Markov Chain Investigation of Discretization Schemes and Computational Cost Reduction in Modeling Photon Multiple Scattering

Abstract

Establishing fast and reversible photon multiple scattering algorithms remains a modeling challenge for optical diagnostics and noise reduction purposes, especially when the scattering happens within the intermediate scattering regime. Previous work has proposed and verified a Markov chain approach for modeling photon multiple scattering phenomena through turbid slabs. The fidelity of the Markov chain method has been verified through detailed comparison with Monte Carlo models. However, further improvement to the Markov chain method is still required to improve its performance in studying multiple scattering. The present research discussed the efficacy of non-uniform discretization schemes and analyzed errors induced by different schemes. The current work also proposed an iterative approach as an alternative to directly carrying out matrix inversion manipulations, which would significantly reduce the computational costs. The benefits of utilizing non-uniform discretization schemes and the iterative approach were confirmed and verified by comparing the results to a Monte Carlo simulation.

Numerical Study of Turbid Slab Optical Properties Reconstruction from Multiple Scattering Signals Using Time-Based Markov Chain Model

“…To further demonstrate the error from the Markov chain approximation, we incorporated the relative error of each model, as shown in Figure 3. As we introduced in our previous work [17], the Markov chain model had the worst performance near 90 • , around which the signal was hard to capture by the sensors in the experiments. Therefore, in Figure 3, we just compared the results at 135 • (corresponding to the maximal point in 90 • -180 • ) to demonstrate the error of the time-based model, and we also confirmed that this error was representative of the overall error of the Markov chain models.…”

“…In the equations, P(z m , z n , θ i ) is the probability for the photon that starts its propagation at z m with the propagating direction of θ i and finishes at z n during a specific time step; P(θ i , θ j , n) is the probability for the photon that propagates in the direction of θ i , snth layer, and then propagates in the new direction of θ j ; and Γ n (α) is the phase function in the nth layer. The implications of Equations ( 2)-( 4) have been introduced elsewhere [15][16][17][18], and we will briefly introduce these equations. Equation ( 2) describes the transition probability T from one state (z m , θ i ) to another state (z n , θ j ).…”

“…With the structure above, the time-based Markov chain method can simulate the timeresolved angular distributions for transmitted photons, which is more accessible than counting the scattering order of the photons practically. More mathematical details can be found in [15][16][17][18].…”

“…In previous works, the authors proposed a Markov chain-based model [15][16][17][18] to calculate the transmitted angular distribution through turbid slabs. In this model, multiple scattering was converted into a matrix (transition matrix), which represented the likelihood of a photon to transit from one state (location and propagation angle) to another.…”

A hybrid full-wave Markov chain approach to calculating radio-frequency wave scattering from scrape-off layer filaments

“…In practice, it is often faster to evaluate the first relation and truncate the series at a finite when the solution is sufficiently converged (Yang et al. 2018).…”