离散坐标法在计算生物组织内光场空间角分布中的应用

从辐射传输理论出发,研究了准直光照下层状生物组织内漫射光场的角分布.在辐射传输方程的基础上,采用离散坐标法得到了描述层状生物组织内漫射光传输问题的微分方程组形式,并用特征值-特征矢量方法对其进行了求解,给出了通解形式.结合边界条件对两类典型生物组织内漫射光场角分布进行了数值计算-各向同性组织和前向散射组织,给出了组织内不同深度处漫射光场空间角分布曲线.通过对计算结果比较分析,得到了生物组织内漫射光场空间角分布随深度的变化规律,及边界效应和光学参量对组织内漫射光场空间角分布的影响.     

Direct solution of the radiative transfer equation for plane-parallel atmospheres

A method is described for solving the monochromatic radiative transfer equation for the case of inhomogeneous, plane-parallel scattering and absorbing atmospheres illuminated by external as well as internal sources. The solution procedure, which is based on a series expansion of the radiation intensity with respect to the angular and spatial coordinates, is analytical in nature and can thus be implemented on small computing facilites. Test calculations were performed for isotropic and Rayleigh scattering atmospheres of various optical thicknesses and single scattering albedos. The results coincide well with data from other methods given in the literature.     

Hybrid finite volume/ finite element method for radiative heat transfer in graded index media

The rays propagate along curved path determined by the Fermat principle in the graded index medium. The radiative transfer equation in graded index medium (GRTE) contains two specific redistribution terms (with partial derivatives to the angular coordinates) accounting for the effect of the curved ray path. In this paper, the hybrid finite volume with finite element method (hybrid FVM/FEM) (P.J. Coelho, J. Quant. Spectrosc. Radiat. Transf., vol. 93, pp. 89–101, 2005) is extended to solve the radiative heat transfer in two-dimensional absorbing-emitting-scattering graded index media, in which the spatial discretization is carried out using a FVM, while the angular discretization is by a FEM. The FEM angular discretization is demonstrated to be preferable in dealing with the redistribution terms in the GRTE. Two stiff matrix assembly schemes of the angular FEM discretization, namely, the traditional assembly approach and a new spherical assembly approach (assembly on the unit sphere of the solid angular space), are discussed. The spherical assembly scheme is demonstrated to give better results than the traditional assembly approach. The predicted heat flux distributions and temperature distributions in radiative equilibrium are determined by the proposed method and compared with the results available in other references. The proposed hybrid FVM/FEM method can predict the radiative heat transfer in absorbing-emitting-scattering graded index medium with good accuracy.     

Generalized form of the direct and adjoint radiative transfer equations

The main goal of this paper is to give a rigorous derivation of the generalized form of the direct (also referenced as forward) and adjoint radiative transfer equations. The obtained expressions coincide with expressions derived by Ustinov [Adjoint sensitivity analysis of radiative transfer equation: temperature and gas mixing ratio weighting functions for remote sensing of scattering atmospheres in thermal IR. JQSRT 2001;68:195-211]. However, in contrast to [Ustinov EA. Adjoint sensitivity analysis of radiative transfer equation: temperature and gas mixing ratio weighting functions for remote sensing of scattering atmospheres in thermal IR. JQSRT 2001;68:195-211] we formulate the generalized form of the direct radiative transfer operator fully independent from its adjoint. To illustrate the application of the derived adjoint radiative transfer operator we consider the angular interpolation problem in the framework of the discrete ordinate method widely used to solve the radiative transfer equation. It is shown that under certain conditions the usage of the solution of the adjoint radiative transfer equation for the angular interpolation of the intensity can be computationally more efficient than the commonly used source function integration technique.     

Frequency domain optical tomography using a conjugate gradient method without line search

A conjugate gradient method without line search (CGMWLS) is presented. This method is used to retrieve the local maps of absorption and scattering coefficients inside the tissue-like test medium, with the synthetic data. The forward problem is solved with a discrete-ordinates finite-difference method based on the frequency domain formulation of radiative transfer equation. The inversion results demonstrate that the CGMWLS can retrieve simultaneously the spatial distributions of optical properties inside the medium within a reasonable accuracy, by reducing cross-talk between absorption and scattering coefficients.     

多维梯度折射率介质内辐射传递的扩散近似

推导多维梯度折射率介质内稳态辐射传递的扩散近似方程.使用有限元法对扩散近似进行离散和求解,利用两个二维半透明介质的稳态辐射传递问题验证该扩散近似的精度及适用性.算例考虑介质为均匀折射率及梯度折射率两种情况.利用扩散近似分别求解辐射平衡时的边界热流、介质内温度场分布,并与辐射传递方程的求解结果进行对比分析.结果表明:介质折射率变化、散射特性、光学厚度及散射反照率均直接影响扩散近似的精度;在光学厚及强散射条件下,该扩散近似可以作为一种快速算法应用于梯度折射率介质稳态辐射传递的求解.     

Complete matrix solution of radiative transfer equation for PILE of horizontally homogeneous slabs

This article covers the analytical solution of the discretized radiative transfer equation in the matrix form. The equation is discretized according to the discrete ordinates method. The solution is based on the representation of the light field in a scattering medium as a superposition of an anisotropic and a smooth regular parts. The first of them is calculated analytically using the smoothness of the solution angular spectrum. The regular part is obtained from a radiative transfer equation boundary problem with the anisotropic part as a source function by discrete ordinates method with a scaling transformation and a matrix-operator method applied. There is no limitation of the scattering law in a medium.     

Radiative transfer with internal reflection via the converged discrete ordinates method

In response to the challenge of establishing highly accurate solutions to the plane layer radiative transfer equation with the simplest of methods, the converged discrete ordinates method is presented. With this algorithm of only finite difference, quadrature and acceleration, we show how to obtain highly accurate intensities for radiative transfer in a finite layer with internal surface reflection. The method features angular smoothing and angular interpolation through “faux” quadrature. In addition, a manufactured solution demonstrates the high accuracy of the method for forward peaked scattering. We consider scattering in a heterogeneous medium as a final demonstration.     

Collision cascade evolution in media with energy independent scattering: spatial,energy, and angular distribution

The evolution of an ion induced collision cascade in a solid medium is studied by means of a DPl-approximation to the linear transport equation. Infinite medium and half space geometries are considered. Special attention is given to the effect of the anisotropy of the energy independent scattering cross section. We present results on the spatial distribution of particles moving at different energies, and the energy and angle distribution at the target surface. The spatial distributions are found to obey simple scaling laws; the energy and angular distributions are independent of the form of the scattering cross section, unless it is very strongly forward peaked.     

光学厚度对颗粒媒质的多重散射的影响

基于辐射传播方程及多重散射基础理论,利用Fraunhofer近似,对颗粒媒质的多重散射光强进行了计算。研究了光学厚度对多重散射的影响,揭示了多重光散射的角分布特征,光强大小随光学厚度的变化,以及单散射引起的误差等方面的规律。为实际的颗粒媒质的光学测量等提供了理论根据。     

Comparison of the radiative transfer equations for constant and spatially varying refraction

The radiative transfer equation for scattering media with constant refraction index (RTE) and the radiative transfer equation for scattering media with spatially varying refraction index (RTEvri) are compared by using the principle of conservation of energy. It is shown that the RTEvri, not only accounts for the spatial variations of refraction index, but also contains a term that accounts for the divergence of the rays. The latter term is missing in the RTE. A corrected RTE is proposed.     

Transmission of radiation through an aerosol medium

The properties of radiation through an aerosol medium have been achieved. This has been done by employing Mie scattering theory to calculate the radiation transfer scattering parameters in the form of extinction, absorption and scattering efficiencies. The equation of radiative transfer for the heat flux through a plane parallel atmosphere of aerosol has been solved. The aerosol size distributions are found in practical systems. Average efficiencies over size distribution for spherical particles of complex refractive index are calculated. Therefore, the radiative properties of stratospheric aerosols have been done. The obtained results found to be in a good agreement with the previous work.     

Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation

The radiative transfer equation can be utilized in optical tomography in situations in which the more commonly applied diffusion approximation is not valid. In this paper, an image reconstruction method based on a frequency domain radiative transfer equation is developed. The approach is based on a total variation output regularized least squares method which is solved with a Gauss–Newton algorithm. The radiative transfer equation is numerically solved with a finite element method in which both the spatial and angular discretizations are implemented in piecewise linear bases. Furthermore, the streamline diffusion modification is utilized to improve the numerical stability. The approach is tested with simulations. Reconstructions from different cases including domains with low-scattering regions are shown. The results show that the radiative transfer equation can be utilized in optical tomography and it can produce good quality images even in the presence of low-scattering regions.     

A probabilistic study of the influence of parameter uncertainty on solutions of the radiative transfer equation

The influence of uncertainty in the absorption and scattering coefficients on the solution and associated parameters of the radiative transfer equation is studied using polynomial chaos theory. The uncertainty is defined by means of uniform and log-uniform probability distributions. By expanding the radiation intensity in a series of polynomial chaos functions we may reduce the stochastic transfer equation to a set of coupled deterministic equations, analogous to those that arise in multigroup neutron transport theory, with the effective multigroup transfer scattering coefficients containing information about the uncertainty. This procedure enables existing transport theory computer codes to be used, with little modification, to solve the problem. Applications are made to a transmission problem and a constant source problem in a slab. In addition, we also study the rod model for which exact analytical solutions are readily available. In all cases, numerical results in the form of mean, variance and sensitivity are given that illustrate how absorption and scattering coefficient uncertainty influences the solution of the radiative transfer equation.     

Least-squares finite element formulations for one-dimensional radiative transfer

We present least-squares-based finite element formulations for the numerical solution of the radiative transfer equation in its first-order primitive variable form. The use of least-squares principles leads to a variational unconstrained minimization problem in a setting of residual minimization. In addition, the resulting linear algebraic problem will always have a symmetric positive definite coefficient matrix, allowing the use of robust and fast iterative methods for its solution. We consider space-angle coupled and decoupled formulations. In the coupled formulation, the space-angle dependency is represented by two-dimensional finite element expansions and the least-squares functional minimized in the continuous space-angle domain. In the decoupled formulation the angular domain is represented by discrete ordinates, the spatial dependence represented by one-dimensional finite element expansions, and the least-squares functional minimized continuously in space domain and at discrete locations in the angle domain. Numerical examples are presented to demonstrate the merits of the formulations in slab geometry, for absorbing, emitting, anisotropically scattering mediums, allowing for spatially varying absorption and scattering coefficients. For smooth solutions in space-angle domain, exponentially fast decay of error measures is demonstrated as the p-level of the finite element expansions is increased. The formulations represent attractive alternatives to weak form Galerkin finite element formulations, typically applied to the more complicated second-order even- and odd-parity forms of the radiative transfer equation.     

An inverse source problem in radiative transfer

The spherical-harmonics method is used to develop a solution to an inverse source problem in radiative transfer. It is assumed that, with the exception of the inhomogeneous source term, all aspects of the radiation-transport problem are known, and we seek to determine the inhomogeneous source term from specified angular distributions of radiation exiting the two surfaces of a homogeneous plane-parallel medium. Anisotropic scattering is included in the monochromatic radiative-transfer model and general reflecting boundary conditions are considered.     

Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model

The coupled radiative transport-diffusion model can be used as light transport model in situations in which the diffusion equation is not a valid approximation everywhere in the domain. In the coupled model, light propagation is modelled with the radiative transport equation in sub-domains in which the approximations of the diffusion equation are not valid, such as within low-scattering regions, and the diffusion approximation is used elsewhere in the domain. In this paper, an image reconstruction method for diffuse optical tomography based on using the coupled radiative transport-diffusion model is developed. In the approach, absorption and scattering distributions are estimated by minimising a regularised least-squares error between the measured data and solution of the coupled model. The approach is tested with simulations. Reconstructions from different cases including domains with low-scattering regions are shown. The results show that the coupled radiative transport-diffusion model can be utilised in image reconstruction problem of diffuse optical tomography and that it produces as good quality reconstructions as the full radiative transport equation also in the presence of low-scattering regions.     

Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems

In graded index medium, the ray goes along a curved path determined by Fermat principle, and the curved ray-tracing is very difficult and complex. To avoid the complicated and time-consuming computation of curved ray trajectory, the methods not based on ray-tracing technique need to be developed for the solution of radiative transfer in graded index medium. For this purpose, in this paper the streaming operator along a curved ray trajectory in original radiative transfer equation for graded index medium is transformed and expressed in spatial and angular ordinates and the radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems are derived. The conservative and the non-conservative forms of radiative transfer equation for three-dimensional graded index medium are given, which can be used as base equations to develop the numerical simulation methods, such as finite volume method, discrete ordinates method, and finite element method, for radiative transfer in graded index medium in cylindrical and spherical coordinate systems.     

Optical properties of pigmented coatings taking into account particle interactions

A T-matrix approach is used to obtain the orientation-averaged scattering and absorption cross sections of randomly oriented particle clusters, and the average angular distribution of the radiation scattered by them. The coefficients involved in the expansion of the phase function are obtained from this T-matrix approach, and used in a multiple scattering formalism to characterize the angular distribution of the diffuse radiation propagating through a particulate coating perpendicularly illuminated with collimated visible radiation. Asymmetry between forward and backward propagating diffuse radiation intensities is taken into account by means of this multiple scattering approach, which is based on solving the radiative transfer equation for successive scattering order contributions. A four-flux model is applied to compute the reflectance in terms of wavelength of the incident radiation and particle concentration. An application of the formalism is carried out to predict the optical properties of titanium dioxide pigmented polymer coatings, in terms of the pigment volume fraction and the degree of aggregation.     

Parallel Jacobian-free Newton Krylov solution of the discrete ordinates method with flux limiters for 3D radiative transfer

The present study introduces a parallel Jacobian-free Newton Krylov (JFNK) general minimal residual (GMRES) solution for the discretized radiative transfer equation (RTE) in 3D, absorbing, emitting and scattering media. For the angular and spatial discretization of the RTE, the discrete ordinates method (DOM) and the finite volume method (FVM) including flux limiters are employed, respectively. Instead of forming and storing a large Jacobian matrix, JFNK methods allow for large memory savings as the required Jacobian-vector products are rather approximated by semiexact and numerical formulations, for which convergence and computational times are presented. Parallelization of the GMRES solution is introduced in a combined memory-shared/memory-distributed formulation that takes advantage of the fact that only large vector arrays remain in the JFNK process. Results are presented for 3D test cases including a simple homogeneous, isotropic medium and a more complex non-homogeneous, non-isothermal, absorbing–emitting and anisotropic scattering medium with collimated intensities. Additionally, convergence and stability of Gram–Schmidt and Householder orthogonalizations for the Arnoldi process in the parallel GMRES algorithms are discussed and analyzed. Overall, the introduction of JFNK methods results in a parallel, yet scalable to the tested 2048 processors, and memory affordable solution to 3D radiative transfer problems without compromising the accuracy and convergence of a Newton-like solution.