have

Dynamical machine learning volumetric reconstruction of objects’ interiors from limited angular views

“…The next level of complexity arises when diffraction and multiple scattering must be taken into account in the forward model; then, the Born or Rytov expansions and the Lippmann-Schwinger integral equation 14 – 18 are more appropriate. These follow from the scalar Helmholtz equation using different forms of expansion for the scattered field 19 .…”

Structured illumination microscopy artifacts caused by illumination scattering

“…Finally, by solving equation (8), the complex amplitude ( ) generated by the sample scattering, and the subsequent total light field ( ) can be obtained. We used the Lippmann-Schwinger model (LS model) [22][23][24] to calculate the scattered field ( ) for the incident illumination in different orientations and phases. The LS model is a superior nonlinear forward model to approximate models such as Born or Rytov, while it can generate accurate estimations.…”

Adaptive Regularization for Three-Dimensional Optical Diffraction Tomography

“…Finally, given u within Ω,G : C N → C M provides the scattered field y sc . We refer the reader to [5] for details concerning the implementation of G andG.…”

“…We first reconstructed the sample using the method described in [4] by minimizing (5). The initial guess was the solution provided by the Fig.…”

“…The reconstruction of the RI consists in solving an inverse scattering problem which is particularly challenging for several reasons. First, wave scattering is a nonlinear phenomenon and the most faithful forward model for ODT is iterative [2][3][4][5]. Second, the wavevector of the incident wave is limited to a small cone around the optical axis ( Figure 1).…”