Spherical Harmonic Representation Research Articles

CRNN-based spatial active noise control in spherical harmonics domain

In this work, an active noise control (ANC) in the spherical harmonics (SH) domain, along with a learning model, is developed. The traditional ANCs are built on the adaptive signal processing framework and fail in the presence of nonlinear distortions. Further, these algorithms are limited to the horizontal plane only. In contrast, the proposed work develops the ANC in the 3D space using the SH representation with the spherical microphone array (SMA) as the error microphone. The SH decomposition calculates the SH pressure coefficients from the SMA recordings. Subsequently, a convolutional recurrent neural network (CRNN) is trained to estimate the real and imaginary spectrograms from the SH coefficients as the input features. The output of the CRNN is the canceling signal that eliminates or attenuates the primary noise in the ANC system. A delay-compensated method addresses ANC system latency. According to simulations and experiments, the proposed learning-based spatial ANC reduces wideband noise and generalizes to untrained noises.

Mie scattering with 3D angular spectrum method.

Mie theory is a powerful method to model electromagnetic scattering from a multilayered sphere. Usually, the incident beam is expanded to its vector spherical harmonic representation defined by beam shape coefficients, and the multilayer sphere scattering is obtained by the T-matrix method. However, obtaining the beam shape coefficients for arbitrarily shaped incident beams has limitations on source locations and requires different methods when the incident beam is defined inside or outside the computational domain or at the scatterer surface. We propose a 3D angular spectrum method for defining beam shape coefficients from arbitrary source field distributions. This method enables the placement of the sources freely within the computational domain without singularities, allowing flexibility in beam design. We demonstrate incident field synthesis and spherical scattering by comparing morphology-dependent resonances to known values, achieving excellent matching and high accuracy. The proposed method has significant benefits for optical systems and inverse beam design. It allows for the analysis of electromagnetic forward/backward propagation between optical elements and spherical targets using a single method. It is also valuable for optical force beam design and analysis.

Adjoint-based control of three dimensional Stokes droplets

We develop a continuous adjoint formulation and implementation for controlling the deformation of clean, neutrally buoyant droplets in Stokes flow through farfield velocity boundary conditions. The focus is on dynamics where surface tension plays an important role through the Young-Laplace law. To perform the optimization, we require access to first-order gradient information, which we obtain from the linearized sensitivity equations and their corresponding adjoint by applying shape calculus to the space-time tube formed by the interface evolution. We show that the adjoint evolution equation can be efficiently expressed through a scalar adjoint transverse field. The optimal control problem is discretized by high-order boundary integral methods using Quadrature by Expansion coupled with a spherical harmonic representation of the droplet surface geometry. We show the accuracy and stability of the scheme on several tracking-type control problems.

Capturing Statistical Isotropy Violation with Generalized Isotropic Angular Correlation Functions of Cosmic Microwave Background Anisotropy

The exquisitely measured maps of fluctuations in the cosmic microwave background (CMB) present the possibility of systematically testing the principle of statistical isotropy of the Universe. A systematic approach based on strong mathematical formulation allows any nonstatistical isotropic (nSI) feature to be traced to the nature of physical effects or observational artifacts. Bipolar spherical harmonics (BipoSH) representation has emerged as an overarching general formalism for quantifying the departures from statistical isotropy for a field on a 2D sphere. We adopt a little-known reduction of the BipoSH functions, dubbed minimal harmonics in the original paper by Manakov et al. We demonstrate that this reduction technique of BipoSH leads to a new generalized set of isotropic angular correlation functions referred to here as minimal BipoSH functions that are observable quantifications of nSI features in a sky map. This paper presents a novel observable quantification of deviation from statistical isotropy in terms of generalized angular correlation functions that are compact and complementary to the BipoSH spectra that generalize the angular power spectrum of CMB fluctuations.

A method for direct in-space thrust estimation from low-acceleration orbital maneuvers

This paper presents a method for performing direct in-space thrust estimation for low-acceleration propulsion systems using measurements of the spacecraft’s position taken during an orbital maneuver. The method is based on the ensemble Kalman update which does not require linearization of the spacecraft dynamics nor does it require Gaussian distributions for parameter uncertainties and measurement noise, allowing for a more general approach to thrust estimation. In addition, modeling error, such as that caused by the truncation of a spherical-harmonics representation of the Earth’s gravitational field, can be explicitly accounted for by representing the error with Gaussian processes. Simulated experiments show that uncertainty in the propulsive acceleration magnitude on the order of 0.1 mum/s^2 (1sigma) can be achieved at an orbit altitude of approximately 410 km with temporally-sparse measurements even in the presence of uncertain atmospheric drag, and Monte Carlo analysis demonstrates the consistency of the inference results. Trends in the estimate of the propulsive acceleration with the true acceleration value are explored empirically and theoretically in order to allow for generalization of the results. The outcome of this work is a systematic approach to direct in-space thrust estimation that can support the final steps of development for future in-space electric propulsion systems or the calibration of a thruster during a mission.

Spherical Harmonic Representation of Energetic Neutral Atom Flux Components Observed by IBEX

The Interstellar Boundary Explorer (IBEX) images the heliosphere by observing energetic neutral atoms (ENAs). The IBEX-Hi instrument on board IBEX provides full-sky maps of ENA fluxes produced in the heliosphere and very local interstellar medium through charge exchange of suprathermal ions with interstellar neutral atoms. The first IBEX-Hi results showed that, in addition to the anticipated globally distributed flux (GDF), a narrow and bright emission from a circular region in the sky, dubbed the IBEX ribbon, is visible in all energy steps. While the GDF is mainly produced in the inner heliosheath, ample evidence indicates that the ribbon forms outside the heliopause in the regions where the interstellar magnetic field is perpendicular to the lines of sight. The IBEX maps produced by the mission team distribute the observations into 6° × 6° rectangle pixels in ecliptic coordinates. The overlap of the GDF and ribbon components complicates qualitative analyses of each source. Here, we find the spherical harmonic representation of the IBEX maps, separating the GDF and ribbon components. This representation describes the ENA flux components in the sky without relying on any pixelization scheme. Using this separation, we discuss the temporal evolution of each component over the solar cycle. We find that the GDF is characterized by larger spatial scale structures than those of the ribbon. However, we identify two isolated, small-scale signals in the GDF region that require further study.

Spherical harmonics representation of the gravitational phase shift

We investigate the general relativistic phase of an electromagnetic wave as it propagates in the gravitational field of the Earth, which is modeled as an isolated, weakly aspherical gravitating body. We introduce coordinate systems to describe light propagation in the Earth's vicinity along with the relevant coordinate transformations, and discuss the transformations between proper and coordinate times. We represent the Earth's gravitational field using Cartesian symmetric trace-free (STF) mass multipole moments. The light propagation equation is solvable along the trajectory of a light ray to all STF orders $\ensuremath{\ell}$. Although we focus primarily on the quadrupole ($\ensuremath{\ell}=2$), octupole ($\ensuremath{\ell}=3$), and hexadecapole ($\ensuremath{\ell}=4$) cases, our approach is valid to all orders. We express the STF moments via spherical harmonic coefficients of various degree and order, ${C}_{\ensuremath{\ell}k},{S}_{\ensuremath{\ell}k}$. The result is the gravitational phase shift expressed in terms of the spherical harmonics. These results are new. We also consider contributions due to tides and the Earth's rotation. We estimate the characteristic magnitudes of each term of the resulting overall gravitational phase shift. The terms assessed are either large enough to impact current-generation clocks or will become significant as future-generation clocks offer greater sensitivity. Our formulation is useful for many practical and scientific applications, including space-based time and frequency transfers, relativistic geodesy and navigation, as well as quantum communication links and space-based tests of fundamental physics.

Spherical harmonics representation of the steady-state membrane potential shift induced by tDCS in realistic neuron models

Objective. We provide a systematic framework for quantifying the effect of externally applied weak electric fields on realistic neuron compartment models as captured by physiologically relevant quantities such as the membrane potential or transmembrane current as a function of the orientation of the field. Approach. We define a response function as the steady-state change of the membrane potential induced by a canonical external field of 1 V m−1 as a function of its orientation. We estimate the function values through simulations employing reconstructions of the rat somatosensory cortex from the Blue Brain Project. The response of different cell types is simulated using the NEURON simulation environment. We represent and analyze the angular response as an expansion in spherical harmonics. Main results. We report membrane perturbation values comparable to those in the literature, extend them to different cell types, and provide their profiles as spherical harmonic coefficients. We show that at rest, responses are dominated by their dipole terms (), in agreement with experimental findings and compartment theory. Indeed, we show analytically that for a passive cell, only the dipole term is nonzero. However, while minor, other terms are relevant for states different from resting. In particular, we show how and terms can modify the function to induce asymmetries in the response. Significance. This work provides a practical framework for the representation of the effects of weak electric fields on different neuron types and their main regions—an important milestone for developing micro- and mesoscale models and optimizing brain stimulation solutions.

Coarse-grained methods for heterogeneous vesicles with phase-separated domains: Elastic mechanics of shape fluctuations, plate compression, and channel insertion

We develop coarse-grained particle approaches for studying the elastic mechanics of vesicles with heterogeneous membranes having phase-separated domains. We perform simulations both of passive shape fluctuations and of active systems where vesicles are subjected to compression between two plates or subjected to insertion into narrow channels. Analysis methods are developed for mapping particle configurations to continuum fields with spherical harmonics representations. Heterogeneous vesicles are found to exhibit rich behaviors where the heterogeneity can amplify surface two-point correlations, reduce resistance during compression, and augment vesicle transport times in channels. The developed methods provide general approaches for characterizing the mechanics of coarse-grained heterogeneous systems taking into account the roles of thermal fluctuations, geometry, and phase separation.

Real-time sound synthesis of pass-by noise: comparison of spherical harmonics and time-varying filters

This paper proposes and compares two sound synthesis techniques to render a moving source for a fixed receiver position based on indoor pass-by noise measurements. The approaches are based on the time-varying infinite impulse response (IIR) filtering and spherical harmonics (SH) representation. The central contribution of the work is a framework for realistic moving source sound synthesis based on transfer functions measured using static far-field microphone arrays. While the SHs require a circular microphone array and a free-field propagation (delay, geometric spread), the IIR filtering relies on far-field microphones that correspond to the propagation path of the moving source. Both frameworks aim to provide accurate sound pressure levels in the far-field that comply with standards. Moreover, the frameworks can be extended to additional sources and filters (e.g. sound barriers) to create different moving source scenarios by removing the room size constraint. The results of the two sound synthesis approaches are preliminary evaluated and compared on a vehicle pass-by noise dataset and it is shown that both approaches are capable of accurately and efficiently synthesize a moving source.

Spherical harmonic representation of the observed directional wave front in the time domain.

Sound field reproduction algorithms require loudspeaker directivity, which is usually measured at discrete frequencies. A time domain model of loudspeaker directivity benefits broadband applications. This Letter proposes the concept of a directional wave front in the time domain, which could be linked to loudspeaker impulse responses measured on a spherical surface. The observed signal in the time domain at the boundary of a spherical region due to a propagating directional wave front is illustrated using a geometric model. Based on the geometric model, the spherical harmonic decomposition of the observed signal in the time domain is also derived.

Quasinormal modes and stability of accelerating Reissner-Norsdtröm AdS black holes

We investigate the numerical stability of accelerating AdS black holes against linear scalar perturbations. In particular, we study the evolution of a probe non-minimally coupled scalar field on Schwarzschild and Reissner-Nordström AdS black holes with small accelerations by computing the quasinormal modes of the perturbation spectrum. We decompose the scalar field Klein-Gordon equation and study the eigenvalue problem for its angular and radial-temporal parts using different numerical methods. The angular part is written in terms of the Heun solution and expanded through the Frobenius method which turns out to give eigenvalues qualitatively similar to the ones obtained through the spherical harmonics representation. The radial-temporal evolution renders a stable field profile which is decomposed in terms of damped and purely imaginary oscillations of the quasinormal modes. We calculate the respective frequencies for different spacetime parameters showing the existence of a fine-structure in the modes, for both real and imaginary parts, which is not present in the non-accelerating AdS black holes. Our results indicate that the Schwarzschild and Reissner-Nordström AdS black holes with small accelerations are stable against linear scalar perturbations.

Densely Sampled Global Dynamic Topographic Observations and Their Significance

AbstractTopography and bathymetry are principally supported by some combination of crustal and sub‐crustal density variations. However, dynamic topography is generated by vertical deflection of the Earth's surface as a result of mantle convection. Isolating and quantifying observable dynamic topography yields valuable information about mantle processes. Here, we investigate global dynamic topography by calculating residual depth anomalies throughout the oceanic realm and residual topographic anomalies across the continents. We correct for sedimentary and crustal loading by exploiting a variety of seismologic datasets that include seismic reflection profiles, wide‐angle/refraction surveys, and receiver functions. In this way, an extensively revised and augmented global compilation of 10,874 oceanic residual depth measurements and 3,777 continental residual topographic measurements is constructed. In the oceanic realm, the methodology has been revised to improve accuracy and resolution. First, quartz/clay content of the sedimentary column is adjusted to remove minor skewness of residual depth anomalies as a function of plate age. Secondly, variation of bulk density as a function of crustal thickness is taken into account. Our global compilation is used to generate spherical harmonic representations of observable dynamic topography out to degree 40 (i.e., ∼1,000 km). Resultant spectra demonstrate that dynamic topographic power varies linearly with inverse wavenumber. The spectral slope directly reflects the way by which dynamic topography is generated by Stokes' flow. Our global results are consistent with independent and diverse geologic markers of uplift and subsidence together with Neogene‐Quaternary intraplate basalt magmatism.

Spherical Harmonic Decomposition of a Sound Field Using Microphones on a Circumferential Contour Around a Non-Spherical Baffle

Spherical harmonic (SH) representations of sound fields are usually obtained from microphone arrays with rigid spherical baffles whereby the microphones are distributed over the entire surface of the baffle. We present a method that overcomes the requirement for the baffle to be spherical. Furthermore, the microphones can be placed along a circumferential contour around the baffle. This greatly reduces the required number of microphones for a given spatial resolution compared to conventional spherical arrays. Our method is based on the analytical solution for SH decomposition based on observations along the equator of a rigid sphere that we presented recently. It incorporates a calibration stage in which the microphone signals due to a suitable set of calibration sound fields are projected onto the SH decomposition of those same sound fields on the surface of a notional rigid sphere by means of a linear filtering operation. The filter coefficients are computed from the calibration data via a least/squares fit. We present an evaluation of the method based on the application of binaural rendering of the SH decomposition of the signals from an 18/element array that uses a human head as the baffle and that provides 8th ambisonic order. We analyse the accuracy and robustness of our method based on simulated data as well as based on measured data from a prototype.

Statistical shape analysis of the tricuspid valve in hypoplastic left heart sydrome.

Hypoplastic left heart syndrome (HLHS) is a congenital heart disease characterized by incomplete development of the left heart. Children with HLHS undergo a series of operations which result in the tricuspid valve (TV) becoming the only functional atrioventricular valve. Some of those patients develop tricuspid regurgitation which is associated with heart failure and death and necessitates further surgical intervention. Repair of the regurgitant TV, and understanding the connections between structure and function of this valve remains extremely challenging. Adult cardiac populations have used 3D echocardiography (3DE) combined with computational modeling to better understand cardiac conditions affecting the TV. However, these structure-function analyses rely on simplistic point-based techniques that do not capture the leaflet surface in detail, nor do they allow robust comparison of shapes across groups. We propose using statistical shape modeling and analysis of the TV using Spherical Harmonic Representation Point Distribution Models (SPHARM-PDM) in order to generate a reproducible representation, which in turn enables high dimensional low sample size statistical analysis techniques such as principal component analysis and distance weighted discrimination. Our initial results suggest that visualization of the differences in regurgitant vs. non-regurgitant valves can precisely locate populational structural differences as well as how an individual regurgitant valve differs from the mean shape of functional valves. We believe that these results will support the creation of modern image-based modeling tools, and ultimately increase the understanding of the relationship between valve structure and function needed to inform and improve surgical planning in HLHS.

MagIC v5.10: a two-dimensional message-passing interface (MPI) distribution for pseudo-spectral magnetohydrodynamics simulations in spherical geometry

Abstract. We discuss two parallelization schemes for MagIC, an open-source, high-performance, pseudo-spectral code for the numerical solution of the magnetohydrodynamics equations in a rotating spherical shell. MagIC calculates the non-linear terms on a numerical grid in spherical coordinates, while the time step updates are performed on radial grid points with a spherical harmonic representation of the lateral directions. Several transforms are required to switch between the different representations. The established hybrid parallelization of MagIC uses message-passing interface (MPI) distribution in radius and relies on existing fast spherical transforms using OpenMP. Our new two-dimensional MPI decomposition implementation also distributes the latitudes or the azimuthal wavenumbers across the available MPI tasks and compute cores. We discuss several non-trivial algorithmic optimizations and the different data distribution layouts employed by our scheme. In particular, the two-dimensional distribution data layout yields a code that strongly scales well beyond the limit of the current one-dimensional distribution. We also show that the two-dimensional distribution implementation, although not yet fully optimized, can already be faster than the existing finely optimized hybrid parallelization when using many thousands of CPU cores. Our analysis indicates that the two-dimensional distribution variant can be further optimized to also surpass the performance of the one-dimensional distribution for a few thousand cores.

A stable and accurate immersed boundary method for simulating vesicle dynamics via spherical harmonics

In this paper, we improve our previous immersed boundary (IB) method for 3D triangulated vesicle in unsteady Navier-Stokes flow (Seol et al., 2016 [31]) from several aspects. Firstly, we adopt spherical harmonic representation for approximating vesicle configuration. By applying spectral differentiation, we are able to obtain high accuracy of geometric quantities such as the mean and Gaussian curvatures, and the surface Laplacian of mean curvature, which is not achievable via triangulation. The vesicle membrane (interface) immersed in 3D Newtonian fluid ensures the surface incompressibility constraint; thus, an unknown elastic tension acting as Lagrange multiplier must be introduced along the interface. To efficiently solve the problem, a logarithmic formulation of approximate elastic tension is explicitly utilized in a nearly incompressible interface approach. Then in computing the elastic tension force, we propose to use the divergence form instead of the commonly used non-divergence one. By doing so, we find that numerical stability can be improved significantly during vesicle relaxation and its transient motions. Moreover, to maintain the interfacial mesh quality, a mesh control technique via filtering of interfacial tangential velocity is coupled within the nearly incompressible interface approach. Upon these improvements, a series of numerical tests on the present scheme is performed to verify numerical accuracy, stability, and convergence of our method. As for practical experiments, the tank-treading and tumbling motions of prolate vesicle in shear flow are extensively studied by varying some dimensionless parameters such as the reduced volume, bending capillary number, viscosity contrast, and the Reynolds number. We further study three types of vesicle shapes, namely, bullet, parachute, and croissant in rectangular Poiseuille flow.

Operational Multi‐GNSS Global Ionosphere Maps at GFZ Derived From Uncombined Code and Phase Observations

AbstractGlobal navigation satellite system (GNSS) networks with multi‐frequency data can be used to monitor the activity of the Earth's ionosphere and to generate global maps of the vertical total electron content (VTEC). This article introduces and evaluates operational GFZ VTEC maps. The processing is based on a rigorous least squares approach using uncombined code and phase observations, and does not entail leveling techniques. A single‐layer model with a spherical harmonic VTEC representation is used. The solutions are generated in a daily post‐processing mode with GPS, GLONASS, and Galileo data, and are provided for the period since the beginning of 2000. A comparison of the GFZ VTEC maps with the final combined International GNSS Service (IGS) product shows a high consistency with the solutions of the IGS analysis centers. A validation with about four years of Jason‐3 altimetry‐derived VTEC data is provided, in which the GFZ solution has the smallest bias of 1.2 TEC units compared to the solutions of the IGS analysis centers, and with 3.0 TEC units one of the smallest standard deviations.

Isotropic non-Lipschitz regularization for sparse representations of random fields on the sphere

In this paper, we consider an infinite-dimensional isotropic non-Lipschitz optimization problem with ℓ 2 , p \ell _{2,p} ( 0 > p > 1 0>p>1 ) regularizer for random fields on the unit sphere with spherical harmonic representations. The regularizer not only gives a group sparse solution, but also preserves the isotropy of the regularized random field represented by the solution. We present first order and second order necessary optimality conditions for local minimizers of the optimization problem. We also derive two lower bounds for the nonzero groups of stationary points, which are used to prove that the infinite-dimensional optimization problem can be reduced to a finite-dimensional problem. Moreover, we propose an iteratively reweighted algorithm for the finite-dimensional problem and prove its convergence. Finally, numerical experiments on Cosmic Microwave Background data are presented to show the efficiency of the non-Lipschitz regularization.

Spheroidal ambisonics: Spatial audio in spheroidal bases.

Ambisonics is an established framework to capture and reproduce spatial sound fields based on the spherical harmonics representation [Gerzon (1973). J. Audio Eng. Soc. 21(1), 2-10]. A generalization-spheroidal ambisonics-based on spheroidal wave functions is proposed for use with spheroidal microphone arrays. Analytical conversion from spheroidal ambisonics to spherical ambisonics are derived to ensure compatibility with the existing ambisonics ecosystem. Numerical experiments verify spheroidal ambisonics encoding and transcoding for spatial sound field recording. The sound field reconstructed from the transcoded coefficients has a zone of accurate reconstruction which is prolonged towards the long axis of a prolate spheroidal microphone array.