Spherical Harmonics Approximation Research Articles

Analytical and numerical study of the ground-track resonances of Dawn orbiting Vesta

The aim of Dawn mission is the acquisition of data from orbits around two bodies (4) Vesta and (1) Ceres, the two most massive asteroids.Due to the low thrust propulsion, Dawn will slowly cross and transit through ground-track resonances, where the perturbations on Dawn orbit may be significant. In this context, to safety go the Dawn mission from the approach orbit to the lowest science orbit, it is essential to know the properties of the crossed resonances.This paper analytically investigates the properties of the major ground-track resonances (1:1, 1:2, 2:3 and 3:2) appearing for Vesta orbiters: location of the equilibria, aperture of the resonances and period at the stable equilibria. We develop a general method using an averaged Hamiltonian formulation with a spherical harmonic approximation of the gravity field. If the values of the gravity field coefficient change, our method stays correct and applicable. We also discuss the effect of one uncertainty on the C20 and C22 coefficients on the properties of the 1:1 resonance. These results are checked by numerical tests. We determine that the increase of the eccentricity appearing in the 2:3 resonance is due to the C22 and S22 coefficients.Our method can be easily adapted to missions similar to Dawn because, contrarily to the numerical results, the analytical formalism stays the same and is valid for a wide range of physical parameters of the asteroid (namely the shape and the mass) as well as for different spacecraft orbits.Finally we numerically study the probability of the capture in resonance 1:1. Our paper reproduces, explains and supplements the results of Tricarico and Sykes (2010).

Cellular neural network to the spherical harmonics approximation of neutron transport equation in x– y geometry. Part I: Modeling and verification for time-independent solution

This paper describes a novel method based on using cellular neural networks (CNN) coupled with spherical harmonics method ( P N ) to solve the time-independent neutron transport equation in x– y geometry. To achieve this, an equivalent electrical circuit based on second-order form of neutron transport equation and relevant boundary conditions is obtained using CNN method. We use the CNN model to simulate spatial response of scalar flux distribution in the steady state condition for different order of spherical harmonics approximations. The accuracy, stability, and capabilities of CNN model are examined in 2D Cartesian geometry for fixed source and criticality problems.

Whole-Body Cerenkov Luminescence Tomography with the Finite Element SP3 Method

Generation of an accurate Cerenkov luminescence imaging model is a current issue of nuclear tomography with optical techniques. The article takes a pro-active approach toward whole-body Cerenkov luminescence tomography. The finite element framework employs the equation of radiative transfer via the third-order simplified spherical harmonics approximation to model Cerenkov photon propagation in a small animal. After this forward model is performed on a digital mouse with optical property heterogeneity and compared with the Monte Carlo method, we investigated the whole body reconstruction algorithm along a regularization path via coordinate descent. The endpoint of the follow-up study is the in vivo application, which provides three-dimensional biodistribution of the radiotracer uptake in the mouse from measured partial boundary currents. The combination of the forward and inverse model with elastic-net penalties is not only validated by numerical simulation, but it also effectively demonstrates in vivo imaging in small animals. Our exact reconstruction method enables optical molecular imaging to best utilize Cerenkov radiation emission from the decay of medical isotopes in tissues.

基于辐射传输方程三阶简化球谐近似模型的时域荧光扩散层析图像重建方法

基于辐射传输方程三阶简化球谐近似模型的时域荧光扩散层析图像重建方法

Evaluation of the simplified spherical harmonics approximation in bioluminescence tomography through heterogeneous mouse models

In vivo bioluminescence imaging (BLI) has played a more and more important role in biomedical research of small animals. Bioluminescence tomography (BLT) further translates the BLI optical information into three-dimensional bioluminescent source distribution, which could greatly facilitate applications in related studies. Although the diffusion approximation (DA) is one of the most widely-used forward models, higher-order approximations are still needed for in vivo small animal imaging. In this work, as a relatively accurate and higher-order approximation theory, the performance of the simplified spherical harmonics approximation (SPN) in BLT is evaluated detailedly in heterogeneous small animals. In the numerical validations, the SPN based results demonstrate better imaging quality compared with diffusion approximation heterogeneously under various source locations over wide optical domain. Although the evaluation for the effects of the optical property mismatch indicates the sensitivity of SPN is similar with DA model in the source localization, it may offer improved performance with much less artifacts. In what follows, heterogeneous experimental BLT reconstructions using in vivo mouse further evaluate the capability of the higher-order method for practical biomedical applications.

Application of the Chebyshev polynomial (T N and U N ) approximation to reflected slab geometry in the neutron transport equation and computation of critical half thicknesses

Abstract The transport of one-speed neutrons has been studied in isotropically scattering slab using ultraspherical polynomial approximation whose sub-cases are spherical harmonics approximation and Chebyshev polynomial approximation of first and second kind with reflective boundaries. An integral equation for the total flux has been derived and solved numerically for different values of c and reflection coefficients. In the solution, Marshak type vacuum boundary conditions were used. The critical thicknesses obtained by the Chebyshev polynomial method are in good agreement with literature values. Computations were made by using the TN, UN and PN approximations for comparison.

A parallel adaptive finite element simplified spherical harmonics approximation solver for frequency domain fluorescence molecular imaging

Fluorescence molecular imaging/tomography may play an important future role in preclinical research and clinical diagnostics. Time- and frequency-domain fluorescence imaging can acquire more measurement information than the continuous wave (CW) counterpart, improving the image quality of fluorescence molecular tomography. Although diffusion approximation (DA) theory has been extensively applied in optical molecular imaging, high-order photon migration models need to be further investigated to match quantitation provided by nuclear imaging. In this paper, a frequency-domain parallel adaptive finite element solver is developed with simplified spherical harmonics (SPN) approximations. To fully evaluate the performance of the SPN approximations, a fast time-resolved tetrahedron-based Monte Carlo fluorescence simulator suitable for complex heterogeneous geometries is developed using a convolution strategy to realize the simulation of the fluorescence excitation and emission. The validation results show that high-order SPN can effectively correct the modeling errors of the diffusion equation, especially when the tissues have high absorption characteristics or when high modulation frequency measurements are used. Furthermore, the parallel adaptive mesh evolution strategy improves the modeling precision and the simulation speed significantly on a realistic digital mouse phantom. This solver is a promising platform for fluorescence molecular tomography using high-order approximations to the radiative transfer equation.

Sparsity-Promoting Tomographic Fluorescence Imaging With Simplified Spherical Harmonics Approximation

Fluorescence molecular tomography has become a promising technique for in vivo small animal imaging and has many potential applications. Due to the ill-posed and the ill-conditioned nature of the problem, Tikhonov regularization is generally adopted to stabilize the solution. However, the result is usually over-smoothed. In this letter, the third-order simplified spherical harmonics approximation to radiative transfer equation is utilized to model the photon propagation within biological tissues. Considering the sparsity of the fluorescent sources, we replace Tikhonov method with an iteratively reweighted scheme. By dynamically updating the weight matrix, L1-norm regularization can be approximated, which can promote the sparsity of the solution. Simulation study shows that this method can preserve the sparsity of the fluorescent sources within heterogeneous medium, even with very limited measurement data.

In Vivo Mouse Bioluminescence Tomography with Radionuclide-Based Imaging Validation

IntroductionBioluminescence imaging, especially planar bioluminescence imaging, has been extensively applied in in vivo preclinical biological research. Bioluminescence tomography (BLT) has the potential to provide more accurate imaging information due to its 3D reconstruction compared with its planar counterpart.MethodsIn this work, we introduce a positron emission tomography (PET) radionuclide imaging-based strategy to validate the BLT results. X-ray computed tomography, PET, spectrally resolved bioluminescence imaging, and surgical excision were performed on a tumor xenograft mouse model expressing a bioluminescent reporter gene.ResultsWith different spectrally resolved measured data, the BLT reconstructions were acquired based on the third-order simplified spherical harmonics (SP3) approximation and the diffusion approximation (DA). The corresponding tomographic images were obtained for validation of bioluminescence source reconstruction.ConclusionOur results show the strength of PET imaging compared with other validation methods for BLT and improved source localization accuracy based on the SP3 approximation compared with the diffusion approximation.

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

We present a simplified spherical harmonics approximation for the time-domain radiative transfer equation including the source-divergence effect. This leads to a set of coupled partial differential equations (PDEs) of the parabolic type that model diffuse light propagation in biological-tissue-like media. We introduce a finite element approach for solving these PDEs, thereby obtaining the time-dependent spatial profile of the fluence. We compare the results with the diffusion equation and Monte Carlo simulations. The fluence obtained via our model is shown to reproduce well the Monte Carlo results in all cases and improves on the solution of the diffusion equation in homogeneous diffusive-defying media. Our solution also shows more sensitivity than the diffusion equation to changes in the absorption coefficient of small inclusions.

The Convergence Rate for a K-Functional in Learning Theory

It is known that in the field of learning theory based on reproducing kernel Hilbert spaces the upper bounds estimate for a -functional is needed. In the present paper, the upper bounds for the -functional on the unit sphere are estimated with spherical harmonics approximation. The results show that convergence rate of the -functional depends upon the smoothness of both the approximated function and the reproducing kernels.

Image reconstruction in diffuse optical tomography based on simplified spherical harmonics approximation

The use of higher order approximations to the Radiative transport equation, through simplified spherical harmonics expansion (SP(N)) in optical tomography are presented. It is shown that, although the anisotropy factor can be modeled in the forward problem, its sensitivity to the measured boundary data is limited to superficial regions and more importantly, due to uniqueness of the inverse problem it cannot be determined using frequency domain data. Image reconstruction through the use of higher ordered models is presented. It is demonstrated that at higher orders (for example SP7) the image reconstruction becomes highly under-determined due to the large increase in the number of unknowns which cannot be adequately recovered. However, reconstruction of diffuse parameters, namely optical absorption and reduced scatter have shown to be more accurate where only the sensitivity matrix used in the inverse problem is based on SP(N) method and image reconstruction is limited to these two diffuse parameters.

Bispectrum covariance in the flat-sky limit

<i>Aims. <i/>To probe cosmological fields beyond the Gaussian level, three-point statistics can be used, all of which are related to the bispectrum. Hence, measurements of CMB anisotropies, galaxy clustering, and weak gravitational lensing alike have to rely upon an accurate theoretical background concerning the bispectrum and its noise properties. If only small portions of the sky are considered, it is often desirable to perform the analysis in the flat-sky limit. We aim at a formal, detailed derivation of the bispectrum covariance in the flat-sky approximation, focusing on a pure two-dimensional Fourier-plane approach.<i>Methods. <i/>We define an unbiased estimator of the bispectrum, which takes the average over the overlap of annuli in Fourier space, and compute its full covariance. The outcome of our formalism is compared to the flat-sky spherical harmonic approximation in terms of the covariance, the behavior under parity transformations, and the information content. We introduce a geometrical interpretation of the averaging process in the estimator, thus providing an intuitive understanding.<i>Results. <i/>Contrary to foregoing work, we find a difference by a factor of two between the covariances of the Fourier-plane and the spherical harmonic approach. We argue that this discrepancy can be explained by the differing behavior with respect to parity. However, in an exemplary analysis it is demonstrated that the Fisher information of both formalisms agrees to high accuracy. Via the geometrical interpretation we are able to link the normalization in the bispectrum estimator to the area enclosed by the triangle configuration at consideration as well as to the Wigner symbol, which leads to convenient approximation formulae for the covariances of both approaches.

Spectrally resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation

Bioluminescence imaging has been extensively applied to in vivo small animal imaging. Quantitative three-dimensional bioluminescent source information obtained by using bioluminescence tomography can directly and much more accurately reflect biological changes as opposed to planar bioluminescence imaging. Preliminary simulated and experimental reconstruction results demonstrate the feasibility and promise of bioluminescence tomography. However, the use of multiple approximations, particularly the diffusion approximation theory, affects the quality of in vivo small animal-based image reconstructions. In the development of new reconstruction algorithms, high-order approximation models of the radiative transfer equation and spectrally resolved data introduce new challenges to the reconstruction algorithm and speed. In this paper, an SP3-based (the third-order simplified spherical harmonics approximation) spectrally resolved reconstruction algorithm is proposed. The simple linear relationship between the unknown source distribution and the spectrally resolved data is established in this algorithm. A parallel version of this algorithm is realized, making BLT reconstruction feasible for the whole body of small animals especially for fine spatial domain discretization. In simulation validations, the proposed algorithm shows improved reconstruction quality compared with diffusion approximation-based methods when high absorption, superficial sources and detection modes are considered. In addition, comparisons between fine and coarse mesh-based BLT reconstructions show the effects of numerical errors in reconstruction image quality. Finally, BLT reconstructions using in vivo mouse experiments further demonstrate the potential and effectiveness of the SP3-based reconstruction algorithm.

Experimental bioluminescence tomography with fully parallel radiative-transfer-based reconstruction framework

Bioluminescence imaging is a very sensitive imaging modality, used in preclinical molecular imaging. However, in its planar projection form, it is non-quantitative and has poor spatial resolution. In contrast, bioluminescence tomography (BLT) promises to provide three dimensional quantitative source information. Currently, nearly all BLT reconstruction algorithms in use employ the diffusion approximation theory to determine light propagation in tissues. In this process, several approximations and assumptions that are made severely affect the reconstruction quality of BLT. It is therefore necessary to develop novel reconstruction methods using high-order approximation models to the radiative transfer equation (RTE) as well as more complex geometries for the whole-body of small animals. However, these methodologies introduce significant challenges not only in terms of reconstruction speed but also for the overall reconstruction strategy. In this paper, a novel fully-parallel reconstruction framework is proposed which uses a simplified spherical harmonics approximation (SPN). Using this framework, a simple linear relationship between the unknown source distribution and the surface measured photon density can be established. The distributed storage and parallel operations of the finite element-based matrix make SPN-based spectrally resolved reconstruction feasible at the small animal whole body level. Performance optimization of the major steps of the framework remarkably improves reconstruction speed. Experimental reconstructions with mouse-shaped phantoms and real mice show the effectiveness and potential of this framework. This work constitutes an important advance towards developing more precise BLT reconstruction algorithms that utilize high-order approximations, particularly second-order self-adjoint forms to the RTE for in vivo small animal experiments.

FRODOCK: a new approach for fast rotational protein–protein docking

Prediction of protein-protein complexes from the coordinates of their unbound components usually starts by generating many potential predictions from a rigid-body 6D search followed by a second stage that aims to refine such predictions. Here, we present and evaluate a new method to effectively address the complexity and sampling requirements of the initial exhaustive search. In this approach we combine the projection of the interaction terms into 3D grid-based potentials with the efficiency of spherical harmonics approximations to accelerate the search. The binding energy upon complex formation is approximated as a correlation function composed of van der Waals, electrostatics and desolvation potential terms. The interaction-energy minima are identified by a novel, fast and exhaustive rotational docking search combined with a simple translational scanning. Results obtained on standard protein-protein benchmarks demonstrate its general applicability and robustness. The accuracy is comparable to that of existing state-of-the-art initial exhaustive rigid-body docking tools, but achieving superior efficiency. Moreover, a parallel version of the method performs the docking search in just a few minutes, opening new application opportunities in the current 'omics' world. http://sbg.cib.csic.es/Software/FRODOCK/

Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations

The accuracy of the commonly used diffusion approximation as used in diffuse optical tomography is known to be limited in cases involving strong absorption and in these situations a higher ordered approximation is necessary. In this study, a light transport model has been developed based upon the three-dimensional frequency-domain simplified spherical harmonics (SPN) approximation for orders up to N = 7. The SPN data are tested against a semi-infinite multi-layered Monte Carlo model. It has been shown that the SPN approximation for higher orders (N >1) provides an increase in accuracy over the diffusion equation specifically near sources and at boundaries of regions with increased optical absorption. It is demonstrated that the error of fluence calculated near the sources between the diffusion approximation and the SPN model (N = 7) can be as large as 60%, therefore limiting the use of the diffusion approximation for small animal imaging and in situations where optical changes near sources are critical for tomographic reconstructions.

Representing the light field in finite three-dimensional spaces from sparse discrete samples

We present a method for measurement and reconstruction of light fields in finite spaces. Using a custom-made device called a plenopter, we can measure spatially and directionally varying radiance distribution functions from a real-world scene up to their second-order spherical harmonics approximations. Interpolating between measurement points, we can recover this function for arbitrary points of a scene. We visualized the global structure of the light field using light tubes, which gives an intuitive description of the flux propagation throughout three-dimensional scenes and provides information about the quality of light in the scenes. Our second-order reconstructions are sufficient to render convex matte objects and therefore have a direct interest for computer graphics applications.

Unified treatment of the P (λ) n approximation to solve the reflected slab criticality problem with strong anisotropy

Abstract Ultraspherical polynomial P (λ) n approximations are used in reflected slab criticality calculations for strongly anisotropic scattering. A unique formulation is developed for reflected slab criticality conditions whose special cases are spherical harmonics approximations, Chebyshev polynomial approximations of first and second kind. It is shown in the calculations that all ultraspherical polynomial approximations are equivalent regarding the convergence of the solution of one-speed neutron transport equation for various degrees of anisotropy and cross section parameters. Although some ultraspherical polynomials approximations converge in the mean as the order of approximation is increased and some converge uniformly, they all converge to the same critical thickness which implies their equi-convergence. It is also demonstrated that the first order approximation P (λ) 1 would give the exact critical thickness for a certain value of the parameter λ.

Near-infrared light heating of a slab by embedded nanoparticles

Exposed to spectral and uniform light, the heating of a one-dimensional, conducting and radiatively participating medium due to embedded absorbing and scattering nanoparticles is solved. Spherical harmonics approximation is used to solve the radiative transfer equation and the finite difference explicit method is used to find the temperature distribution in a generic slab having both boundaries subjected to convection. The host medium is transparent to spectral radiation and the temperature distribution is obtained when the temperature of the irradiated boundary reaches a desired point specified to ensure that any temperature in the medium does not exceed its melting temperature. It is found that the variation of the concentration and configuration of the embedded nanoparticles, particularly gold nanoshells, changes the radiative transfer spectrum, which leads to an alteration in the local heat generation spectrum and the resulting temperature distribution in the medium. It is shown that a gold nanoshell configuration with a great amount of scattering increases the internal diffuse radiation, which creates a more even radiative distribution, while a configuration with a great amount of absorption promotes a high amount of absorption in the entry region and very little in the rear region, leading to the formation of a large temperature gradient between the two boundaries. The present study provides a framework from which the photothermal heating of nanoparticle mixtures in non-transparent host media may be applied.