Spherical Harmonic Basis Research Articles

Statistical estimation of full-sky radio maps from 21cm array visibility data using Gaussian Constrained Realisations

Abstract An important application of next-generation wide-field radio interferometers is making high dynamic range maps of radio emission. Traditional deconvolution methods like CLEAN can give poor recovery of diffuse structure, prompting the development of wide-field alternatives like Direct Optimal Mapping and m-mode analysis. In this paper, we propose an alternative Bayesian method to infer the coefficients of a full-sky spherical harmonic basis for a drift-scan telescope with potentially thousands of baselines, that can precisely encode the uncertainties and correlations between the parameters used to build the recovered image. We use Gaussian Constrained Realisations (GCR) to efficiently draw samples of the spherical harmonic coefficients, despite the very large parameter space and extensive sky-regions of missing data. Each GCR solution provides a complete, statistically-consistent gap-free realisation of a full-sky map conditioned on the available data, even when the interferometer’s field of view is small. Many realisations can be generated and used for further analysis and robust propagation of statistical uncertainties. In this paper, we present the mathematical formalism of the spherical harmonic GCR-method for radio interferometers. We focus on the recovery of diffuse emission as a use case, along with validation of the method against simulations with a known diffuse emission component.

Assembling algorithm for Green's tensors and absorbing boundary conditions for Galbrun's equation in radial symmetry

Solar oscillations can be modeled by Galbrun's equation which describes Lagrangian wave displacement in a self-gravitating stratified medium. For spherically symmetric backgrounds, we construct an algorithm to compute efficiently and accurately the coefficients of the Green's tensor of the time-harmonic equation in vector spherical harmonic basis. With only two resolutions, our algorithm provides values of the kernels for all heights of source and receiver, and prescribes analytically the singularities of the kernels. We also derive absorbing boundary conditions (ABC) to model wave propagation in the atmosphere above the cut-off frequency. The construction of ABC, which contains varying gravity terms, is rendered difficult by the complex behavior of the solar potential in low atmosphere and for frequencies below the Lamb frequency. We carry out extensive numerical investigations to compare and evaluate the efficiency of the ABCs in capturing outgoing solutions. Finally, as an application towards helioseismology, we compute synthetic solar power spectra that contain pressure modes as well as internal-gravity (g-) and surface-gravity (f-) ridges which are missing in simpler approximations of the wave equation. For purpose of validation, the locations of the ridges in the synthetic power spectra are compared with observed solar modes.

MolSym: A Python package for handling symmetry in molecular quantum chemistry.

A consideration of the point group symmetry of molecules is often advantageous from a computational efficiency standpoint and sometimes necessary for the correct treatment of chemical physics problems. Many modern electronic structure software packages include a treatment of symmetry, but these are sometimes incomplete or unusable outside of that program's environment. Therefore, we have developed the MolSym package for handling molecular symmetry and its associated functionalities to provide a platform for including symmetry in the implementation and development of other methods. Features include point group detection, molecule symmetrization, arbitrary generation of symmetry element sets and character tables, and symmetry adapted linear combinations of real spherical harmonic basis functions, Cartesian displacement coordinates, and internal coordinates. We present some of the advantages of using molecular symmetry as achieved by MolSym, particularly with respect to Hartree-Fock theory, and the reduction of finite difference displacements in gradient/Hessian computations. This package is designed to be easily integrated into other software development efforts and may be extended to further symmetry applications.

Fourier-Hankel-Abel Nyquist-limited tomography: A spherical harmonic basis function approach to tomographic velocity-map image reconstruction.

A new method for the fully generalized reconstruction of three-dimensional (3D) photoproduct distributions from velocity-map imaging (VMI) projection data is presented. This approach, dubbed Fourier-Hankel-Abel Nyquist-limited TOMography (FHANTOM), builds on recent previous work in tomographic image reconstruction [C. Sparling and D. Townsend, J. Chem. Phys. 157, 114201 (2022)] and takes advantage of the fact that the distributions produced in typical VMI experiments can be simply described as a sum over a small number of spherical harmonic functions. Knowing the solution is constrained in this way dramatically simplifies the reconstruction process and leads to a considerable reduction in the number of projections required for robust tomographic analysis. Our new method significantly extends basis set expansion approaches previously developed for the reconstruction of photoproduct distributions possessing an axis of cylindrical symmetry. FHANTOM, however, can be applied generally to any distribution-cylindrically symmetric or otherwise-that can be suitably described by an expansion in spherical harmonics. Using both simulated and real experimental data, this new approach is tested and benchmarked against other tomographic reconstruction strategies. In particular, the reconstruction of photoelectron angular distributions recorded in a strong-field ionization regime-marked by their extensive expansion in terms of spherical harmonics-serves as a key test of the FHANTOM methodology. With the increasing use of exotic optical polarization geometries in photoionization experiments, it is anticipated that FHANTOM and related reconstruction techniques will provide an easily accessible and relatively low-cost alternative to more advanced 3D-VMI spectrometers.

Nonlinear optimization for compact representation of orientation distributions based on generalized spherical harmonics

An orientation distribution is a necessary input in any crystal plasticity simulation. The computational time involved in crystal plasticity simulations scales linearly with the number of crystal orientations in the input distributions. Reducing the number of crystal orientations in representing the input orientation distributions quantitatively is a critical and necessary requirement for performing computationally efficient crystal plasticity simulations of deformation processes. A procedure for the compaction of orientation distribution functions (ODFs) relying on a spectral representation using series of generalized spherical harmonics (GSH) basis functions was recently developed. Linear fitting of the spectral representation of an ODF containing a compact set of weighted orientations towards a full-size ODF containing many crystal orientations was in the core of the procedure. This paper advances the compaction procedure by replacing the linear with a nonlinear optimization in Matlab for which a suitably defined error, gradient, and Hessian matrix are derived to allow for more efficient, accurate, and greater compactions. The utility of the new procedure is to allow for not only fitting the weights of a compacted set of orientations but to allow for optimizing weighted orientations or a combination of optimizing weights and orientations. The new compaction procedure has been successfully applied to compactions of large ODFs of cubic, hexagonal, and orthorhombic polycrystalline metals. In doing so, the evolution of texture, twinning, and stress-strain are predicted at large plastic strains with compact ODFs to agree with the corresponding full size ODFs using crystal plasticity models. In closing, guidance for effective texture compaction trading off the accuracy and computational gains are provided.

Multi-target field control for matrix gradient coils.

Conventional single-target field control for matrix gradient coils will add control complexity in MRI spatial encoding, such as designing specialized fields and sequences. This complexity can be reduced by multi-target field control, which is realized by optimizing the coil structure according to target fields. Based on the principle of multi-target field control, the X, Y and Z gradient fields can be set as target fields, and all coil elements can then be divided into three groups to generate these fields. An improved simulated annealing algorithm is proposed to optimize the coil element distribution of each group to generate the corresponding target field. In the improved simulated annealing process, two swapping modes are presented, and randomly selected with certain probabilities that are set to 0.25, 0.5 and 0.75, respectively. The flexibility of the final designed structure is demonstrated by a spherical harmonic basis up to the full second order with single-target field control. An experimental platform is built to measure the gradient fields generated by the designed structure with multi-target target control. With three probabilities of swapping modes, three similar coil element distributions are optimized, and their maximum magnetic field errors for generating X, Y and Z gradients are all below 5%. The structure selected for the final design is the one with a probability of 0.75, considering the coil performance and structural symmetry. The maximum error for all target fields generated by single-target field control is also below 5%. The experimental results show that the measured gradient fields along the axes have enough strength and high linearity. With the proposed improved simulated annealing algorithm and swapping modes, multi-target field control for matrix gradient coils is verified and achieved in this study by optimizing the coil element distribution. Moreover, this study provides a solution to simplify the complexity of controlling the matrix gradient coil in spatial encoding.

Dual symplectic classical circuits: An exactly solvable model of many-body chaos

We propose a general exact method of calculating dynamical correlation functions in dual symplectic brick-wall circuits in one dimension. These are deterministic classical many-body dynamical systems which can be interpreted in terms of symplectic dynamics in two orthogonal (time and space) directions. In close analogy with quantum dual-unitary circuits, we prove that two-point dynamical correlation functions are non-vanishing only along the edges of the light cones. The dynamical correlations are exactly computable in terms of a one-site Markov transfer operator, which is generally of infinite dimensionality. We test our theory in a specific family of dual-symplectic circuits, describing the dynamics of a classical Floquet spin chain. Remarkably, expressing these models in the form of a composition of rotations leads to a transfer operator with a block diagonal form in the basis of spherical harmonics. This allows us to obtain analytical predictions for simple local observables. We demonstrate the validity of our theory by comparison with Monte Carlo simulations, displaying excellent agreement with the latter for a choice of observables.

Q-space imaging based on Gaussian radial basis function with Laplace regularization.

To introduce the diffusion signal characteristics presented by spherical harmonics (SH) basis into the q-space imaging method based on Gaussian radial basis function (GRBF) to robustly reconstruct ensemble average diffusion propagator (EAP) in diffusion MRI (dMRI). We introduced the Laplacian regularization of the signal into the dMRI imaging method based on GRBF, and derived the relevant indicators of microstructure imaging and the orientation distribution function (ODF) providing fiber bundle direction information based on EAP. In addition, this method is combined with a multi-compartment model to calculate the diameter of fiber bundle axons. The evaluation of the results included qualitative comparisons and quantitative assessments of the signal fitting. The results show that the proposed method achieves the more significant accuracy improvement in reconstructing signal. Meanwhile, ODFs estimated by the proposed method show the sharper profiles and less spurious peaks, even under the sparse and noisy conditions. In the 36 sets of axon diameter estimation experiments, 34 and 30 sets of results showed that the proposed method reduced the mean and SD of axon diameter estimates, respectively. Moreover, compared with the current state-of-the-art method, the mean and SD of axon diameter estimated by the proposed method are mostly lower, with 32 and 29 of 36 groups. The proposed method outperforms the GRBF regarding signal fitting and the estimation of the EAP and ODF with multi-shell sparse samples. Moreover, it shows the potential to recover important features of microstructures with less uncertainty by using proposed method together with multi-compartment models.

Computation of spherical sector harmonics norm with application to blind source separation.

Spherical microphone arrays (SMAs) are widely being used for source localization and separation. However, it is uneconomical to build a full SMA when sources are present in restricted regions of environment. Hence, a spherical sector microphone array is utilized for blind source separation for the first time. In particular, the norm of the spherical sector harmonics basis function is computed for mixing matrix estimation. The estimated steering vectors are clustered using mean-shift algorithm. The number of sources is estimated automatically from the number of clusters. The developed mathematical framework is verified using various simulations and experiments on real data.

Spatial audio reproduction over ad hoc loudspeaker array using near-field compensation in spherical harmonics domain

Spatial audio reproduction using the loudspeaker array introduces the curvature effect leading to a distorted listening experience when the listener is in the near field. In the near-field, the loudspeakers are approximated as point sources (spherical wave) and amplify the mode vectors. Further, the problem becomes more challenging for the irregular loudspeaker arrangement, which causes uneven energy distribution in the reproduction region. In this context, a near-field compensation is applied to the encoded ambisonics coefficients. An optimization problem is formulated, such as the loudspeaker gains encoded with spherical harmonics basis coefficients should match the target ambisonics coefficients. Further, the in-phase and quadrature components of the energy localization vector are imposed as the constraints to direct maximum energy in the reproduction region. The solution to the optimization problem is obtained using a derivative-free optimization solver. The performance of the proposed methods is evaluated for ITU-R recommended loudspeaker layouts using the technical and perceptual evaluation attributes.

Optimal method for reconstructing polychromatic maps from broadband observations with an asymmetric antenna pattern

Broadband time-ordered data obtained from telescopes with a wavelength-dependent, asymmetric beam pattern can be used to extract maps at multiple wavelengths from a single scan. This technique is especially useful when collecting data on cosmic phenomena such as the cosmic microwave background (CMB) radiation, as it provides the ability to separate the CMB signal from foreground contaminants. We develop a method to determine the optimal linear combinations of wavelengths (``colors'') that can be reconstructed for a given telescope design and the number of colors that are measurable with high signal-to-noise ratio. The optimal colors are found as eigenvectors of a matrix derived from the inverse noise covariance matrix. When the telescope is able to scan the sky isotropically, it is useful to transform to a spherical harmonic basis, in which this matrix has a particularly simple form. We propose using the optimal colors determined from the isotropic case even when the actual scanning pattern is not isotropic (e.g., covers only part of the sky). We perform simulations showing that maps in multiple colors can be reconstructed accurately from both full-sky and partial-sky scans. Although the original motivation for this research comes from mapping the CMB, this method of polychromatic mapmaking will have broader applications throughout astrophysics.

Matrix quantization of gravitational edge modes

Gravitational subsystems with boundaries carry the action of an infinite-dimensional symmetry algebra, with potentially profound implications for the quantum theory of gravity. We initiate an investigation into the quantization of this corner symmetry algebra for the phase space of gravity localized to a region bounded by a 2-dimensional sphere. Starting with the observation that the algebra mathfrak{sdiff} (S2) of area-preserving diffeomorphisms of the 2-sphere admits a deformation to the finite-dimensional algebra mathfrak{su} (N), we derive novel finite-N deformations for two important subalgebras of the gravitational corner symmetry algebra. Specifically, we find that the area-preserving hydrodynamical algebra mathfrak{sdiff}left({S}^2right){oplus}_{mathcal{L}}{mathbb{R}}^{S^2} arises as the large-N limit of mathfrak{sl}left(N,mathbb{C}right)oplus mathbb{R} and that the full area-preserving corner symmetry algebra mathfrak{sdiff}left({S}^2right){oplus}_{mathcal{L}}mathfrak{sl}{left(2,mathbb{R}right)}^{S^2} is the large-N limit of the pseudo-unitary group mathfrak{su} (N, N). We find matching conditions for the Casimir elements of the deformed and continuum algebras and show how these determine the value of the deformation parameter N as well as the representation of the deformed algebra associated with a quantization of the local gravitational phase space. Additionally, we present a number of novel results related to the various algebras appearing, including a detailed analysis of the asymptotic expansion of the mathfrak{su} (N) structure constants, as well as an explicit computation of the full mathfrak{diff} (S2) structure constants in the spherical harmonic basis. A consequence of our work is the definition of an area operator which is compatible with the deformation of the area-preserving corner symmetry at finite N.

The Gravity Force Generated by a Non-Rotating Level Ellipsoid of Revolution with Low Eccentricity as a Series of Spherical Harmonics

The gravity force of a gravity field generated by a non-rotating level ellipsoid of revolution enclosing mass M is given as a solution of a partial differential equation along with a boundary condition of Dirichlet type. The partial differential equation is formulated herein on the basis of the behavior of spherical gravity fields. A classical solution to this equation is represented on the basis of spherical harmonics. The series representation of the solution is exploited in order to conduct a rigorous asymptotic analysis with respect to eccentricity. Finally, the Dirichlet boundary problem is solved for the case of an ellipsoid of revolution (spheroid) with low eccentricity. This has been accomplished on the basis of asymptotic analysis, which resulted in the determination of the coefficients participating in the spherical harmonics expansion. The limiting case of this series expresses the gravity force of a non-rotating sphere.

Local gravity field modeling using gravity difference observations along the line of sight of the GRACE-FO satellites and adjusted spherical cap harmonic basis functions

Local gravity field modeling using gravity difference observations along the line of sight of the GRACE-FO satellites and adjusted spherical cap harmonic basis functions

Spherical-sector harmonics domain processing for wideband source localization using spherical-sector array of directional microphones

The spherical microphone array can be uneconomical for applications where the sources arrive only from a known section of the sphere. For this reason, the spherical-sector harmonics was developed for processing spherical sector array. The orthonormal spherical sector harmonics (SSH) basis functions which accounts for the discontinuity arising from sectioning the sphere have been developed and shown to work for the array of omnidirectional microphones. In this work, the SSH basis functions are applied to far-field wideband sound source localization using spherical-sector array of first-order directional microphones (cardioid microphones). The array manifold interpolation method is used to produce the steered covariance matrix and the MUSIC algorithm applied for the direction of arrival estimation. The root-mean-square error performance of this spherical-sector array of the first-order cardioid microphones is compared against that of the omnidirectional microphones for different directions and signal-to-noise ratio.

Boltzmann equations for astrophysical Stochastic Gravitational Wave Backgrounds scattering off of massive objects

The goal of this work is to present a set of coupled Boltzmann equations describing the intensity and polarisation Stokes parameters of the SGWB. Collision terms (as discussed e.g. in ref. [1]) which account for gravitational Compton scattering off of massive objects, are also included. This set resembles that for the CMB Stokes parameters, but the different spin nature of the gravitational radiation and the physics involved in the scattering process determine crucial differences. In the case of gravitational Compton scattering, due to the Rutherford angular dependence of the cross section, all the SGWB intensity multipoles of order ℓ are scattered out, therefore producing outgoing intensity anisotropies of any order ℓ if they are present in the incoming radiation. On the other hand, as already outlined in [1], SGWB linear polarisation modes can be expanded in a basis of spherical harmonics with m = ±4 and ℓ ≥ 4. This means that SGWB polarisation modes can be generated from unpolarised anisotropic radiation only with m = ±4, therefore requiring at least a hexadecapole anisotropy (ℓ ≥ 4) in the incoming intensity. Assuming a simplified toy model where scattering targets are localised in a small redshift range, we solve analytically the set of coupled Boltzmann equations to get explicit expressions for the intensity and polarisation angular power spectra. We confirm the contribution of the gravitational Compton scattering to the SGWB anisoptropies is extremely small for collisions with massive compact objects (BH and SMBH) in the frequency range of current and upcoming surveys. The system of coupled Boltzmann equations presented here provides a way to accurate estimate the total amount of anisotropies generated by multiple SGWB scattering processes off of massive objects, as well as the interplay between polarisation and intensity, during the GW propagation across the LSS of the universe. These results will be useful for the full treatment of the astrophysical SWGB anisotropies in view of upcoming gravitational waves observatories.

Inverse Source Solutions With Huygens’ Surface Conforming Distributed Directive Spherical Harmonics Expansions

Time-harmonic inverse source solutions are commonly working with electric and magnetic surface current densities defined and discretized on appropriately chosen Huygens’ surfaces. An efficient meshless alternative are spherical harmonics expansions of low-order distributed along the chosen Huygens’ surface, which still possess pretty good spatial localization properties. Similar to expansions with electric and magnetic surface current densities, distributed expansions with Cartesian spherical harmonics (SHCs) or standard vector spherical harmonics (SHVs) are redundant when they are placed on Huygens’ surfaces. This redundancy is reduced by allowing only spherical harmonics, which can be excited by surface current density distributions in a specific plane, and by forming directive spherical harmonics. This results in a considerable reduction of the necessary number of unknowns for representing the sources and improved conditioning of the discretized integral equations. Inspired by the Huygens’ radiator concept, appropriate directive harmonics on the basis of SHCs and SHVs are derived, implemented, and compared in terms of their performances. The validity of the presented techniques is confirmed via inverse source solutions for synthetic and real measurement data.

Expansion of multicenter Coulomb integrals in terms of two-center integrals

Three- and four-center Coulomb integrals in the solid spherical harmonic Gaussian basis are solved by expansion in terms of two-center integrals. The two-electron Gaussian product rule, coupled with the addition theorem for solid spherical harmonics, reduces four-center Coulomb integrals into a linear combination of two-center Coulomb integrals and one-center overlap integrals. With this approach, three- and four-center Coulomb integrals can be reduced to the same form of two-center integrals. Resulting two-center Coulomb integrals can be further simplified into a simpler form, which can be related to the Boys function. Multi-center Coulomb integrals are solved hierarchically: simple two-center Coulomb integrals are used for calculation of more complicated two-center Coulomb integrals, which are used in the calculation of multicenter integrals.

Optimal beamformer design in spherical sector harmonics domain

Beamforming using spherical microphone arrays (SMAs) with processing in spherical harmonics (SH) domain is widely being studied. This is due to the ease of array processing in SH domain with no spatial ambiguity. Despite the widespread applications of SMA in beamforming, its size makes it inconvenient for practical or commercial use. Building of SMA over a rigid sphere is also a challenging task. Additionally, use of the entire sphere comes at the cost of more microphones and signals to process. Hence, it is uneconomic to use full SMA when sources are only present in restricted regions of the environment. Attempts have been made in literature to use hemispherical microphone array for beamforming. Acoustic image principle was utilized to enable application of SH but with greater computational complexity. In this paper, the use of a spherical sector microphone array is proposed for beamforming. An orthonormal spherical sector harmonics (S2H) basis function is used for data model development. An ideal direction-invariant beampattern using S2H basis function is utilized. Subsequently, to make the beamformer robust against sensor noise with a sharp directivity pattern at the direction of arrival and zero elsewhere, a new maximum directivity and white noise gain (MD-WNG) beamformer is designed for a spherical sector microphone array. Performance analysis of the proposed beamformer is presented using various experiments on directivity pattern and array gain. The proposed method is additionally compared with several existing beamforming techniques. The comparison reveals that the S2H based beamforming provides reasonably better results compared to the existing methods.

Correction of residual errors in configuration interaction electronic structure calculations.

Methods for correcting residual energy errors of configuration interaction (CI) calculations of molecules and other electronic systems are discussed based on the assumption that the energy defect can be mapped onto atomic regions. The methods do not consider the detailed nature of excitations but instead define a defect energy per electron that is unique to a specific atom. Defect energy contributions are determined from calculations on diatomic and hydride molecules and then applied to other systems. Calculated energies are compared with experimental thermodynamic and spectroscopic data for a set of 41 mainly organic molecules representing a wide range of bonding environments. The most stringent test is based on a severely truncated virtual space in which higher spherical harmonic basis functions are removed. The errors of the initial CI calculations are large, but in each case, including defect corrections brings calculated CI energies into agreement with experimental values. The method is also applied to a NIST compilation of coupled cluster calculations that employ a larger basis set and no truncation of the virtual space. The corrections show excellent consistency with total energies in very good agreement with experimental values. An extension of the method is applied to dmsn states of Sc, Ti, V, Mn, Cr, Fe, Co, Ni, and Cu, significantly improving the agreement of calculated transition energies with spectroscopic values.