一种改进的光学层析图像重建方法

Alexander D．Klose将联合差分方法用于光学层析图像重建的梯度计算中,但给出的对光学参量的求导算法有局限,他的算法只能实现对边界点光学参量的导数计算,而无法实现对内部点光学参量导数的计算,会导致图像重建失败。在联合差分算法的基础上,研究了针对内部点光学参量的求导方法,给出了一种基于树形结构的对内部点光学参量求导的策略。具体实现时,为了降低计算复杂度,采用近似梯度计算方法。算法的仿真实验结果表明：该方法可以有效地实现对内部点光学参量的导数计算,提出的近似计算方法可降低梯度计算复杂度,提高运算速度,并可得到良好的图像重建质量。     

Optical tomography with the discontinuous Galerkin formulation of the radiative transfer equation in frequency domain

Optical tomography is an inverse method of probing semi-transparent media with the help of light sources. The reconstruction of the optical properties usually employs finite volumes or continuous finite elements formulations of light transport as a forward model for the predictions. In a previous study, we have introduced a generalization of the inversion approach with finite elements formulations by using an integral form of the objective function. The novelty is that the surfaces of the detectors are taken into account in the reconstruction and compatibility is obtained for all finite element formulations. This present paper illustrates this new approach by developing a Discontinuous Galerkin formulation as a forward model for an optical tomography application in the frequency domain framework. Numerical tests are performed to gauge the accuracy of the method in recovering optical properties distribution with a gradient-based algorithm where the adjoint method is used to fastly compute the objective function gradient. It is seen that the reconstruction is accurate and can be affected by noise on the measurements as expected. Filtering of the gradient at each iteration of the reconstruction is used to cope with the ill-posed nature of the inverse problem and to improves the quality and accuracy of the reconstruction.     

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.     

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.     

Optical tomography reconstruction based on gradient tree

The adjoint differentiation scheme was first proposed by Alexander D. Klose for the reconstruction of optical tomography [J Q S R T 2002;72:715-32]. However, the scheme proposed there can only be employed for the computation of derivatives with respect to the optical parameters on the boundary, i.e. those derivatives with respect to the internal parameters cannot be obtained there, and as a result, the correct reconstructed result cannot be obtained with the gradient based optimization algorithms. In our paper, based on the adjoint differentiation scheme, a novel algorithm based on gradient tree is proposed for the computation of the derivatives with respect to both boundary and internal optical parameters. A scheme for approximate gradient calculation is also proposed for speeding up algorithm implementation. Experimental results show that the algorithm proposed performs well and good reconstruction results can be achieved.     

Design sensitivity analysis for sequential structural-acoustic problems

A design sensitivity analysis of a sequential structural-acoustic problem is presented in which structural and acoustic behaviors are de-coupled. A frequency-response analysis is used to obtain the dynamic behavior of an automotive structure, while the boundary element method is used to solve the pressure response of an interior, acoustic domain. For the purposes of design sensitivity analysis, a direct differentiation method and an adjoint variable method are presented. In the adjoint variable method, an adjoint load is obtained from the acoustic boundary element re-analysis, while the adjoint solution is calculated from the structural dynamic re-analysis. The evaluation of pressure sensitivity only involves a numerical integration process for the structural part. The proposed sensitivity results are compared to finite difference sensitivity results with excellent agreement.     

FDFD: A 3D Finite-Difference Frequency-Domain Code for Electromagnetic Induction Tomography

A new 3D code for electromagnetic induction tomography with intended applications to environmental imaging problems has been developed. The approach consists of calculating the fields within a volume using an implicit finite-difference frequency-domain formulation. The volume is terminated by an anisotropic perfectly matched layer region that simulates an infinite domain by absorbing outgoing waves. Extensive validation of this code has been done using analytical and semianalytical results from other codes, and some of those results are presented in this paper. The new code is written in Fortran 90 and is designed to be easily parallelized. Finally, an adjoint field method of data inversion, developed in parallel for solving the fully nonlinear inverse problem for electrical conductivity imaging (e.g., for mapping underground conducting plumes), uses this code to provide solvers for both forward and adjoint fields. Results obtained from this inversion method for high-contrast media are encouraging and provide a significant improvement over those obtained from linearized inversion methods.     

一种光学层析图像的多准则重建方法

光学层析成像是一个病态重建问题,为克服重建过程的病态性,提出将多准则优化理论引入到图像重建中。利用了三个用于光学层析图像重建的准则:平方误差函数、图像熵函数和局部平滑函数。采用向量优化方法将多准则优化问题转化为单准则优化问题求解。为了确定各个目标函数间的权重系数,提出一种动态权重系数求解方法。重建过程目标函数关于光学参量的梯度计算是关键,因此提出一种基于梯度树的计算方法。实验过程中对多准则重建结果和基于平方误差函数的单准则重建结果做了比较,证明该方法能够克服传统的偏重单一目标的单准则重建的不足,有效地重建光学层析图像,提高图像重建质量。     

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.     

光脉冲传输数值模拟的分步小波方法

从信号的多尺度小波分解和正交小波变换出发，将描述光学介质中脉冲传输的非线性薛定谔 方程（NLSE）表示为小波域中的分步算符形式，给出了分步小波算法的迭代公式，导出了线 性算符在小波域中的具体表式，并讨论微分算符的矩阵结构.作为一个例子，用分步小波方 法（SSWM）解NLSE，给出了超短高斯脉冲在光纤中线性和非线性传输的波形演化，并与解析 解和分步傅里叶方法的结果作了比较.结果表明，分步小波方法是研究脉冲在光学介质中传 输的一种有效的数值计算方法. 关键词： 分步小波方法 光脉冲传输 非线性薛定谔方程 多尺度小波分解     

Adjoint wall functions: A new concept for use in aerodynamic shape optimization

The continuous adjoint method for the computation of sensitivity derivatives in aerodynamic optimization problems of steady incompressible flows, modeled through the k–ε turbulence model with wall functions, is presented. The proposed formulation leads to the adjoint equations along with their boundary conditions by introducing the adjoint to the friction velocity. Based on the latter, an adjoint law of the wall that bridges the gap between the solid wall and the first grid node off the wall is proposed and used during the solution of the system of adjoint (to both the mean flow and turbulence) equations. Any high Reynolds turbulence model, other than the k–ε one used in this paper, could also profit from the proposed adjoint wall function technique. In the examined duct flow problems, where the total pressure loss due to viscous effects is used as objective function, emphasis is laid on the accuracy of the computed sensitivity derivatives, rather than the optimization itself. The latter might rely on any descent method, once the objective function gradient has accurately been computed.     

光学CT中的图像重建算法

从理论上研究了利用正向模型求解逆问题的迭代算法，特别是其中Ｊａｃｏｂｉ矩阵的有限元解法，引入了图像恢复了领域广泛采用的规则化方法用于处理逆问题中的奇异性问题；作为总结，最后给出了ＦＥＭ法求解光学ＣＴ问题的完整步骤。     

Linearization of radiative transfer with respect to surface properties

The linearization of radiative transfer with respect to surface properties in the UV and visible part of the solar spectrum is presented. The proposed method is a rigorous extension of the radiative perturbation theory with respect to surface properties. Given the forward and adjoint intensity field, analytical expressions are provided for the linearization of any observable related to the radiation field with respect to surface properties characterized by Minnaert's and Lambertian bidirectional reflection distribution function. For the considered surface reflection characteristics, we also discuss an extension of the reduction approach of Chandrasekhar as an alternative linearization method. The suitability of both approaches for the combined retrieval of trace gas and surface properties from the backscattered sunlight in the UV and visible part of the spectrum is discussed. The authors come to the conclusion that the perturbation theory, for this purpose, represents the superior method because of its general applicability to any parameter characterizing the optical properties of the atmosphere and the underlying surface.     

Adjoint-based sensitivity analysis of flames

Simulation of chemically reacting flows using detailed chemistry introduces a large number of chemistry model parameters. While not all significantly affect the target outcomes of a simulation, the parameters that do are not always known a priori. In order to improve simulations for specified target outcomes, termed quantities of interest (QoIs), the sensitivity of these QoIs to the model parameters are needed. However, evaluating the sensitivities is computationally expensive, especially for complex fuels that may involve many parameters. For these simulations, the forward sensitivity method requires the solution of an additional number of governing equations proportional to the number of parameters. Here, an adjoint sensitivity approach is formulated where the computational cost scales as the number of QoIs and not the number of parameters. Specifically, adjoint equations are derived for laminar, incompressible, variable density reacting flow and applied to hydrogen flame simulations. From the solution of the corresponding adjoint equations, sensitivity of the QoIs to chemistry model parameters is calculated. The one-dimensional simulation results show that the adjoint sensitivity results closely match those of forward sensitivity methods, thus providing validation of the adjoint method. The two-dimensional simulation results indicate the most sensitive parameters for two QoIs, flame tip temperature and NOx emission. For these tests, the adjoint method reduces computational expense compared to forward sensitivity methods by a factor proportional to the number of QoIs over the number of parameters, here 2/172. Such savings can be more drastic for cases that involve complex fuels, such as combustion of jet fuel, requiring thousands of chemistry model parameters. Further, this sensitivity information can be used in development of experiments by pointing out which are the critical chemistry model parameters.     

荧光分子断层成像正向问题的并行计算

针对荧光分子断层成像中相应于激发光和发射光的两个正向方程必须串行求解的实际情况,提出了一种可同时对两个扩散方程进行求解的并行算法。其思想是通过引入乘子矩阵对耦合方程进行解耦来实现并行计算,并利用有限元方法进行了二维数值模拟,将算法求解所得结果与基于串行方法,以Ralf B．Schulz等提出的并行算法所得到的数值模拟结果进行了综合比较。实验表明,该算法一方面适合于任何大小的斯托克斯频移条件,具有更广泛的适应性;另一方面提高了荧光分子断层成像正向问题的求解速度和精度,从而有利于整个荧光分子断层成像的快速精确求解。     

Output-based space–time mesh adaptation for the compressible Navier–Stokes equations

This paper presents an output-based adaptive algorithm for unsteady simulations of convection-dominated flows. A space–time discontinuous Galerkin discretization is used in which the spatial meshes remain static in both position and resolution, and in which all elements advance by the same time step. Error estimates are computed using an adjoint-weighted residual, where the discrete adjoint is computed on a finer space obtained by order enrichment of the primal space. An iterative method based on an approximate factorization is used to solve both the forward and adjoint problems. The output error estimate drives a fixed-growth adaptive strategy that employs hanging-node refinement in the spatial domain and slab bisection in the temporal domain. Detection of space–time anisotropy in the localization of the output error is found to be important for efficiency of the adaptive algorithm, and two anisotropy measures are presented: one based on inter-element solution jumps, and one based on projection of the adjoint. Adaptive results are shown for several two-dimensional convection-dominated flows, including the compressible Navier–Stokes equations. For sufficiently-low accuracy levels, output-based adaptation is shown to be advantageous in terms of degrees of freedom when compared to uniform refinement and to adaptive indicators based on approximation error and the unweighted residual. Time integral quantities are used for the outputs of interest, but entire time histories of the integrands are also compared and found to converge rapidly under the proposed scheme. In addition, the final output-adapted space–time meshes are shown to be relatively insensitive to the starting mesh.     

用于频域光学相干层析成像的深度分辨色散补偿方法

提出一种深度分辨的色散补偿方法,用于增强频域光学相干层析成像的纵向分辨率。将频域光学相干层析成像采集的干涉谱信号傅里叶变换到空域,获取对应不同深度的干涉谱条纹相位,通过对其进行调整实现对被测样品不同深度处的色散的精确补偿。避免传统方法中采用统一色散系数进行色散补偿所带来的过补偿与欠补偿误差,可以有效消除色散引起的频域光学相干层析成像系统点扩展函数的展宽和扭曲。模拟和实验结果表明,基于深度分辨的色散补偿方法在样品的全深度探测范围内可以达到较佳的补偿效果,可有效提高光学相干层析成像系统的纵向分辨率。     

基于光强传播方程的相位恢复

对于圆形孔径的光学系统,利用光强传播方程进行相位恢复时很难得到相位的边界径向斜率值,另外,要获得精确的圆域边界采样值也并非易事。为了克服上述困难,提出了一种相位恢复的改进方法,即改变了方程的表示形式、计算区域和边界条件,并用多重网格法进行求解获得重构相位,最后再将其修正。为了对重构相位进行修正,还得到了重构相位与原始相位之间泽尼克系数的传递矩阵。对均匀照明的情形进行了仿真,发现该方法不仅可以避免复杂的边界条件,减少运算时间,而且还能够较好地恢复出原始波前的泽尼克系数,即便在加噪的情况下,修正相位与原始相位的均方根误差也在可以接受的范围内。     

Fast intersections on nested tetrahedrons (FINT): An algorithm for adaptive finite element based distributed parameter estimation

A variety of biomedical imaging techniques such as optical and fluorescence tomography, electrical impedance tomography, and ultrasound imaging can be cast as inverse problems, wherein image reconstruction involves the estimation of spatially distributed parameter(s) of the PDE system describing the physics of the imaging process. Finite element discretization of imaged domain with tetrahedral elements is a popular way of solving the forward and inverse imaging problems on complicated geometries. A dual-adaptive mesh-based approach wherein, one mesh is used for solving the forward imaging problem and the other mesh used for iteratively estimating the unknown distributed parameter, can result in high resolution image reconstruction at minimum computation effort, if both the meshes are allowed to adapt independently. Till date, no efficient method has been reported to identify and resolve intersection between tetrahedrons in independently refined or coarsened dual meshes. Herein, we report a fast and robust algorithm to identify and resolve intersection of tetrahedrons within nested dual meshes generated by 8-similar subtetrahedron subdivision scheme. The algorithm exploits finite element weight functions and gives rise to a set of weight functions on each vertex of disjoint tetrahedron pieces that completely cover up the intersection region of two tetrahedrons. The procedure enables fully adaptive tetrahedral finite elements by supporting independent refinement and coarsening of each individual mesh while preserving fast identification and resolution of intersection. The computational efficiency of the algorithm is demonstrated by diffuse photon density wave solutions obtained from a single- and a dual-mesh, and by reconstructing a fluorescent inclusion in simulated phantom from boundary frequency domain fluorescence measurements.