Spherical Harmonics Approximation Research Articles

Experimental Comparison of Continuous-Wave and Frequency-Domain Fluorescence Tomography in a Commercial Multi-Modal Scanner.

The performance evaluation of a variety of small animal tomography measurement approaches and algorithms for recovery of fluorescent absorption cross section has not been conducted. Herein, we employed an intensified CCD system installed in a commercial small animal CT (Computed Tomography) scanner to compare image reconstructions from time-independent, continuous wave (CW) measurements and from time-dependent, frequency domain (FD) measurements in a series of physical phantoms specifically designed for evaluation. Comparisons were performed as a function of (1) number of projections, (2) the level of preprocessing filters used to improve the signal-to-noise ratio (SNR), (3) endogenous heterogeneity of optical properties, as well as in the cases of (4) two fluorescent targets and (5) a mouse-shaped phantom. Assessment of quantitative recovery of fluorescence absorption cross section was performed using a fully parallel, regularization-free, linear reconstruction algorithm with diffusion approximation (DA) and high order simplified spherical harmonics ( SPN) approximation to the radiative transport equation (RTE). The results show that while FD measurements may result in superior image reconstructions over CW measurements, data acquisition times are significantly longer, necessitating further development of multiple detector/source configurations, improved data read-out rates, and detector technology. FD measurements with SP3 reconstructions enabled better quantitative recovery of fluorescent target strength, but required increased computational expense. Despite the developed parallel reconstruction framework being able to achieve more than 60 times speed increase over sequential implementation, further development in faster parallel acceleration strategies for near-real time and real-time image recovery and more precise forward solution is necessary.

Electron Transport Coefficients and Effective Ionization Coefficients in SF6-O2 and SF6-Air Mixtures Using Boltzmann Analysis

The electron drift velocity, electron energy distribution function (EEDF), density-normalized effective ionization coefficient and density-normalized longitudinal diffusion velocity are calculated in SF6-O2 and SF6-Air mixtures. The experimental results from a pulsed Townsend discharge are plotted for comparison with the numerical results. The reduced field strength varies from 40 Td to 500 Td (1 Townsend=10−17 V·cm2) and the SF6 concentration ranges from 10% to 100%. A Boltzmann equation associated with the two-term spherical harmonic expansion approximation is utilized to gain the swarm parameters in steady-state Townsend. Results show that the accuracy of the Boltzmann solution with a two-term expansion in calculating the electron drift velocity, electron energy distribution function, and density-normalized effective ionization coefficient is acceptable. The effective ionization coefficient presents a distinct relationship with the SF6 content in the mixtures. Moreover, the E/Ncr values in SF6-Air mixtures are higher than those in SF6-O2 mixtures and the calculated value E/Ncr in SF6-O2 and SF6-Air mixtures is lower than the measured value in SF6-N2. Parametric studies conducted on these parameters using the Boltzmann analysis offer substantial insight into the plasma physics, as well as a basis to explore the ozone generation process.

Prediction of the critical reduced electric field strength for carbon dioxide and its mixtures with 50% O2 and 50% H2 from Boltzmann analysis for gas temperatures up to 3500 K at atmospheric pressure

This paper provides theoretical calculations that predict the dielectric breakdown properties of carbon dioxide (CO2) and its mixtures with 50% O2 and 50% H2 for a gas temperature range of 300–3500 K at 0.1 MPa. CO2 is one of the most likely candidates for an environment-friendly arc-quenching medium to replace SF6 in high-voltage circuit breakers. Initially, the electron energy distribution function (EEDF) is derived by solving the Boltzmann equation under the zero-dimensional two-term spherical harmonic approximation. Then the reduced ionization and attachment coefficients are obtained, based on the calculated EEDF. Finally, the critical reduced electric field strength (E/N)cr, which is defined as the value for which total ionization reactions are equal to total attachment reactions, is obtained and analysed. The results demonstrate the superior breakdown properties of a 50% CO2–50% O2 mixture to those of both pure CO2 and 50% CO2–50% H2. Nearly no deviation in (E/N)cr is found in a 50% CO2–50% O2 mixture for gas temperatures up to 2500 K, and although there is clear reduction as the gas temperature is increased further to 3500 K, the value remains higher than that of pure CO2.

Adaptive hp finite element method for fluorescence molecular tomography with simplified spherical harmonics approximation

Recently, the simplified spherical harmonics equations (SPN) model has attracted much attention in modeling the light propagation in small tissue geometries at visible and near-infrared wavelengths. In this paper, we report an efficient numerical method for fluorescence molecular tomography (FMT) that combines the advantage of SPN model and adaptive hp finite element method (hp-FEM). For purposes of comparison, hp-FEM and h-FEM are, respectively applied to the reconstruction process with diffusion approximation and SPN model. Simulation experiments on a 3D digital mouse atlas and physical experiments on a phantom are designed to evaluate the reconstruction methods in terms of the location and the reconstructed fluorescent yield. The experimental results demonstrate that hp-FEM with SPN model, yield more accurate results than h-FEM with diffusion approximation model does. The phantom experiments show the potential and feasibility of the proposed approach in FMT applications.

Hybrid radiosity-SP3 equation based bioluminescence tomography reconstruction for turbid medium with low- and non-scattering regions

To provide an ideal solution for a specific problem of gastric cancer detection in which low-scattering regions simultaneously existed with both the non- and high-scattering regions, a novel hybrid radiosity-SP3 equation based reconstruction algorithm for bioluminescence tomography was proposed in this paper. In the algorithm, the third-order simplified spherical harmonics approximation (SP3) was combined with the radiosity equation to describe the bioluminescent light propagation in tissues, which provided acceptable accuracy for the turbid medium with both low- and non-scattering regions. The performance of the algorithm was evaluated with digital mouse based simulations and a gastric cancer-bearing mouse based in situ experiment. Primary results demonstrated the feasibility and superiority of the proposed algorithm for the turbid medium with low- and non-scattering regions.

Integrative Approaches in Computational Biomedical Imaging 2013

Increasingly wider availability of biomedical imaging modalities, such as X-ray computed tomography (CT), magnetic resonance imaging (MRI), ultrasonic imaging, positron emission tomography (PET), single-photon emission computed tomography (SPECT), and optical imaging, have led to significant progresses in both biomedical research and clinical practice. Advancements in imaging technology and image informatics have facilitated and enabled quantitative understanding of the biological structures, functional organisms, and pathological mechanisms. This special issue presents 14 latest research contributions in computational biomedical imaging. The focuses have been on the computational principles, strategies, methods, and algorithms that create, extract, and understand imaging data of biomedical significance. The first group of papers has showcased the development and applications of rigorous optimization techniques in biomedical imaging. The work based on dual unscented Kalman filtering provides a reliable and efficient platform to understand the BOLD signals from fMRI. In the other paper, the robust discrete particle swarm optimization algorithm is adopted to solve the branch-cut phase unwrapping problem of MRI data. Without the need for a regularization penalty term, a semigreed method for bioluminescence tomography reconstruction is fast and stable for small animal imaging. The second group of papers has addressed image segmentation and tissue characterization problems, ever popular among image analysis researchers. The work of L. Li et al. applies machine learning techniques to characterize lymph node metastasis in gastric cancer from gemstone spectral imaging with much improved outcomes, while G. V. Sanchez. Ferrero et al. make new advances in ultrasonic speckle characterization using generalized Gama distribution and generalized Gama mixture model. In recent years, physically meaningful modeling and simulation have become integral components for model-constrained imaging and model-based image understanding. Thus, the third group of papers has continued the stride in this direction. The propagation of myocardial electrical activation is simulated using a monodomain model, without the need of explicit mesh constraints. In similar spirit, one work performs forward modeling of 3D optical molecular imaging with simplified spherical harmonics approximation using extended finite element method. The final group of papers has highlighted some new imaging modalities and their potential use in clinical practice. The further development of ultrasound-based quantitative imaging of Young's modulus with high accuracy is presented in this issue. And the novel Diffusion Kurtosis Imaging shows that more detailed classification of brain tissues is clinical significance. We appreciate the many high quality submissions to this special issue from leading researchers in computational biomedical imaging. Because of space limitation, we can only accept a small number of most exciting works. We sincerely hope that the remaining interesting research outcomes will be soon published in other venues.

基于变量分离近似稀疏重构和简化球谐近似的生物发光断层成像

基于变量分离近似稀疏重构和简化球谐近似的生物发光断层成像

Doubling–Adding Method for Delta-Four-Stream Spherical Harmonic Expansion Approximation in Radiative Transfer Parameterization

Abstract Though the single-layer solutions have been found for the δ-four-stream spherical harmonic expansion method (SHM) in radiative transfer, there is lack of a corresponding doubling–adding method (4SDA), which enables the calculation of radiative transfer through a vertically inhomogeneous atmosphere with multilayers. The doubling–adding method is based on Chandrasekhar's invariance principle, which was originally developed for discrete ordinates approximation. It is shown that the invariance principle can also be applied to SHM and δ-four-stream spherical harmonic expansion doubling–adding method (δ-4SDA) is proposed in this paper. The δ-4SDA method has been systematically compared to the δ-Eddington doubling–adding method (δ-2SDA), the δ-two-stream discrete ordinates doubling–adding method (δ-2DDA), and δ-four-stream discrete ordinates doubling–adding method (δ-4DDA). By applying δ-4SDA to a realistic atmospheric profile with gaseous transmission considered, it is found that the accuracy of δ-4SDA is superior to δ-2SDA or δ-2DDA, especially for the cloudy/aerosol conditions. It is shown that the relative errors of δ-4SDA are generally less than 1% in both heating rate and flux, while the relative errors of both δ-2SDA and δ-2DDA can be over 6%. Though δ-4DDA is slightly more accurate than δ-4SDA in heating rates, both of them are accurate enough to obtain the cloud-top solar heating. Here δ-4SDA is superior to δ-4DDA in computational efficiency. It is found that the error of aerosol radiative forcing can be up to 3 W m−2 by using δ-2SDA at the top of the atmosphere (TOA); such error is substantially reduced by applying δ-4SDA. In view of the overall accuracy and computational efficiency, δ-4SDA is suitable for application in climate models.

Light transport in turbid media with non-scattering, low-scattering and high absorption heterogeneities based on hybrid simplified spherical harmonics with radiosity model

Modeling light propagation in the whole body is essential and necessary for optical imaging. However, non-scattering, low-scattering and high absorption regions commonly exist in biological tissues, which lead to inaccuracy of the existing light transport models. In this paper, a novel hybrid light transport model that couples the simplified spherical harmonics approximation (SPN) with the radiosity theory (HSRM) was presented, to accurately describe light transport in turbid media with non-scattering, low-scattering and high absorption heterogeneities. In the model, the radiosity theory was used to characterize the light transport in non-scattering regions and the SPN was employed to handle the scattering problems, including subsets of low-scattering and high absorption. A Neumann source constructed by the light transport in the non-scattering region and formed at the interface between the non-scattering and scattering regions was superposed into the original light source, to couple the SPN with the radiosity theory. The accuracy and effectiveness of the HSRM was first verified with both regular and digital mouse model based simulations and a physical phantom based experiment. The feasibility and applicability of the HSRM was then investigated by a broad range of optical properties. Lastly, the influence of depth of the light source on the model was also discussed. Primary results showed that the proposed model provided high performance for light transport in turbid media with non-scattering, low-scattering and high absorption heterogeneities.

Study of the dielectric breakdown properties of hot SF6–CF4 mixtures at 0.01–1.6 MPa

The dielectric breakdown properties of SF6–CF4 mixtures were investigated at different ratios of SF6, 0.01–1.6 MPa, and gas temperatures up to 3000 K. Initially, the equilibrium compositions of SF6–CF4 mixtures were calculated by minimizing the Gibbs free energy under the assumptions of local thermodynamic and chemical equilibrium. Then the electron energy distribution function was obtained based on those data by solving the Boltzmann equation under the zero-dimensional two-term spherical harmonic approximation. Finally, the critical reduced electric field strength (E/N)cr of SF6–CF4 mixtures, which is defined as the value for which total ionization reaction is equal to total attachment reaction, were determined and analyzed. The results confirm the superior breakdown properties of pure SF6 at relatively low gas temperatures. However, for higher gas temperatures (i.e., T > 2200 K at 0.4 MPa), the (E/N)cr in SF6–CF4 mixtures are obviously higher than that in pure SF6 and the values of (E/N)cr increase with the reduction of the ratio of SF6.

A compact frequency-domain photon migration system for integration into commercial hybrid small animal imaging scanners for fluorescence tomography

The work presented herein describes the system design and performance evaluation of a miniaturized near-infrared fluorescence (NIRF) frequency-domain photon migration (FDPM) system with non-contact excitation and homodyne detection capability for small animal fluorescence tomography. The FDPM system was developed specifically for incorporation into a Siemens micro positron emission tomography/computed tomography (microPET/CT) commercial scanner for hybrid small animal imaging, but could be adapted to other systems. Operating at 100 MHz, the system noise was minimized and the associated amplitude and phase errors were characterized to be ±0.7% and ±0.3°, respectively. To demonstrate the tomographic ability, a commercial mouse-shaped phantom with 50 µM IRDye800CW and 68Ga containing inclusion was used to associate PET and NIRF tomography. Three-dimensional mesh generation and anatomical referencing was accomplished through CT. A third-order simplified spherical harmonics approximation (SP3) algorithm, for efficient prediction of light propagation in small animals, was tailored to incorporate the FDPM approach. Finally, the PET-NIRF target co-localization accuracy was analyzed in vivo with a dual-labeled imaging agent targeting orthotopic growth of human prostate cancer. The obtained results validate the integration of time-dependent fluorescence tomography system within a commercial microPET/CT scanner for multimodality small animal imaging.

Solution of the Radiative Transfer Equation in Three-Dimensional Participating Media Using a Hybrid Discrete Ordinates: Spherical Harmonics Method

In this article, a new hybrid solution to the radiative transfer equation (RTE) is proposed. Following the modified differential approximation (MDA), the radiation intensity is first split into two components: a “wall” component, and a “medium” component. Traditionally, the wall component is determined using a viewfactor-based surface-to-surface exchange formulation, while the medium component is determined by invoking the first-order spherical harmonics (P1) approximation. Recent studies have shown that although the MDA approach is accurate over a large range of optical thicknesses, it is prohibitive for complex three-dimensional geometry with obstructions, both from a computational efficiency as well as memory standpoint. The inefficiency stems from the use of the viewfactor-based approach for determination of the wall-emitted component. In this work, instead, the wall component is determined directly using the control angle discrete ordinates method (CADOM). The new hybrid method was validated for both two-dimensional (2D) and three-dimensional (3D) geometries against benchmark Monte Carlo results for gray media in which the optical thickness was varied over a large range. In all cases, the accuracy of the hybrid method was found to be within a few percent of Monte Carlo results, and comparable to the solutions of the RTE obtained directly using CADOM. Finally, the new hybrid method was explored for 3D nongray media in the presence of reflecting walls and various scattering albedos. As a noteworthy advantage, irrespective of the conditions used, it was always found to be computationally more efficient than standalone CADOM and up to 15 times more efficient than standalone CADOM for optically thick media with strong scattering.

Cellular neural network to the spherical harmonics approximation of neutron transport equation in x–y geometry: Part II: Transient simulation

In an earlier paper we utilized a novel method using cellular neural networks (CNNs) coupled with spherical harmonics method to solve the steady state neutron transport equation in x–y geometry. Here, the previous work is extended to the study of time-dependent neutron transport equation. To achieve this goal, an equivalent electrical circuit based on a second-order form of time-dependent neutron transport equation and one equivalent group of neutron precursor density is obtained by the CNN method. The CNN model is used to simulate step and ramp perturbation transients in a typical 2D core.

A boundary element–response matrix method for the multigroup criticality problems in the SP3 approximation

The 3D xyz multigroup BERM-SP3 transport method is here presented. This method, based on the simplified spherical harmonics approximation (SPN), with N=3, and the assumption of linearly anisotropic scattering, makes use of a Response Matrix (RM) solution procedure, coupled with the Boundary Element Method (BEM), by which the SP3 partial differential equations are reduced to a system of boundary integral equations in terms of partial currents. Numerical problems, all endowed with a complete set of data and the reference results obtained by well assessed transport codes such as the discrete ordinate code TORT and the Monte Carlo code MCNP, illustrate the accuracy and efficiency of the method.

Extended Finite Element Method with Simplified Spherical Harmonics Approximation for the Forward Model of Optical Molecular Imaging

An extended finite element method (XFEM) for the forward model of 3D optical molecular imaging is developed with simplified spherical harmonics approximation (SPN). In XFEM scheme of SPN equations, the signed distance function is employed to accurately represent the internal tissue boundary, and then it is used to construct the enriched basis function of the finite element scheme. Therefore, the finite element calculation can be carried out without the time-consuming internal boundary mesh generation. Moreover, the required overly fine mesh conforming to the complex tissue boundary which leads to excess time cost can be avoided. XFEM conveniences its application to tissues with complex internal structure and improves the computational efficiency. Phantom and digital mouse experiments were carried out to validate the efficiency of the proposed method. Compared with standard finite element method and classical Monte Carlo (MC) method, the validation results show the merits and potential of the XFEM for optical imaging.

Integrative Approaches in Computational Biomedical Imaging

Increasingly wider availability of biomedical imaging modalities, such as X-ray computed tomography (CT), magnetic resonance imaging (MRI), ultrasonic imaging, positron emission tomography (PET), single-photon emission computed tomography (SPECT), and optical imaging, have led to significant progresses in both biomedical research and clinical practice. Advancements in imaging technology and image informatics have facilitated and enabled quantitative understanding of the biological structures, functional organisms, and pathological mechanisms. This special issue presents 14 latest research contributions in computational biomedical imaging. The focuses have been on the computational principles, strategies, methods, and algorithms that create, extract, and understand imaging data of biomedical significance. The first group of papers has showcased the development and applications of rigorous optimization techniques in biomedical imaging. The work based on dual unscented Kalman filtering provides a reliable and efficient platform to understand the BOLD signals from fMRI. In the other paper, the robust discrete particle swarm optimization algorithm is adopted to solve the branch-cut phase unwrapping problem of MRI data. Without the need for a regularization penalty term, a semigreed method for bioluminescence tomography reconstruction is fast and stable for small animal imaging. The second group of papers has addressed image segmentation and tissue characterization problems, ever popular among image analysis researchers. The work of L. Li et al. applies machine learning techniques to characterize lymph node metastasis in gastric cancer from gemstone spectral imaging with much improved outcomes, while G. V. Sanchez. Ferrero et al. make new advances in ultrasonic speckle characterization using generalized Gama distribution and generalized Gama mixture model. In recent years, physically meaningful modeling and simulation have become integral components for model-constrained imaging and model-based image understanding. Thus, the third group of papers has continued the stride in this direction. The propagation of myocardial electrical activation is simulated using a monodomain model, without the need of explicit mesh constraints. In similar spirit, one work performs forward modeling of 3D optical molecular imaging with simplified spherical harmonics approximation using extended finite element method. The final group of papers has highlighted some new imaging modalities and their potential use in clinical practice. The further development of ultrasound-based quantitative imaging of Young's modulus with high accuracy is presented in this issue. And the novel Diffusion Kurtosis Imaging shows that more detailed classification of brain tissues is clinical significance. We appreciate the many high quality submissions to this special issue from leading researchers in computational biomedical imaging. Because of space limitation, we can only accept a small number of most exciting works. We sincerely hope that the remaining interesting research outcomes will be soon published in other venues.

Comparison between radiative transfer theory and the simplified spherical harmonics approximation for a semi-infinite geometry

In this study, the third-order simplified spherical harmonics equations (SP3), an approximation of the radiative transfer equation, are solved for a semi-infinite geometry considering the exact simplified spherical harmonics boundary conditions. The obtained Green's function is compared to radiative transfer calculations and the diffusion theory. In general, it is shown that the SP3 equations provide better results than the diffusion approximation in media with high absorption coefficient values but no improvement is found for small distances to the source.

Sparse tensor spherical harmonics approximation in radiative transfer

The stationary monochromatic radiative transfer equation is a partial differential transport equation stated on a five-dimensional phase space. To obtain a well-posed problem, boundary conditions have to be prescribed on the inflow part of the domain boundary. We solve the equation with a multi-level Galerkin FEM in physical space and a spectral discretization with harmonics in solid angle and show that the benefits of the concept of sparse tensor products, known from the context of sparse grids, can also be leveraged in combination with a spectral discretization. Our method allows us to include high spectral orders without incurring the “curse of dimension” of a five-dimensional computational domain. Neglecting boundary conditions, we find analytically that for smooth solutions, the convergence rate of the full tensor product method is retained in our method up to a logarithmic factor, while the number of degrees of freedom grows essentially only as fast as for the purely spatial problem. For the case with boundary conditions, we propose a splitting of the physical function space and a conforming tensorization. Numerical experiments in two physical and one angular dimension show evidence for the theoretical convergence rates to hold in the latter case as well.

Diffuse light propagation in biological media by a time-domain parabolic simplified spherical harmonics approximation with ray-divergence effects: erratum

Diffuse light propagation in biological media by a time-domain parabolic simplified spherical harmonics approximation with ray-divergence effects: erratum

Hybrid discrete ordinates—spherical harmonics solution to the Boltzmann Transport Equation for phonons for non-equilibrium heat conduction

The Boltzmann Transport Equation (BTE) for phonons has found prolific use for the prediction of non-equilibrium heat conduction phenomena in semiconductor materials. This article presents a new hybrid formulation and associated numerical procedures for solution of the BTE for phonons. In this formulation, the phonon intensity is first split into two components: ballistic and diffusive. The governing equation for the ballistic component is solved using two different established methods that are appropriate for use in complex geometries, namely the discrete ordinates method (DOM), and the control angle discrete ordinates method (CADOM). The diffusive component, on the other hand, is determined by invoking the first-order spherical harmonics (or P 1) approximation, which results in a Helmholtz equation with Robin boundary conditions. Both governing equations, referred to commonly as the ballistic-diffusive equations (BDE), are solved using the unstructured finite-volume procedure. Results of the hybrid method are compared against benchmark Monte Carlo results, as well as solutions of the BTE using standalone DOM and CADOM for two two-dimensional transient heat conduction problems at various Knudsen numbers. Subsequently, the method is explored for a large-scale three-dimensional geometry in order to assess convergence and computational cost. It is found that the proposed hybrid method is accurate at all Knudsen numbers. From an efficiency standpoint, the hybrid method is found to be superior to direct solution of the BTE both for steady state as well as for unsteady non-equilibrium heat conduction calculations with the computational gains increasing with increase in problem size.