Optical tomography using the time-independent equation of radiative transfer—Part 2: inverse model

Optical tomography is a novel imaging modality that is employed to reconstruct cross-sectional images of the optical properties of highly scattering media given measurements performed on the surface of the medium. Recent advances in this field have mainly been driven by biomedical applications in which near-infrared light is used for transillumination and reflectance measurements of highly scattering biological tissues. Many of the reconstruction algorithms currently utilized for optical tomography make use of model-based iterative image reconstruction (MOBIIR) schemes. The imaging problem is formulated as an optimization problem, in which an objective function is minimized. In the simplest case the objective function is a normalized-squared error between measured and predicted data. The predicted data are obtained by using a forward model that describes light propagation in the scattering medium given a certain distribution of optical properties.In part I of this two-part study, we presented a forward model that is based on the time-independent equation of radiative transfer. Using experimental data we showed that this transport-theory-based forward model can accurately predict light propagation in highly scattering media that contain void-like inclusions. In part II we focus on the details of our image reconstruction scheme (inverse model). A crucial component of this scheme involves the efficient and accurate determination of the gradient of the objective function with respect to all optical properties. This calculation is performed using an adjoint differentiation algorithm that allows for fast calculation of this gradient. Having calculated this gradient, we minimize the objective function with a gradient-based optimization method, which results in the reconstruction of the spatial distribution of scattering and absorption coefficients inside the medium. In addition to presenting the mathematical and numerical background of our code, we present reconstruction results based on experimentally obtained data from highly scattering media that contain void-like regions. These types of media play an important role in optical tomographic imaging of the human brain and joints.     

An efficient and robust reconstruction method for optical tomography with the time-domain radiative transfer equation

An efficient and robust method based on the complex-variable-differentiation method (CVDM) is proposed to reconstruct the distribution of optical parameters in two-dimensional participating media. An upwind-difference discrete-ordinate formulation of the time-domain radiative transfer equation is well established and used as forward model. The regularization term using generalized Gaussian Markov random field model is added in the objective function to overcome the ill-posed nature of the radiative inverse problem. The multi-start conjugate gradient method was utilized to accelerate the convergence speed of the inverse procedure. To obtain an accurate result and avoid the cumbersome formula of adjoint differentiation model, the CVDM was employed to calculate the gradient of objective function with respect to the optical parameters. All the simulation results show that the CVDM is efficient and robust for the reconstruction of optical parameters.     

Numerical developments for short-pulsed Near Infra-Red laser spectroscopy. Part II: inverse treatment

Indirect optical spectroscopy or tomography, that is, mapping of optical properties in scattering and absorption inside a medium given a set of measurements at the boundaries, is highly dependent on the radiative transfer model used to track radiative energy propagation in semi-transparent materials. In the first part of this study, a numerical tool adapted for treating radiative transfer in the frame of short-pulsed laser beam interaction with non-homogeneous matter has been presented. In this paper, it is intended to show how such numerical tools can undergo inversion through adjoint treatment or reverse differentiation.Adjoint models, as well as reverse differentiation, are used in order to allow an efficient computation of the gradient, in the unknown optical parameters space, of an objective or cost function estimating the residual between data obtained at the boundary and predictions by numerical simulations. This gradient is a crucial indication as to update, through line minimization, the set of internal optical properties of the medium.First, the theoretical background of the inverse treatments, both reverse differentiation and adjoint model, for the transient radiative transfer equation model introduced in Part I is developed. Second, different reconstruction configurations are presented. Time-dependent sampling and time filtering effects of the measurements are addressed. Image reconstructions from simulated data are achieved for material phantoms of simple geometry.     

Accurate reconstruction of the optical parameter distribution in participating medium based on the frequency-domain radiative transfer equation

Reconstructing the distribution of optical parameters in the participating medium based on the frequency-domain radiative transfer equation(FD-RTE) to probe the internal structure of the medium is investigated in the present work.The forward model of FD-RTE is solved via the finite volume method(FVM). The regularization term formatted by the generalized Gaussian Markov random field model is used in the objective function to overcome the ill-posed nature of the inverse problem. The multi-start conjugate gradient(MCG) method is employed to search the minimum of the objective function and increase the efficiency of convergence. A modified adjoint differentiation technique using the collimated radiative intensity is developed to calculate the gradient of the objective function with respect to the optical parameters. All simulation results show that the proposed reconstruction algorithm based on FD-RTE can obtain the accurate distributions of absorption and scattering coefficients. The reconstructed images of the scattering coefficient have less errors than those of the absorption coefficient, which indicates the former are more suitable to probing the inner structure.     

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.     

Estimation of the optical properties of seawater from measurements of exit radiance

An inverse analysis for simultaneous estimation of the radiation phase function, single scattering albedo and optical thickness in natural waters, from the knowledge of the exit radiance measurements, is presented. A forward and an inverse model are utilized in our analysis. The forward model uses an analytical discrete-ordinates method for solving the radiative transfer equation and the inverse model contains an algorithm for least-squares estimation that is iteratively solved for retrieving the desired optical properties. The experimental data are simulated with synthetic data corrupted with noise. The results show that the optical properties, with the exception of the optical thickness, can be recovered with high accuracy, even for data with up to 10% noise.     

Adjoint problem of radiative transfer for a pseudo-spherical atmosphere and general viewing geometries

We formulate the adjoint radiative transfer for a pseudo-spherical atmosphere and various retrieval scenarios. The single scattering radiance is computed in a spherical atmosphere by using the source integration technique, while for the multiple scattering radiance we formulate an one-dimensional adjoint radiative transfer equation in a plane-parallel atmosphere. The adjoint solution of the radiative transfer equation is obtained by employing the discrete ordinate method with matrix exponential. We provide an abbreviated derivation of our formalism as well as a discussion of the numerical implementation of the theory.     

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.     

基于稳态辐射传输方程的小动物尺寸组织体扩散光学层析法

在Newton-Raphson逆模型框架下,发展了基于稳态辐射传输方程光子输运正模型和代数重建技术的扩散光学层析图像重建算法,并针对小尺寸组织体内部吸收和散射系数同时重建情况下的标准代数重建技术算法进行了改进,数值结果表明改进后的代数重建技术算法重建图像效果优于标准代数重建技术算法.     

Forward Monte Carlo computations of fully polarized microwave radiation in non-isotropic media

A completely forward Monte Carlo radiative transfer code has been developed with biasing techniques to efficiently solve the polarized radiative transfer equation for the full Stokes vector. The code has been adapted to accommodate plane parallel/3-D vertically/horizontally inhomogeneous scattering atmospheres in Cartesian geometries. Particular attention has been paid in stochastically treating the propagation, the emission and the scattering through anisotropic media particularly suited for clouds containing perfectly or partially oriented particles. Our modelling is very appealing because all its biasing techniques do not introduce unphysical Stokes vector. Numerical results and comparisons with benchmark tests are presented for verification.     

Noninvasive determination of tissue optical properties based on radiative transfer theory

We present a feasibility study of a new method for determining the tissue optical properties, including the absorption and scattering coefficients and the scattering asymmetry factor. A state-of-the-art radiative transfer model for the coupled air/tissue system, based on rigorous radiative transfer theory, is used in our forward modeling simulations. The concept of the effective photon penetration depth is introduced and used to help determine the depth below, which information about the tissue will not be available through noninvasive imaging of a biological tissue using reflected diffuse light. Simulation results show that for accurate determination of tissue optical properties, one can use radiative transfer theory in conjunction with measurements of reflected radiances as well as other existing techniques.     

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.     

Meshless approach for coupled radiative and conductive heat transfer in one-dimensional graded index medium

A meshless local Petrov-Galerkin (MLPG) approach is employed for solving the coupled radiative and conductive heat transfer in a one-dimensional slab with graded index media. The angular distribution term in discrete ordinate equation of radiative transfer within a one-dimensional graded index slab is discretized by a step scheme, and the meshless approach for radiative transfer is based on the discrete ordinate equation. A moving least-squares approximation is used to construct the shape function. Two particular test cases for coupled radiative and conductive heat transfer within a one-dimensional graded index slab are examined to verify this new approximate method. The temperatures and the radiative heat fluxes are obtained. The results are compared with the other benchmark approximate solutions. By comparison, the results show that the MLPG approach has a good accuracy in solving the coupled radiative and conductive heat transfer in one-dimensional graded index media.     

Radiative properties of scattering and absorbing dense media: theory and experimental study

We investigate the validity of the radiative transfer equation to model transmission of light through an absorbing and scattering medium. Assuming that radiative transfer equation is valid, the inverse scattering problem for non-polarized radiative transfer in one-dimensional absorbing and scattering media is solved using a parameter identification method. We discuss how to identify the albedo, phase function and extinction coefficient of the medium. We present experimental data that confirm that this approach is robust and can be used to make reliable predictions of the behavior of scattering absorbing systems.     

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 through inhomogeneous turbid media: implementation of the adjoint perturbation approach at the first order

Recently, the advantages of application of the perturbation technique which is based on joint use of both the direct and adjoint solutions of the radiative transfer equation to solve and analyze some 1D problems of atmospheric physics has been demonstrated. In this paper this technique is applied to problems of radiative transfer through spatially inhomogeneous scattering and absorbing media. This technique is shown to allow one both to obtain the solution with reasonable accuracy and to get physical insight into the problem under consideration. The accuracy of the perturbation technique is demonstrated through comparison with results from the SHDOM simulation code for one problem of cloud optics.     

Analysis of combined conduction and radiation heat transfer in presence of participating medium by the development of hybrid method

The current study addresses the mathematical modeling aspects of coupled conductive and radiative heat transfer in the presence of absorbing, emitting and isotropic scattering gray medium within two-dimensional square enclosure. A blended method where the concepts of modified differential approximation employed by combining discrete ordinate method and spherical harmonics method, has been developed for modeling the radiative transport equation. The gray participating medium is bounded by isothermal walls of two-dimensional enclosure which are considered to be opaque, diffuse and gray. The effect of various influencing parameters i.e., radiation-conduction parameter, surface emissivity, single scattering albedo and optical thickness has been illustrated. The adaptability of the present method has also been addressed.     

Modeling of ultrashort pulsed laser ablation in water and?biological tissues in cylindrical coordinates

A numerical model combining the ultrafast radiative transfer and the ablation rate equation is proposed to investigate the transient process of plasma formation during laser plasma-mediated ablation of absorbing-scattering media. The focus beam propagation governed by the transient equation of radiative transfer is solved by the transient discrete ordinates method to account for scattering effect. The temporal evolution of the free-electron density governed by the ablation rate equation is calculated using a fourth-order Runge–Kutta method to examine various effects such as the multiphoton, chromophore, and cascade ionizations. The threshold of optical breakdown, the shape and maximum length of plasma growth for ablation in water are predicted by the present model and compared with the existing experimental and numerical data. Good agreements have been found. The dynamic process of plasma formation for ablation in the model skin tissue is simulated. A parametric study with regard to the influences of the ionization energy and the critical free-electron density on the ablation threshold of the tissue is conducted.     

Effect of reflecting modes on combined heat transfer within an anisotropic scattering slab

Under various interface reflecting modes, different transient thermal responses will occur in the media. Combined radiative-conductive heat transfer is investigated within a participating, anisotropic scattering gray planar slab. The two interfaces of the slab are considered to be diffuse and semitransparent. Using the ray tracing method, an anisotropic scattering radiative transfer model for diffuse reflection at boundaries is set up, and with the help of direct radiative transfer coefficients, corresponding radiative transfer coefficients (RTCs) are deduced. RTCs are used to calculate the radiative source term in energy equation. Transient energy equation is solved by the full implicit control-volume method under the external radiative-convective boundary conditions. The influences of two reflecting modes including both specular reflection and diffuse reflection on transient temperature fields and steady heat flux are examined. According to numerical results obtained in this paper, it is found that there exits great difference in thermal behavior between slabs with diffuse interfaces and that with specular interfaces for slabs with big refractive index.     

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.