André Liemert

For over thirty years, the definition of the diffusion coefficient in light transport has been the subject of persistent debate. Its canonical expression includes an explicit dependence on absorption, violating a fundamental scaling property of the radiative transfer equation (RTE). In the time domain (TD), evidence shows that the correct definition is independent of absorption, yet absorption-dependent formulations unfortunately remain common in practical data analysis. In the continuous-wave regime, the radial decay of the RTE can be reproduced by an absorption-dependent coefficient, but this does not imply that steady-state transport is genuinely diffusive. Here, this controversy is resolved through an exact consistency test based solely on the requirement that diffusion converges to the RTE in the limit of infinite propagation. In the TD, we prove that the absorption-independent coefficient is both necessary and sufficient for exact long-time convergence. In the continuous-wave regime, we prove analytically that no scalar diffusion coefficient can make diffusion asymptotically equivalent to the RTE when absorption is nonzero: matching the dominant attenuation length fixes one coefficient, whereas matching the residue requires a different one. We further show that this failure is not caused by any finite set of ballistic or few-scattering contributions, and quantify the intrinsic error introduced by incorrect parameterizations in idealized absorption-retrieval scenarios. These results establish the time-domain formulation as the only self-consistent reference and call for a re-examination of how diffusion is applied in optical transport studies

In this paper, we investigate a spline collocation in the context of the spatial-temporal frequency domain radiative transfer equation. For this, we exemplarily consider an isotropically scattering slab that is illuminated by a spatially and temporally modulated light beam. The advantage of this approach compared with the often used P _N method is the overall numerical stability, even in the high spatial and temporal frequency range. This is also important when it is intended to reconstruct solutions in the space-time domain under the use of Fourier transforms. Apart from this, we present closed-form solutions for the single-scattered radiance. They can be used to approximate the complete solution for optically thin scattering media and to estimate their sample thickness independently of the optical properties. For the case of a half-space medium, we remark on a resonance effect that can appear under certain conditions on the spatial and temporal modulation frequency. Numerical experiments are carried out for verification and illustration purposes.

In the context of autonomous driving, the augmentation of existing data through simulations provides an elegant solution to the challenge of capturing the full range of adverse weather conditions in training datasets. However, existing physics-based augmentation models typically rely on single scattering approximations to predict light propagation under unfavorable conditions, such as fog. This can prevent the reproduction of important signal characteristics encountered in a real-world environment. Consequently, in this work, Monte Carlo simulations are employed to assess the relevance of multiple-scattered light to the detected LiDAR signal in different types of fog, with scattering phase functions calculated from Mie theory considering real particle size distributions. Bidirectional path tracing is used within the self-developed GPU-accelerated Monte Carlo software to compensate for the unfavorable photon statistics associated with the limited detection aperture of the LiDAR geometry. To validate the Monte Carlo software, an analytical solution of the radiative transfer equation for the time-resolved radiance in terms of scattering orders is derived, thereby providing an explicit representation of the double-scattered contributions. The results of the simulations demonstrate that the shape of the detected signal can be significantly impacted by multiple-scattered light, depending on LiDAR geometry and visibility. In particular, double-scattered light can dominate the overall signal at low visibilities. This indicates that considering higher scattering orders is essential for improving AI-based perception models.

In this work, we introduce a novel variance reduction approach utilising the integral formulation of the radiative transfer equation to calculate the radiance in a planar symmetric slab geometry. Due to its integral nature, our method offers a fundamental advantage over well-established variance reduction methods such as the local estimate technique. As opposed to the local estimate procedure, photons add to the overall radiance not only at specific points of interaction but also throughout each consecutive path element; hence, our variance reduction approach can be thought of as an integral local estimate method. This facilitates a substantial enhancement in statistical efficiency, especially in scenarios where only a small number of scattering events or a high attenuation along the detection paths is to be anticipated. To evaluate the overall performance of the integral approach, we incorporated it into a self-developed GPU-accelerated Monte Carlo software, together with a conventional local estimate implementation adapted to slab geometry for a comprehensive comparison.

Significance: Code verification is an unavoidable step prior to using a Monte Carlo (MC) code. Indeed, in biomedical optics, a widespread verification procedure for MC codes is still missing. Analytical benchmarks that can be easily used for the verification of different MC routines offer an important resource. Aim: We aim to provide a two-step verification procedure for MC codes enabling the two main tasks of an MC simulator: (1) the generation of photons’ trajectories and (2) the intersections of trajectories with boundaries separating the regions with different optical properties. The proposed method is purely based on elementary analytical benchmarks, therefore, the correctness of an MC code can be assessed with a one-sample t-test. Approach: The two-step verification is based on the following two analytical benchmarks: (1) the exact analytical formulas for the statistical moments of the spatial coordinates where the scattering events occur in an infinite medium and (2) the exact invariant solutions of the radiative transfer equation for radiance, fluence rate, and mean path length in media subjected to a Lambertian illumination. Results: We carried out a wide set of comparisons between MC results and the two analytical benchmarks for a wide range of optical properties (from non-scattering to highly scattering media, with different types of scattering functions) in an infinite non-absorbing medium (step 1) and in a non-absorbing slab (step 2). The deviations between MC results and exact analytical values are usually within two standard errors (i.e., t-tests not rejected at a 5% level of significance). The comparisons show that the accuracy of the verification increases with the number of simulated trajectories so that, in principle, an arbitrary accuracy can be obtained. Conclusions: Given the simplicity of the verification method proposed, we envision that it can be widely used in the field of biomedical optics.

In this paper, some explicit analytical solutions for single-scattered radiance in a half-space medium under consideration of a reflecting boundary are derived. We consider both a unidirectional beam source as well as an isotropic point source. In addition to direct applications within optical tomography and computer graphics, the obtained solutions are also needed when solving the radiative transport equation after the separation of the unscattered and single-scattered contribution. Comparisons between the derived analytical solutions and the Monte Carlo method display excellent agreement.

Accurate and efficient solutions of the three dimensional radiative transport equation were derived in all domains for the case of layered scattering media. Index mismatched boundary conditions based on Fresnel’s equations were implemented. Arbitrary rotationally symmetric phase functions can be applied to characterize the scattering in the turbid media. Solutions were derived for an obliquely incident beam having arbitrary spatial profiles. The derived solutions were successfully validated with Monte Carlo simulations and partly compared with analytical solutions of the diffusion equation.