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.     

快速多极边界元法用于扩散光学断层成像研究

在求解扩散光学断层成像中的正向问题时, 目前普遍采用有限元法, 但是随着实际模型规模的增大, 有限元法的计算量问题日益显著, 而边界元法则由于可以降低计算维度使计算量减少而备受关注. 本文以均匀的高散射介质为模型, 研究了将快速多极边界元法用于扩散光学断层成像的正向问题. 快速多极边界元法利用核函数的多极展开, 将常规边界元法中系数矩阵和迭代矢量的乘积项等价为相应四叉树结构的一次递归, 再结合广义最小残量法进行迭代求解. 将计算结果和蒙特卡罗法的模拟结果进行了比较, 表明利用快速多极边界元法的模拟结果和蒙特卡罗法的结果有很好的一致性. 研究结果验证了快速多极边界元法可以用于扩散光学断层成像, 为其大规模和实时成像带来可观的前景. 关键词： 扩散光学断层成像 边界元法 快速多极边界元法     

等离子体中Fokker-Planck方程的有限元模拟

利用有限元方法设计了一套相对简单明了的求解Fokker-Planck方程的方案.这个方案不必严格限制计算格点的步长和时间步长,就可以确保分布函数的非负性和粒子数的守恒.通过一维程序模拟,进一步证实了这个方案的可靠性.对于多维问题的分析和一维问题完全一样,所以非常容易将其推广到多维问题.     

The operator method for solving the fractional Fokker-Planck equation

The operator method has been used to solve the fractional Fokker-Planck equation (FPE) which recently formulated as a model for the anomalous transport process. Two classes of special interest of fractional F-P equations coming from plasma physics and charged particle transport problem has been considered. It is shown that the mean square-displacement 〈x²(t)〉 satisfy the universal power law characterized the anomalous time evolution i.e. .     

The Fokker-Planck equation in the second-order pitch angle approximation and its exact solution

The diffusive particle propagation and its pitch angle scattering is studied using kinetic equation of the Fokker-Planck form. The case is considered when charged particles preferable propagate along the strong mean magnetic field direction and undergo the pitch angle scattering with respect to it. The paper deals with solution of the equation for particle distribution function in the second-order approximation in the pitch angle. The exact analytical solution is obtained in an integral form. The well-known solution in the first-order pitch angle approximation can be restored performing the small time limit in the result. Unlike the first-order solution the obtained solution in the second approximation rightly shows that the pitch angle diffusion is closely connected with the particle transport along the mean magnetic field. The expression for particle density for the point instantaneous unidirectional source also has been obtained.     

Eigentheory of the inhomogeneous Fokker-Planck equation

Ambiguities that occur in the existing eigentheory of the inhomogeneous Fokker-Planck equation are resolved. The eigenfunction expansion is shown to be identical to the known exact solution, generalizing an earlier result for the space-homogeneous case.Work partially supported by the NSF.     

融合结构先验信息的稳态扩散光学断层成像重建算法研究

扩散光学断层成像作为一种无辐射损伤、低成本的光学在体成像技术,有着良好的应用前景,但具有空间分辨率低、难以定量的缺陷.为了提高扩散光学断层成像的分辨率,实现光学参数分布的精确重建,基于有限元方法,提出了融合结构先验信息的稳态扩散光学断层成像重建算法.该算法以扩散近似作为成像模型,通过软先验的Laplace 正则化方法引入由MicroCT提供的空间结构信息.采用伴随法计算Jacobian矩阵,Levenberg-Marquardt方法用来进行迭代优化.仿真结果表明该算法不仅能获得精确的光学参数值分布,而且显著地提高了迭代收敛的速度.     

The Fokker-Planck equation for arbitrary nonlinear noise

We present the Fokker-Planck equation for arbitrary nonlinear noise terms. The white noise limit is taken as the zero correlation time limit of the Ornstein-Uhlenbeck process. The drift and diffusion coefficients of the Fokker-Planck equation are given by triple integrals of the fluctuations. We apply the Fokker-Planck equation to the active rotator model with a fluctuating potential barrier which depends nonlinearly on an additive noise. We show that the nonlinearity may be transformed into the correlation of linear noise terms.     

Estimation of optical abnormalities in breast phantom by diffuse equation

Optical imaging is an emerging method for bio-imaging. The advantages of this imaging provide non-ionizing and safe radiation, non-invasive and functional medical imaging. Due to diffusion of photons inside biological tissues, its image processing is complicated. So in spite of these advantages, this imaging method has not been progressed like ultrasound imaging and magnetic resonance imaging (MRI). Also, the penetration depth of photons inside biological phantom is low. To overcome this problem, the complicated diffusion of photons through tissue must be modeled. The diffuse equation can be applied to simulate photons through turbid media like biological tissues. In this paper, the diffuse equation is used to study propagation of diffuse photons. The green function method is applied to solve this equation, and then the optical properties of abnormalities in breast phantom are estimated. This fast method can be applied for image processing.     

Half-range completeness for the Fokker-Planck equation with an external force

This paper deals with a generalization of a classical result obtained by R. Beals and V. Protopopescu for the Fokker-Planck equations to the case in which a constant external force is present.     

Exact solutions of a Fokker-Planck equation

Exact explicit solutions are given for a one-dimensional Fokker-Planck equation with a particular potential form involving hyperbolic functions. This potential contains four arbitrary parameters that can be chosen so that the potential is bistable. The solutions also contain parameters that can be chosen so that the initial distribution is approximately Gaussian, centered either at the unstable potential maximum or in the neighborhood of the secondary minimum. The use of the solutions to approximate solutions for other potentials is considered.     

Solution of the Fokker-Planck equation for the shock wave problem

An eigenexpansion solution of the time-independent Brownian motion Fokker-Planck equation is given for a situation in which the external acceleration is a step function. The solution describes the heavy-species velocity distribution function in a binary mixture undergoing a shock wave, in the limit of high dilution of the heavy species and negligible width of the light-gas internal shock. The diffusion solution is part of the eigenexpansion. The coefficients of the series of eigenfunctions are obtained analytically with transcendentally small errors of order exp(–1/M), whereM 1 is the mass ratio. Comparison is made with results from a hypersonic approximation.     

A least square finite element formulation of the collimated irradiation in frequency domain for optical tomography applications

This paper presents an extension of the least square finite element formulation associated to the discrete ordinates method to solve collimated irradiation problems in frequency domain. The features of the method are shown with a separation of the intensity into its collimated and scattered parts for a better handling of discontinuities due to the boundary conditions of Dirichlet type used in optical tomography applications. Numerical tests are used to gauge the accuracy of the model in both isotropic and anisotropic scattering media, with and without frequency modulation. The results show that the method is accurate compared to some reference solutions.     

大口径光学元件环境热稳定性的有限元分析

 讨论了靶场光学元件在环境热载荷作用下的变形分析理论和数学描述,采用有限元分析软件ANSYS建立了靶场反射镜的模型,用靶场实测环境温度变化作为载荷,计算得到了反射镜在靶场温度变化0.3 ℃时,垂直镜面方向的变形及其在平行于镜面平面内的转角漂移。结果表明：在当前的温控条件下,光学元件在环境热载荷作用下的变形满足稳定性设计要求。并计算了几种环境温度变化下反射镜的变形和转角漂移。初步的结果表明：环境温度变化与反射镜的转角漂移成正比。     

Numerical solution of the Fokker-Planck equation with variable coefficients

We study the numerical solution of the Fokker-Planck equation. This equation gives a good approximation to the radiative transport equation when scattering is peaked sharply in the forward direction which is the case for light propagation in tissues, for example. We derive first the numerical solution for the problem with constant coefficients. This numerical solution is constructed as an expansion in plane wave solutions. Then we extend that result to take into account coefficients that vary spatially. This extension leads to a coupled system of initial and final value problems. We solve this system iteratively. Numerical results show the utility of this method.     

Three-dimensional diffuse optical imaging of hand joints: System description and phantom studies

In this paper we describe a three-dimensional (3D) continuous wave (CW) diffuse optical tomography (DOT) system and present 3D volumetric reconstruction studies using this DOT system with simple phantom models that simulate hand joints. The CCD-based DOT system consists of 64×64 source/detector fiber optic channels, which are arranged in four layers, forming a cylindrical fiber optic/tissue interface. Phantom experiments are used to evaluate system performance with respective to axial spatial resolution, optical contrast and target position for detection of osteoarthritis where cartilage is the primary target region of interest. These phantom studies suggest that we are able to quantitatively resolve a 2 mm thick “cartilage” and qualitatively resolve a 1 mm thick “cartilage” using our 3D reconstruction approach. Our results also show that optical contrast of 3:1–7:1 between the “disease cartilage” and normal cartilage can be quantitatively recovered. Finally, the target position along axial direction on image reconstruction is studied. All the images are obtained using our 3D finite-element-based reconstruction algorithm.     

Finite element method for the two-dimensional atmospheric radiative transfer

The finite element method is applied to the solution of the two-dimensional atmospheric radiative transfer. The analysis is mainly focussed on the derivation of the cell or element equation. The Galerkin method and several hybrid methods using the integral and finite difference form of the radiative transfer equation are employed to obtain the cell equation. The assembled system of equations relating the radiances at the lower and upper boundary of the domain is solved by a direct method.     

On the solution of Fokker-Planck equation for electron transport

Two different techniques have been used to solve the Fokker-Planck equation for electron transport in infinite homogeneous medium namely, maximum entropy and flux-limited approach. The solutions obtained for the scalar flux function φ₀(x,s) by both methods are numerically compared.     

Solvable models of the Fokker-Planck equation: An approach based on the Gel'fand-Levitan method

From a given solvable Fokker-Planck equation one can construct a number of other solvable models for diffusion in a stable or bistable potential fields using the Gel'fand-Levitan method of the inverse scattering theory. The simplest way of achieving this is to change the lowest eigenvalue and/or the normalization of the lowest eigenfunction of the ordinary differential equation obtained by separating the time-dependent part. For these cases it is shown that the new probability distribution is expressible in terms of integrals involving the original probability distribution and the Gel'fand-Levitan kernel. The possibility of changing the lowest eigenvalue enables one to find bistable potential fields which would correspond to a well-defined long time relaxation rate for the probability current.     

The kinetic boundary layer for the Fokker-Planck equation with absorbing boundary

The Fokker-Planck equation for the distribution of position and velocity of a Brownian particle is a particularly simple linear transport equation. Its normal solutions and an apparently complete set of stationary boundary layer solutions can be determined explicitly. By a numerical algorithm we select linear combinations of them that approximately fulfill the boundary condition for a completely absorbing plane wall, and that approach a linearly increasing position space density far from the wall. Various aspects of these approximate solutions are discussed. In particular we find that the extrapolated asymptotic density reaches zero at a distance x_M beyond the wall. We find x_M=1.46 in units of the velocity persistence length of the Brownian particle. This study was motivated by certain problems in the theory of diffusion-controlled reactions, and the results might be used to test approximate theories employed in that field.