High-order Spherical Harmonics Research Articles

Quantifying molecular deformation in polymer melts by a generalized Zimm plot approach

The Zimm plot has been widely used to characterize the molecular dimensions of polymers from small-angle scattering experiments, where the reciprocal intensity is analyzed as a function of the square of the magnitude of the scattering wavevector Q. This work explores the benefits of analyzing the reciprocal scattering intensity from deformed polymers, extending the original Zimm plot to anisotropic materials. In the small-angle limit, a tensorial extension of the Guinier law is found for the gyration tensor and the reciprocal single-chain structure factor. In the high-Q limit, application of the spherical harmonic expansion technique to the reciprocal structure factor permits direct model-independent analysis of spatially dependent molecular deformation of polymers. Additionally, the contributions from high-order spherical harmonics become insignificant in the reciprocal-intensity representation. The proposed generalized Zimm plot approach is demonstrated computationally with the affine deformation model and the Rouse model, and experimentally with small-angle neutron scattering measurements of deformed polystyrene melts.

Describing the characteristics of light field in architectural spaces using spherical harmonic function

The direction, density, and diffusivity of light are important indicators of spatial characteristics in describing a three-dimensional light environment. Mury presented a method for describing, measuring, and visualizing the structure of light fields using spherical harmonics in terms of changes in the density and direction of light in three-dimensional space. We extended this study by using higher-order spherical harmonics, which would represent more diverse characteristics of the light environment. We also quantitatively described the light environment as numerical values and investigated the correspondence between these numerical values and human perceptual quantities. As a result, we confirmed that there is a certain degree of correspondence between the “complexity” quantified by the spherical harmonic and the “complexity” perceived by people when observing real space.

Angular resolution of the search for anisotropic stochastic gravitational-wave background with terrestrial gravitational-wave detectors

We consider an anisotropic search for the stochastic gravitational-wave (GW) background by decomposing the gravitational-wave sky into its spherical harmonics components. Previous analyses have used the diffraction limit to define the highest-order spherical harmonics components used in this search. We investigate whether the angular resolution of this search is indeed diffraction-limited by testing our ability to detect and localize simulated GW signals. We show that while using low-order spherical harmonics modes is optimal for initially detecting GW sources, the detected sources can be better localized with higher-order spherical harmonics than expected based on the diffraction limit argument. Additionally, we discuss how the ability to recover simulated GW sources is affected by the number of detectors in the network, the frequency range over which the search is performed, and the method by which the covariance matrix of the GW skymap is regularized. While we primarily consider point-source signals in this study, we briefly apply our methodology to spatially-extended sources and discuss potential future modifications of our analysis for such signals.

A mixed-order modeling approach for head-related transfer function in the spherical harmonic domain

Head-related transfer function (HRTF) contains all the relevant cues that humans rely on to perceive the direction of a sound source, and its modeling has been widely studied in the past years. Motivated by the different degree of fluctuations in HRTF magnitude spectrum across frequency, in this paper, a mixed-order modeling approach in the spherical harmonic domain is proposed in which HRTF magnitude spectrum with low fluctuation is modeled by the low-order spherical harmonics and that with high fluctuation is modeled by the high-order spherical harmonics. Furthermore, a data-driven optimization approach is presented to determine the transition frequency that separates HRTF magnitude spectrum into two ranges with different degrees of fluctuations. The effectiveness of the proposed mixed-order modeling approach is examined using FABIAN HRTF database through listening tests. Results show that this mixed-order modeling approach enables to reduce the order of spherical harmonics to 8 in the low frequencies and 18 in the high frequencies without perceptual degradation in sound quality and localization.

A fieldmap-driven few-channel shim coil design for MRI of the human brain

We exploit the inter-subject similarity of inhomogeneous static magnetic field patterns arising in the human brain under MRI examination to design a small set of shim coils providing performance equivalent to numerous coils based on high-order Spherical Harmonics corrections. A hundred brain B 0-maps were first collected at 3 T. Ideal subject-specific electric current density stream functions are then computed with low power constraints, on a cylindrical surface. This step is repeated over tens of brain maps so that a Principal Component Analysis can be applied to the stream functions; the main components result in the small set of coils. Both 50-subject hold-out and 10-fold cross-validation are employed to evaluate consistency of the proposed system performance over a posteriori subjects. Simulations show that only three cylindrical coils manage to capture the principal magnetic field profiles in the human brain, thus providing a better static field inhomogeneity mitigation than that obtained from 16 unlimited-power high-order Spherical Harmonics coils, with inhomogeneity greatly reduced in the pre-frontal cortex compared to 2nd-order shimmed baseline field acquisitions. The approach provides a very reduced channel count system for mitigating complex B 0-inhomogeneity patterns. Thus, a compact, cost-effective system could be conceived and driven by relatively low-budget electronics. The method should therefore have a strong impact in both ultra-high and portable low-field MRI/MRS. Moreover, this technique can be applied to the design of shim coils addressing anatomies other than the brain.

Dynamic B0 shimming for multiband imaging using high order spherical harmonic shims.

To describe and implement a strategy for dynamic slice-by-slice and multiband B0 shimming using spherical harmonic shims in the human brain at 7T. For thin axial slices, spherical harmonic shims can be divided into pairs of shims (z-degenerate and non-z-degenerate) that are spatially degenerate, such that only ½ of the shims (non-z-degenerate) are required for single slice optimizations. However, when combined, the pairs of shims can be used to simultaneously generate the same in-plane symmetries but with different amplitudes as a function of their z location. This enables multiband shimming equivalent to that achievable by single slice-by-slice optimization. All data were acquired at 7T using a spherical harmonic shim insert enabling shimming up through 4th order with two additional 5th order shims (1st-4th+). Dynamic shim updating was achieved using a 10A shim power supply with 2 ms ramps and constrained optimizations to minimize eddy currents. In groups of eight subjects, we demonstrated that: 1) dynamic updating using 1st-4th+ order shims reduced the SD of the B0 field over the whole brain from 32.4 ± 2.6 and 24.9 ± 2 Hz with 1st-2nd and 1st-4th+ static global shimming to 15.1 ± 1.7 Hz; 2) near equivalent performance was achieved when dynamically updating only the non-z-degenerate shims (14.3 ± 1.5 Hz), or when a using multiband shim factor of 2, MBs = 2, and all shims (14.4 ± 2.0 Hz). High order spherical harmonics provide substantial improvements over static global shimming and enable dynamic multiband shimming with near equivalent performance to that of dynamic slice-by-slice shimming. This reduces distortion in echo planar imaging.

A novel analytic continuation power series solution for the perturbed two-body problem

Inspired by the original developments of recursive power series by means of Lagrange invariants for the classical two-body problem, a new analytic continuation algorithm is presented and studied. The method utilizes kinematic transformation scalar variables differentiated to arbitrary order to generate the required power series coefficients. The present formulation is extended to accommodate the spherical harmonics gravity potential model. The scalar variable transformation essentially eliminates any divisions in the analytic continuation and introduces a set of variables that are closed with respect to differentiation, allowing for arbitrary-order time derivatives to be computed recursively. Leibniz product rule is used to produce the needed arbitrary-order expansion variables. With arbitrary-order time derivatives available, Taylor series-based analytic continuation is applied to propagate the position and velocity vectors for the nonlinear two-body problem. This foundational method has been extended to also demonstrate an effective variable step size control for the Taylor series expansion. The analytic power series approach is demonstrated using trajectory calculations for the main problem in satellite orbit mechanics including high-order spherical harmonics gravity perturbation terms. Numerical results are presented to demonstrate the high accuracy and computational efficiency of the produced solutions. It is shown that the present method is highly accurate for all types of studied orbits achieving 12–16 digits of accuracy (the extent of double precision). While this double-precision accuracy exceeds typical engineering accuracy, the results address the precision versus computational cost issue and also implicitly demonstrate the process to optimize efficiency for any desired accuracy. We comment on the shortcomings of existing power series-based general numerical solver to highlight the benefits of the present algorithm, directly tailored for solving astrodynamics problems. Such efficient low-cost algorithms are highly needed in long-term propagation of cataloged RSOs for space situational awareness applications. The present analytic continuation algorithm is very simple to implement and efficiently provides highly accurate results for orbit propagation problems. The methodology is also extendable to consider a wide variety of perturbations, such as third body, atmospheric drag and solar radiation pressure.

High-order spherical harmonics-nodal collocation scheme for the numerical solution of the time-dependent radiative transfer equation.

The validity of the spherical harmonics-nodal collocation approximation to the stationary Boltzmann equation has been established in multidimensional neutron transport problems. This is a high-order approximation method that allows a coarse spatial discretization without losing accuracy. We extend the method to solve the time-dependent radiative transfer equation for absorbing media with anisotropic scattering, also incorporating into the model reflecting boundary conditions, due to refractive index mismatching. The formalism is then applied to numerical test cases in one and two spatial dimensions that, using typical values in optical tomography for the physical parameters, check the accuracy and convergence of the method.

Automated retinofugal visual pathway reconstruction with multi-shell HARDI and FOD-based analysis

Diffusion MRI tractography provides a non-invasive modality to examine the human retinofugal projection, which consists of the optic nerves, optic chiasm, optic tracts, the lateral geniculate nuclei (LGN) and the optic radiations. However, the pathway has several anatomic features that make it particularly challenging to study with tractography, including its location near blood vessels and bone–air interface at the base of the cerebrum, crossing fibers at the chiasm, somewhat-tortuous course around the temporal horn via Meyer's Loop, and multiple closely neighboring fiber bundles. To date, these unique complexities of the visual pathway have impeded the development of a robust and automated reconstruction method using tractography. To overcome these challenges, we develop a novel, fully automated system to reconstruct the retinofugal visual pathway from high-resolution diffusion imaging data. Using multi-shell, high angular resolution diffusion imaging (HARDI) data, we reconstruct precise fiber orientation distributions (FODs) with high order spherical harmonics (SPHARM) to resolve fiber crossings, which allows the tractography algorithm to successfully navigate the complicated anatomy surrounding the retinofugal pathway. We also develop automated algorithms for the identification of ROIs used for fiber bundle reconstruction. In particular, we develop a novel approach to extract the LGN region of interest (ROI) based on intrinsic shape analysis of a fiber bundle computed from a seed region at the optic chiasm to a target at the primary visual cortex. By combining automatically identified ROIs and FOD-based tractography, we obtain a fully automated system to compute the main components of the retinofugal pathway, including the optic tract and the optic radiation. We apply our method to the multi-shell HARDI data of 215 subjects from the Human Connectome Project (HCP). Through comparisons with post-mortem dissection measurements, we demonstrate the retinotopic organization of the optic radiation including a successful reconstruction of Meyer's loop. Then, using the reconstructed optic radiation bundle from the HCP cohort, we construct a probabilistic atlas and demonstrate its consistency with a post-mortem atlas. Finally, we generate a shape-based representation of the optic radiation for morphometry analysis.

Development of high-order PN models for radiative heat transfer in special geometries and boundary conditions

The high-order spherical harmonics (PN) method for 2-D Cartesian domains is extracted from the 3-D formulation. The number of equations and intensity coefficients reduces to (N+1)2/4 in the 2-D Cartesian formulation compared with N(N+1)/2 for the general 3-D PN formulation. The Marshak boundary conditions are extended to solve problems with nonblack and mixed diffuse-specular surfaces. Additional boundary conditions for specified radiative wall flux, for symmetry/specular reflection boundaries have also been developed. The mathematical details of the formulations and their implementation in the OpenFOAM finite volume based CFD software platform are presented. The accuracy and computational cost of the 2-D Cartesian PN are compared with that of the 3-D PN solver and a Photon Monte Carlo solver for a square enclosure, as well as a 45° wedge geometry with variable radiative properties. The new boundary conditions have been applied for both test cases, and the boundary condition for mixed diffuse-specular surfaces is further illustrated by numerical examples of a rectangular geometry enclosed by walls with different surface characteristics.

The generation of arbitrary order, non-classical, Gauss-type quadrature for transport applications

A method is presented, based upon the Stieltjes method (1884), for the determination of non-classical Gauss-type quadrature rules, and the associated sets of abscissae and weights. The method is then used to generate a number of quadrature sets, to arbitrary order, which are primarily aimed at deterministic transport calculations. The quadrature rules and sets detailed include arbitrary order reproductions of those presented by Abu-Shumays in [4,8] (known as the QR sets, but labelled QRA here), in addition to a number of new rules and associated sets; these are generated in a similar way, and we label them the QRS quadrature sets.The method presented here shifts the inherent difficulty (encountered by Abu-Shumays) associated with solving the non-linear moment equations, particular to the required quadrature rule, to one of the determination of non-classical weight functions and the subsequent calculation of various associated inner products. Once a quadrature rule has been written in a standard form, with an associated weight function having been identified, the calculation of the required inner products is achieved using specific variable transformations, in addition to the use of rapid, highly accurate quadrature suited to this purpose. The associated non-classical Gauss quadrature sets can then be determined, and this can be done to any order very rapidly. In this paper, instead of listing weights and abscissae for the different quadrature sets detailed (of which there are a number), the MATLAB code written to generate them is included as Appendix D.The accuracy and efficacy (in a transport setting) of the quadrature sets presented is not tested in this paper (although the accuracy of the QRA quadrature sets has been studied in [12,13]), but comparisons to tabulated results listed in [8] are made. When comparisons are made with one of the azimuthal QRA sets detailed in [8], the inherent difficulty in the method of generation, used there, becomes apparent, with the highest order tabulated sets showing unexpected anomalies.Although not in an actual transport setting, the accuracy of the sets presented here is assessed to some extent, by using them to approximate integrals (over an octant of the unit sphere) of various high order spherical harmonics. When this is done, errors in the tabulated QRA sets present themselves at the highest tabulated orders, whilst combinations of the new QRS quadrature sets offer some improvements in accuracy over the original QRA sets.Finally, in order to offer a quick, visual understanding of the various quadrature sets presented, when combined to give product sets for the purposes of integrating functions confined to the surface of a sphere, three-dimensional representations of points located on an octant of the unit sphere (as in [8,12]) are shown.

Implementation of High-Order Spherical Harmonics Methods for Radiative Heat Transfer on openfoam

A general formulation of the spherical harmonics (PN) methods was developed recently to expand the method to high orders of PN. The set of N(N + 1)/2 three-dimensional second-order elliptic PDEs formulation and their Marshak boundary conditions for arbitrary geometries are implemented in the openfoam finite volume based cfd software. The results are verified for four cases, including a 1D slab, a 2D square enclosure, a 3D cylindrical enclosure, and an axisymmetric flame. All cases have strongly varying radiative properties, and the results are compared with exact solutions and solutions from the photon Monte Carlo method (PMC).

Two-dimensional axisymmetric formulation of high order spherical harmonics methods for radiative heat transfer

The spherical harmonics (PN) method is a radiative transfer equation solver, which approximates the radiative intensity as a truncated series of spherical harmonics. For general 3-D configurations, N(N+1)/2 intensity coefficients must be solved from a system of coupled second-order elliptic PDEs. In 2-D axisymmetric applications, the number of equations and intensity coefficients reduces to (N+1)2/4 if the geometric relations of the intensity coefficients are taken into account. This paper presents the mathematical details for the transformation and its implementation on the OpenFOAM finite volume based CFD software platform. The transformation and implementation are applicable to any arbitrary axisymmetric geometry, but the examples to test the new formulation are based on a wedge grid, which is the most common axisymmetric geometry in CFD simulations, because OpenFOAM and most other platforms do not have true axisymmetric solvers. Two example problems for the new axisymmetric PN formulation are presented, and the results are verified with that of the general 3-D PN solver, a Photon Monte Carlo solver and exact solutions.

HIGH-ORDER SPHERICAL HARMONICS METHOD FOR RADIATIVE TRANSFER IN SPHERICALLY SYMMETRIC PROBLEMS

A matrix formulation of the spherical harmonics method to predict radiative transfer in participating layer and layered media within spherical geometry is presented. This formulation combines forward finite-difference spatial discretization and the conjugate gradient squared methods to solve the resulting partial differential equations of radiative intensity moments. Henceforth, a high-order spherical harmonics solution has been obtained without difficulty. Comparisons with other methods are carried out for boundary radiative fluxes, transmittance, and reflectance associated with radiative heat transfer through homogeneous/inhomogeneous, isotropic/anisotropic participating spherical layer and layered media. The comparisons show excellent agreement between exact and very high-order spherical harmonics predictions. It was found that a high order of the PN approximation is necessary to produce accurate results at the inner boundary of hollow spherically symmetric media, while low- or moderate-order of the PN approximation is sufficient to obtained accurate results at the outer boundary of both hollow and solid spherically symmetric media.

Using nonlinear programming to correct leakage and estimate mass change from GRACE observation and its application to Antarctica

The Gravity Recovery And Climate Experiment (GRACE) mission has been providing high quality observations since its launch in 2002. Over the years, fruitful achievements have been obtained and the temporal gravity field has revealed the ongoing geophysical, hydrological and other processes. These discoveries help the scientists better understand various aspects of the Earth. However, errors exist in high degree and order spherical harmonics, which need to be processed before use. Filtering is one of the most commonly used techniques to smooth errors, yet it attenuates signals and also causes leakage of gravity signal into surrounding areas. This paper reports a new method to estimate the true mass change on the grid (expressed in equivalent water height or surface density). The mass change over the grid can be integrated to estimate regional or global mass change. This method assumes the GRACE‐observed apparent mass change is only caused by the mass change on land. By comparing the computed and observed apparent mass change, the true mass change can be iteratively adjusted and estimated. The problem is solved with nonlinear programming (NLP) and yields solutions which are in good agreement with other GRACE‐based estimates.

A Spectral Theory Formulation for Elastostatics by Means of Tensor Spherical Harmonics

Consider a set of (N+1)-phase concentric spherical ensemble consisting of a core region encased by a sequence of nested spherical layers. Each phase is spherically isotropic and is functionally graded (FG) in the radial direction. Determination of the elastic fields when the outermost spherical surface is subjected to a nonuniform loading and the constituent phases are subjected to some prescribed nonuniform body force and eigenstrain fields is of interest. When the outermost layer is an unbounded medium with zero eigenstrain and body force fields, then an N-phase multi-inhomogeneous inclusion problem is realized. Based on higher-order spherical harmonics, presenting a three-dimensional strain formulation with a robust form of compatibility equations, a spectral theory of elasticity in the spherical coordinate system is developed. Application of the established spectral theory leads to the exact closed-form solution when the elastic moduli of each phase vary as power-law functions of radius.

Optimal secondary source position in exterior spherical acoustical holophony

Exterior spherical acoustical holophony is a branch of spatial audio reproduction that deals with the rendering of a given free-field radiation pattern (the primary field) by using a compact spherical loudspeaker array (the secondary source). More precisely, the primary field is known on a spherical surface surrounding the primary and secondary sources and, since the acoustic fields are described in spherical coordinates, they are naturally subjected to spherical harmonic analysis. Besides, the inverse problem of deriving optimal driving signals from a known primary field is ill-posed because the secondary source cannot radiate high-order spherical harmonics efficiently, especially in the low-frequency range. As a consequence, a standard least-squares solution will overload the transducers if the primary field contains such harmonics. Here, this is avoided by discarding the strongly decaying spherical waves, which are identified through inspection of the radiation efficiency curves of the secondary source. However, such an unavoidable regularization procedure increases the least-squares error, which also depends on the position of the secondary source. This paper deals with the above-mentioned questions in the context of far-field directivity reproduction at low and medium frequencies. In particular, an optimal secondary source position is sought, which leads to the lowest reproduction error in the least-squares sense without overloading the transducers. In order to address this issue, a regularization quality factor is introduced to evaluate the amount of regularization required. It is shown that the optimal position improves significantly the holophonic reconstruction and maximizes the regularization quality factor (minimizes the amount of regularization), which is the main general contribution of this paper. Therefore, this factor can also be used as a cost function to obtain the optimal secondary source position.

An Efficient Approach to Include Full-Band Effects in Deterministic Boltzmann Equation Solver Based on High-Order Spherical Harmonics Expansion

We present an efficient method to include full-band-structure effects for the case of a silicon conduction band in a deterministic Boltzmann equation solver based on the high-order spherical harmonics expansion method. This method employs the exact density of states and the group velocity obtained from band structure calculations, and it eliminates the modulus of the wave vector in the formulation such that an explicit invertible dispersion relation is not required. While the present method does not require additional central-processing-unit time and memory, compared with the analytic band model, the simulation results are significantly improved and in excellent agreement with those from the full-band Monte Carlo simulations and from an approach based on an invertible anisotropic band that matches several moments of the group velocity of the full band structure.

Optical properties of a metal nanosphere with spatially dispersive permittivity

To model elastic and inelastic light scattering on a metal nanosphere with spatially dispersive permittivity, an extension of the Lorenz–Mie theory is applied. The theory takes into account longitudinal vector spherical harmonics inside the sphere and determines the generalized Mie coefficients using a condition of vanishing electron flow through the sphere surface. In general, this condition is distinct from the conventional additional boundary condition of the continuity of the normal component of the electric field. Therefore, contrary to the common belief, the problem of divergence of the local density of electromagnetic states at the surface of an absorbing sphere is not solved by considering the spatial dispersion of the permittivity. We illustrate the theory by a study of the optical properties of a silver nanosphere using a hydrodynamic model for the dielectric function of the electron gas. Predictions of the nonlocal theory differ markedly from those of the local one if the sphere’s radius or the distance to the surface is smaller than a few nanometers. In particular, we demonstrate a shift of the Fröhlich resonance of nanometer-sized Ag particles caused by the spatial dispersion. Excitation of high-order spherical harmonics in larger particles is discussed. We show how the spatial dispersion decreases the rate of fluorescence quenching in close proximity to the particle surface.

Diffeomorphic image registration of diffusion MRI using spherical harmonics.

Nonrigid registration of diffusion magnetic resonance imaging (MRI) is crucial for group analyses and building white matter and fiber tract atlases. Most current diffusion MRI registration techniques are limited to the alignment of diffusion tensor imaging (DTI) data. We propose a novel diffeomorphic registration method for high angular resolution diffusion images by mapping their orientation distribution functions (ODFs). ODFs can be reconstructed using q-ball imaging (QBI) techniques and represented by spherical harmonics (SHs) to resolve intra-voxel fiber crossings. The registration is based on optimizing a diffeomorphic demons cost function. Unlike scalar images, deforming ODF maps requires ODF reorientation to maintain its consistency with the local fiber orientations. Our method simultaneously reorients the ODFs by computing a Wigner rotation matrix at each voxel, and applies it to the SH coefficients during registration. Rotation of the coefficients avoids the estimation of principal directions, which has no analytical solution and is time consuming. The proposed method was validated on both simulated and real data sets with various metrics, which include the distance between the estimated and simulated transformation fields, the standard deviation of the general fractional anisotropy and the directional consistency of the deformed and reference images. The registration performance using SHs with different maximum orders were compared using these metrics. Results show that the diffeomorphic registration improved the affine alignment, and registration using SHs with higher order SHs further improved the registration accuracy by reducing the shape difference and improving the directional consistency of the registered and reference ODF maps.