Spherical Harmonic Decomposition Research Articles

Dynamics and active mixing of a droplet in a Stokes trap

Particle trapping and manipulation have a wide range of applications in biotechnology and engineering. Recently, a flow-based, particle-trapping device called the Stokes trap was developed for trapping and control of small particles in the intersection of multiple branches in a microfluidic channel. This device can also be used to perform rheological experiments to determine the viscoelastic response of an emulsion or suspension. We show that besides these applications, the various flow modes produced by the Stokes trap are able to manipulate drop shapes and induce active mixing inside droplets. To this end, we analyse the dynamics of a droplet in a Stokes trap through boundary-integral simulations. We also explore the dynamic response of drop shape with respect to distinct external flow modes, which allows us to perform numerical experiments such as step strain and oscillatory extension. A linear controller is used to manipulate drop position, and the drop deformation is characterized by a spherical-harmonic decomposition. For small drop deformations, we observe a linear superposition of harmonics, which, surprisingly, seems to hold even for moderate deformations. This result indicates that such a device can be used for shape control of droplets. We also investigate how the different flow modes may be combined to induce mixing inside the droplets. The transient combination of modes produces an effective chaotic mixing, which is characterized by a mixing number. The mixing inside the droplet can be further enhanced for lower viscosity ratios and low, but non-zero capillary number and flow frequencies.

Phasor field based photoacoustics: A virtual orthogonal wavefield model for nonnegative photoacoustic computational tomography

The phasor field technique has substantially aided the advancement of the time of flight imaging field and sparked intense scientific research interest during the past several years. Recently, phasor field approach was brought to the field of photoacoustic imaging and results in greatly enhanced image SNR and less artifact interference. This paper aims to establish the foundation for a comprehensive physical framework for phasor field based photoacoustics. To demonstrate the rigorousness of the phasor field based photoacoustics, this work gives a derivation of its virtual forward propagation and backward phasor field reconstruction. Phasor field based photoacoustic system works by creating chunks of initial orthogonal photoacoustic vibration, propagating the virtual orthogonal phasor waves, and performing backward phasor field reconstruction. We present here a formal study of phasor field framework utilizing spherical harmonic function decomposition based point spread function (PSF) analysis and provide a rigorous theoretical explanation for its nonnegative imaging capability of phasor field photoacoustic imaging. Simulation and experimental results reported in the paper support the effectiveness of the phasor field photoacoustic method for nonnegative imaging reconstruction. Moreover, the virtual nature of phasor field method enables flexibility and compatibility with different reconstructive mechanism for optimized imaging performance. Phasor field method the potential to advance the photoacoustic imaging techniques across multiple fields, benefiting various applications and enabling new avenues of research and development.

Breakup prediction of oscillating droplets under turbulent flow

Droplet and bubble breakup phenomena under turbulent flow have been widely studied for over 70 years through theoretical, experimental and numerical approaches. Despite significant progress, the dynamics of breakup remains an active research topic. The present study considers a database of breakup events under turbulent conditions. The database contains neutrally buoyant droplets obtained for a single moderate turbulent Reynolds number and a Weber number near the critical Weber number of the droplets, allowing for long time evolution prior to droplet breakage. Each droplet in the database follows a consistent behavior: the droplet undergoes (1) an initial shape deformation, then (2) an extended oscillatory regime, until (3) a critical large deformation starts, leading to (4) a capillary-driven breakup. While the first two stages have been investigated in our previous work using spherical harmonics decomposition, this study focuses on the last two stages, describing them qualitatively and quantitatively by means of seven different shape parameters. These shape parameters are based on different information of the specific droplet such as its surface, its volume or its averaged mean curvature, and provide a multidimensional analysis of the droplet breakup. They are used to measure the time spent in the final two stages prior to breakup, as well as the transition between these stages. Additionally, we establish breakup thresholds for each shape parameter to predict neutrally buoyant droplet breakup under turbulent flow conditions.

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.

Scale‐dependent impact of Aeolus winds on a global forecast system

AbstractThe European Space Agency's Aeolus mission, launched in August 2018, provides the first global horizontal line‐of‐sight wind profile measurements. Previous studies have shown that Aeolus winds in global forecast systems improve the overall forecast skill, especially in the upper tropospheric tropics and in other data‐sparse regions. In this study, we use a series of observing system experiments with the latest version of the reprocessed Aeolus wind product (2B11) to better characterize the locations and drivers of improved skill from Aeolus with Environment and Climate Change Canada's Global Deterministic Prediction System. Observing system experiments that test the impact of Aeolus winds and the impact of all operational wind observations are carried out, covering the period summer 2019 and winter 2019–2020. Assimilation of operational winds improves the tropospheric wind forecast over the Tropics by a reduction of 8% in the forecast error, and adding the Aeolus winds to the assimilations results in an extra 0.7–0.9%. Aeolus wind impacts are improvements are 0.7–0.9% for the Arctic, and 0.4–0.6% over the Northern and Southern Hemisphere extratropics. The scale dependence of these impacts is investigated using spatial spectra (spherical harmonic decomposition). The improvement is quantified using the difference of the 250 hPa kinetic energy forecast error spectra between experiments. The operational winds largely improve the forecast of planetary scale to intermediate scale for spherical wave numbers between 1 and 20 in the short‐range forecasts. The operational wind impact decreases as the forecast lead time increases. On the other hand, the impact of Aeolus is mostly seen in the intermediate to large scale range with a peak around spherical wave number 8. The Aeolus‐related improvement around this wave number increases with forecast lead time. This analysis suggests that Aeolus winds provide estimates of the wind state that are valuable and complementary to that provided from current operational winds.

Physics-informed neural network assisted spherical microphone array signal processing

Thanks to their rotational symmetry that facilitates three-dimensional signal processing, spherical microphone arrays are the common array apertures used for spatial audio and acoustic applications. However, practical implementations of spherical microphone arrays suffer from two issues. First, at high frequency range, a large number of sensors are needed to accurately capture a sound field. Second, the accompanying signal processing algorithm, i.e., the spherical harmonic decomposition method, requires a variable radius array or a rigid surface array to circumvent the spherical Bessel function nulls. Such arrays are hard to design and introduce a scattering field. To address these issues, this paper proposes to assist a spherical microphone array with a physics-informed neural network (PINN) for three-dimensional signal processing. The PINN models the sound field around the array based on the sensor measurements and the acoustic wave equation, augmenting the sound field information captured by the array through prediction. This makes it possible to analyze a high frequency sound field with a reduced number of sensors and avoid the spherical Bessel function nulls with a simple single radius open-sphere microphone array.

The Hardy inequality and large time behaviour of the heat equation on [formula omitted]

In this paper we study the large time asymptotic behaviour of the heat equation with Hardy inverse-square potential on corner spaces RN−k×(0,∞)k, k≥0. We first show a new improved Hardy-Poincaré inequality for the quantum harmonic oscillator with Hardy potential. In view of that, we construct the appropriate functional setting in order to pose the Cauchy problem. Then we obtain optimal polynomial large time decay rates and subsequently the first term in the asymptotic expansion of the solutions in L2(RN−k×(0,∞)k). Particularly, we extend and improve the results obtained by Vázquez and Zuazua (J. Funct. Anal. 2000), which correspond to the case k=0, to any k≥0. We emphasize that the higher the value of k the better time decay rates are. We employ a different and simplified approach than Vázquez and Zuazua, managing to remove the usage of spherical harmonics decomposition in our analysis.

A time-domain nearfield beamformer with spherical harmonic decomposition.

This paper proposes a time-domain nearfield beamformer with spherical harmonic decomposition. The beamformer design is separated into two stages: sound field measurement and beamformer coefficient design. This makes it easier for the beamformer to be implemented by different array structures. The beamformer coefficients are further separated into several parts, making it easier to design a beamformer with different characteristics. The time-domain implementation minimizes the latency between the array input and the beamformer response, and the nearfield focusing improves the farfield noise suppression ability of the beamformer. The proposed beamformer could be useful for emerging acoustic applications such as virtual reality and drones, and for further development of more advanced real-time nearfield beamformers. Simulations and experiments examine the performance of the proposed beamformer.

The Sun’s Magnetic Power Spectra over Two Solar Cycles. I. Calibration between SDO/HMI and SOHO/MDI Magnetograms

The Sun’s magnetic field is strongly structured over a broad range of scales. Magnetic spatial power spectral analysis provides a powerful tool for understanding the various scales of magnetic fields and their interactions with plasma motions. We aim to investigate the power spectra using spherical harmonic decomposition of high-resolution Solar and Heliospheric Observatory (SOHO)/Michelson Doppler Imager (MDI) and Solar Dynamics Observatory (SDO)/Helioseismic and Magnetic Imager (HMI) synoptic magnetograms covering three consecutive solar cycle minima, in a series of papers. As the first of the series, we calibrate and analyze the power spectra based on cotemporal SDO/HMI and SOHO/MDI data in this paper. For the first time, we find that the calibration factor r between SOHO/MDI and SDO/HMI varies with the spatial scale l of the magnetic field, where l is the degree of the spherical harmonic. The calibration factor satisfies . With the calibration function, most contemporaneous SOHO/MDI and SDO/HMI magnetograms show consistent power spectra from about 8 Mm to the global scales over about three orders of magnitude. Moreover, magnetic power spectra from SOHO/MDI and SDO/HMI maps show peaks/knees at l ≈ 120, corresponding to the typical supergranular scale (about 35 Mm) constrained from direct velocimetric measurements. This study paves the way for investigating the solar cycle dependence of supergranulation and magnetic power spectra in subsequent studies.

Statistical anisotropy in galaxy ellipticity correlations

As well as the galaxy number density and peculiar velocity, the galaxy intrinsic alignment can be used to test the cosmic isotropy. We study distinctive impacts of the isotropy breaking on the configuration-space two-point correlation functions (2PCFs) composed of the spin-2 galaxy ellipticity field. For this purpose, we build a formalism for general types of the isotropy-violating 2PCFs and a methodology to efficiently compute them by generalizing the polypolar spherical harmonic decomposition approach to the spin-weighted version. As a demonstration, we analyze the 2PCFs when the matter power spectrum has a well-known g *-type isotropy-breaking term (induced by, e.g., dark vector fields). We then confirm that some anisotropic distortions indeed appear in the 2PCFs and their shapes rely on a preferred direction causing the isotropy violation, d̂. Such a feature can be a distinctive indicator for testing the cosmic isotropy. Comparing the isotropy-violating 2PCFs computed with and without the plane parallel (PP) approximation, we find that, depending on d̂, the PP approximation is no longer valid when an opening angle between the directions towards target galaxies is 𝒪(1°) for the density-ellipticity and velocity-ellipticity cross correlations and around 10° for the ellipticity auto correlation. This suggests that an accurate test for the cosmic isotropy requires the formulation of the 2PCF without relying on the PP approximation.

Cell shape characterization, alignment and comparison using FlowShape.

The shape of a cell is tightly controlled, and reflects important processes including actomyosin activity, adhesion properties, cell differentiation and polarization. Hence, it is informative to link cell shape to genetic and other perturbations. However, most currently used cell shape descriptors capture only simple geometric features such as volume and sphericity. We propose FlowShape, a new framework to study cell shapes in a complete and generic way. In our framework a cell shape is represented by measuring the curvature of the shape and mapping it onto a sphere in a conformal manner. This single function on the sphere is next approximated by a series expansion: the spherical harmonics decomposition. The decomposition facilitates many analyses, including shape alignment and statistical cell shape comparison. The new tool is applied to perform a complete, generic analysis of cell shapes, using the early Caenorhabditis elegans embryo as a model case. We distinguish and characterize the cells at the seven-cell stage. Next, a filter is designed to identify protrusions on the cell shape to highlight lamellipodia in cells. Further, the framework is used to identify any shape changes following a gene knockdown of the Wnt pathway. Cells are first optimally aligned using the fast Fourier transform, followed by calculating an average shape. Shape differences between conditions are next quantified and compared to an empirical distribution. Finally, we put forward a highly performant implementation of the core algorithm, as well as routines to characterize, align and compare cell shapes, through the open-source software package FlowShape. The data and code needed to recreate the results are freely available at https://doi.org/10.5281/zenodo.7778752. The most recent version of the software is maintained at https://bitbucket.org/pgmsembryogenesis/flowshape/. Supplementary data are available at Bioinformatics online.

Droplet oscillations in a turbulent flow

The oscillations of an initially unperturbed spherical droplet immersed in a homogeneous and isotropic turbulent background flow are investigated through spherical harmonic decomposition. As suggested in the literature, the shape oscillations under turbulent conditions are related to the frequency of droplets oscillating in a fluid without background flow. A series of direct numerical simulations (DNS) of droplets with single deformation modes in a fluid at rest are first performed. The frequency and damping rate are compared with weakly viscous linear theory. Then, a database of 220 droplets deformed under turbulent conditions for a single Weber and Reynolds number is generated with an identical numerical set-up. Each spherical harmonic coefficient shows an oscillatory motion with comparable frequency to the single deformation mode simulations. The power spectrum of the coefficients provides the amount of surface of each mode. After a transient regime, the surface area reaches a stationary saturation level. The saturation level of each mode is linked to the turbulence and the energy stored at the interface. Droplets after a high deformation are studied with and without background flow. As expected, the physics of relaxation is driven by capillary forces.

Complete gravitational-waveform amplitude modes for quasicircular compact binaries to the 3.5PN order

We complete the computation of the gravitational waveform amplitude for non-spinning compact binaries to the third and a half post-Newtonian (PN) order in the quasi-circular case. This is done by performing a spin-weighted spherical harmonics decomposition of the amplitude. This computation is achieved using the post-Newtonian-Multipolar-post-Minkowskian formalism. The amplitude modes are written in a PN expanded way as well as a factorized form suitable for EOB template building. Combining this work with previous ones, we provide in the supplemental materials the expressions of the modes including all non-spinning and spinning effects to the 3.5PN order for non-precessing quasi-circular binaries.

Biorthogonal harmonics for the decomposition of gravitational radiation. I. Angular modes, completeness, and the introduction of adjoint-spheroidal harmonics

The estimation of radiative modes is a central problem in gravitational wave theory, with essential applications in signal modeling and data analysis. This problem is complicated by most astrophysically relevant systems not having modes that are analytically tractable. A ubiquitous workaround is to use not modes, but multipole moments defined by spin-weighted spherical harmonics. However, spherical multipole moments are only related to the modes of systems without angular momentum. As a result, they can obscure the underlying physics of astrophysically relevant systems, such as binary black hole merger and ringdown. In such cases, spacetime angular momentum means that radiative modes are not spherical, but spheroidal in nature. Here, we work through various problems related to spheroidal harmonics. We show for the first time that spheroidal harmonics are not only capable of representing arbitrary gravitational wave signals, but that they also possess a kind of orthogonality not used before in general relativity theory. Along the way we present a new class of spin-weighted harmonic functions dubbed ``adjoint-spheroidal'' harmonics. These new functions may be used for the general estimation of spheroidal multipole moments via complete biorthogonal decomposition (in the angular domain). By construction, adjoint-spheroidal harmonics suppress mode-mixing effects known to plague spherical harmonic decomposition; as a result, they better approximate a system's true radiative modes. We discuss potential applications of these results. Lastly, we briefly comment on the challenges posed by the analogous problem with Teukolsky's radial functions.

Normal modes of Proca fields in AdS spacetime

A normal mode analysis for Proca fields in the anti-de Sitter (AdS) spacetime is given. It is found that the equations for the Proca field can be decoupled analytically. This is performed by changing the basis of the vector spherical harmonics decomposition. The normal modes and the normal mode frequencies of the Proca equation in the AdS spacetime are then analytically determined. It is also shown that the Maxwell field can be recovered by taking the massless limit of the Proca field with care so that the nonphysical gauge modes are eliminated.

Three-dimensional iterative reconstruction of pulsed radiation sources using spherical harmonic decomposition.

Neutron and x-ray imaging are essential ways to diagnose a pulsed radiation source. The three-dimensional (3D) intensity distribution reconstructed from two-dimensional (2D) radiation images can significantly promote research regarding the generation and variation mechanisms of pulsed radiation sources. Only a few (≤5) projected images at one moment are available due to the difficulty in building imaging systems for high-radiation-intensity and short-pulsed sources. The reconstruction of a 3D source with a minimal number of 2D images is an ill-posed problem that leads to severe structural distortions and artifacts of the image reconstructed by conventional algorithms. In this paper, we present an iterative method to reconstruct a 3D source using spherical harmonic decomposition. Our algorithm improves the representation ability of spherical harmonic decomposition for 3D sources by enlarging the order of the expansion, which is limited in current analytical reconstruction algorithms. Prior knowledge of the source can be included to obtain a reasonable solution. Numerical simulations demonstrate that the reconstructed image quality of the iterative algorithm is better than that of the analytical algorithm. The iterative method can suppress the effect of noise in the integral projection image and has better robustness and adaptability than the analytical method.

Spherical harmonic decomposition and interpretation of the shapes of the small Saturnian inner moons

Context. The Cassini-Huygens space mission made a series of observations of Saturn’s small satellites during its grand finale stage. These measurements were performed in order to study the shape, geology, and surface composition of the small satellites as well as to study the impact of the environment, in particular the rings, on these small bodies. Aims. The purpose of this study is to focus on the shape analysis of the small Saturnian satellites in order to describe their global figure and large-scale topography, as well as to deduce fundamental quantities, gravity field, and amplitude of the diurnal libration by assuming that the bodies are homogeneous. Methods. We used two approaches in this study. On the one hand, we directly exploited the Cassini images of the small satellites by performing limb measurements and deducing a confidence interval on the shape measurements. On the other hand, we used previously published shape models which combine limb measurements and control points. These shape models were then decomposed and described in spherical harmonics. Results. We found that the shape of the small satellites can be described with a confidence interval between 50 and 150 m. The low degree in spherical harmonics (degree 2) indicated that Telesto, Pandora, Pan, Janus, and Helene have a degree 2 shape close to the Omega sequence, which was defined recently, where the potential is constant along a meridian perpendicular to the longest axis. The degree 2 shape of Epimetheus, on the other hand, is close to the Roche sequence. In contrast, Prometheus, Calypso, and Atlas are in the Low-Brown region. The root mean square spectrum and spherical harmonic maps then allowed us to describe the topography of the satellites, and in particular to highlight equatorial ridges for some satellites including Daphnis. Finally, our estimates of the libration amplitude in the homogeneous case provide values in agreement with previously published librational measurements for Epimetheus while highlighting the proximity of the resonance for Epimetheus, Pandora, and Prometheus. Conclusions. The high resolution images of the internal satellites have allowed us to describe the geology and the geophysics of these bodies. Future comparison of these amplitudes with new librational measurements deduced, for example, by the astrometric method, will allow us to obtain information on the internal structure of these bodies. Similar studies could be carried out on the internal satellites of Jupiter using images from the Europa Clipper (NASA) or JUICE (ESA) missions.

Drone noise directivity and psychoacoustic evaluation using a hemispherical microphone array.

This paper proposes an experimental setup for measuring the sound radiation of a quadrotor drone using a hemispherical microphone array. The measured sound field is decomposed into spherical harmonics, which enables the evaluation of the radiation pattern to non-probed positions. Additionally, the measurement setup allows the assessment of noise emission and psychoacoustic metrics at a wide range of angles. The obtained directivity patterns using a third-order spherical harmonic decomposition (SHD) are shown to exhibit low distortion with respect to the original measurements, therefore, validating the SHD as an adequate representation strategy. Furthermore, the noise emissions are evaluated, and the highest noise emission is observed in the 90 ° azimuth direction. The exterior spherical acoustic holography description is employed to evaluate psychoacoustic metrics at arbitrary far-field positions and validated on a reference microphone. The estimated psychoacoustic metrics are closely related to the target metrics, which allows for sound quality analysis at any point external to the drone.

Spherical harmonics domain-based approach for source localization in presence of directional interference.

This paper presents a learning-based method for source localization in the presence of directional interference under reverberant and noisy conditions. The proposed method operates on the spherical harmonic decomposition of the spherical microphone array recordings to yield spherical harmonics coefficients as the features. An attention mechanism is incorporated through a binary mask that filters out the dominant undesired source components from the features before training. A convolutional neural network is trained to map the phase and magnitude of the filtered coefficients with the location class. Hence, the objective is to develop the binary mask followed by source localization.

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.