Efficient equation-solving integral equation method based on the radiation distribution factor for calculating radiative transfer in 3D anisotropic scattering medium

High-directional resolution radiation intensity (RI) can provide substantial measurement information inside the participating medium, such as the distribution of temperature and physical properties. Therefore, efficiently and accurately solving the radiative transfer equation (RTE) to obtain RI in any direction is the key and challenge of target-detection and inverse-radiation problems. In our previous works [1,2], the integral equation method based on the radiation distribution factor (RDFIEM) was proposed to accurately obtain an arbitrary directional RI. To overcome the inefficiency of the RDFIEM in building a radiation distribution factor (RDF) database from the time-consuming reverse Monte Carlo (RMC) method, a method of equation-solving RDFIEM (ES-RDFIEM) was improved and developed to obtain the solution of RTE in a three-dimensional (3D) anisotropic scattering medium in this work, which can effectively avoid the stochastic ray-tracing process of the traditional RMC method and obtain the value of RDF by solving linear equations directly. The mathematical principles and formulae of ES-RDFIEM are introduced and deduced in detail, whose core idea is to suppose a specified element with a unit blackbody emission, whereas the remaining elements have no energy emission. Subsequently, the linear equations for the RDF, which are only concerned with the physical properties and geometric factors of the radiative system, can be constructed and solved. The computational accuracy and efficiency of ES-RDFIEM and RMC in several cases with different parameters were comprehensively compared. The results showed an excellent agreement between the RDF values and the RI calculated by the two methods. The computational efficiency of ES-RDFIEM was significantly improved compared to RMC, which was almost unaffected by radiation properties.