Spherical Harmonic Domain Research Articles

High-Efficiency Forward Modeling of Gravitational Fields in Spherical Harmonic Domain with Application to Lunar Topography Correction

Gravity forward modeling as a basic tool has been widely used for topography correction and 3D density inversion. The source region is usually discretized into tesseroids (i.e., spherical prisms) to consider the influence of the curvature of planets in global or large-scale problems. Traditional gravity forward modeling methods in spherical coordinates, including the Taylor expansion and Gaussian–Legendre quadrature, are all based on spatial domains, which mostly have low computational efficiency. This study proposes a high-efficiency forward modeling method of gravitational fields in the spherical harmonic domain, in which the gravity anomalies and gradient tensors can be expressed as spherical harmonic synthesis forms of spherical harmonic coefficients of 3D density distribution. A homogeneous spherical shell model is used to test its effectiveness compared with traditional spatial domain methods. It demonstrates that the computational efficiency of the proposed spherical harmonic domain method is improved by four orders of magnitude with a similar level of computational accuracy compared with the optimized 3D GLQ method. The test also shows that the computational time of the proposed method is not affected by the observation height. Finally, the proposed forward method is applied to the topography correction of the Moon. The results show that the gravity response of the topography obtained with our method is close to that of the optimized 3D GLQ method and is also consistent with previous results.

Contrasting Discrepancy in the Sea Level Budget Between the North and South Atlantic Ocean Since 2016

AbstractThe discrepancy in the observed global mean sea level budget increased significantly since 2016, but the budget discrepancy over basin‐scales is unclear. In this contribution, we investigate the sea level budget discrepancies in major basins with observations from satellite altimetry, satellite gravimetry, and Argo floats. During 2016–2020, we find substantial discrepancy of 5.72 ± 0.98 mm/yr over the North Atlantic Ocean, and the basin scale discrepancies are smaller elsewhere. Our analysis suggests that three factors, including the wet tropospheric correction (WTC) effect, deep ocean warming signal, and the contemporary ocean bottom deformation (OBD), together reduce the discrepancy by only 1 mm/yr for the North Atlantic Ocean. We decompose sea level observations into the spherical harmonic domain and observe increased discrepancy in low‐degree variations of C10 and C21 since 2016. These two coefficients result in a contrasting signal between the North and South Atlantic Ocean and contribute to the large discrepancy over the North Atlantic Ocean. We further demonstrate that the C10 and C21 discrepancies are independent of the three factors. However, we find regional salinity biases in the Argo data that reduce the discrepancy for the North Atlantic Ocean. Our findings add to the debate about recent sea level budget and imply that further analysis of the Argo North Atlantic data set may be useful.

Sparse reconstruction methods for wideband source localization in spherical harmonics domain

Wideband acoustic source localization is a challenging task especially in the presence of noise. Generally wideband source localization is done by averaging the Direction of Arrival (DOA) estimates obtained over multiple frequencies or narrow subbands by the method of frequency smoothing. Localization of multiple wideband sources which are correlated is even more challenging. In this work acoustic source localization under these challenging conditions is addressed. A sparse reconstruction framework is developed for wideband source localization in the Spherical Harmonics (SH) domain. The proposed framework jointly computes both the azimuth and elevation. In contrast to earlier methods of source localization in the SH domain, this work utilizes the expression for the SH coefficients of the amplitude density of the plane wave, to develop the sparse reconstruction framework. An expression for Cramér Rao Lower Bound (CRLB) on the DOA using this framework is also derived for the wideband sources. This CRLB is shown to easily generalize for narrowband sources as well. The performance of the proposed method is compared with MUltiple Signal Classification in SH domain (MUSIC-SH), Steered Response Power in SH domain (SRP-SH), MUSIC with Direct Path Dominance test in SH domain (MUSIC-DPD-SH) and Pressure based Compressive Sensing in SH domain (PCS-SH). The performance is evaluated for correlated narrowband and wideband sources through simulations, conducting experiments in an anechoic chamber and under reverberant conditions. Using the proposed framework, it is shown that correlated narrowband and wideband sources can be resolved reasonably well when compared to conventional methods of acoustic source localization in SH domain.

Sparsity-driven loudspeaker gain optimization for sound field reconstruction with spherical microphone array

The paper presents a sparsity-driven method utilizing loudspeakers to reconstruct spatial sound fields using measurements obtained from a spherical microphone array (SMA). Employing spherical harmonics decomposition (SHD), the SMA recordings are characterized in the spherical harmonics domain. The gains for the loudspeakers are determined through an optimization problem, equating spherical harmonics pressure coefficients from primary and secondary sources. Furthermore, the sparsity within the loudspeaker feeds is redefined as a constrained sparse optimization problem, integrating linearity and orthogonality constraints. This method effectively reduces the required loudspeakers while maintaining sound field quality. The Bregman iteration method is applied to solve the constrained optimization problem. Rigorous evaluation based on reconstructed sound fields and objective measures highlights significant enhancements compared to least square and compressed sensing methods.

Multiple source direction-of-arrival estimation applicable under near-, far-, and mixed-field scenarios.

Traditionally, direction-of-arrival (DOA) estimations under near- and far-field scenarios are treated as independent tasks based on the corresponding acoustic model, hence necessitating a proper soundfield detector as an upstream processing tool, whereas there may not be a distinct boundary between different soundfield types, especially the mixed-field scenarios where both near- and far-field sources coexist simultaneously. To handle this issue, this article investigates a multisource DOA estimator that equally localizes multiple near-, far-, and mixed-field sources, not requiring any specialized adjustments. We (i) define a signal-invariant multichannel feature denoted generalized relative harmonic coefficients in the spherical harmonics domain; (ii) derive the analytical expression of this feature and summarize its unique properties, exhibiting consistence for both near- and far-field sources; (iii) estimate source elevation and azimuth using the magnitude and phase parts of this feature, respectively; (iv) detect single-source dominated periods from the mixed measurements based on an investigated distance measure; and (v) count the number of sources and localize their DOAs by clustering the single-source dominated estimates. Extensive experimental results, in both simulated and real-life environments, finally confirm the effectiveness of the proposed algorithm under diverse acoustic scenarios, and a superiority over baseline approaches in localizing mixed-field sources.

Near-Field Source Localization and Beamforming in the Spherical Sector Harmonics Domain

Near-Field Source Localization and Beamforming in the Spherical Sector Harmonics Domain

Single-source zones detection using relative harmonic coefficients under diverse acoustic scenarios

Accurate detection of single-source zones, where only a single source is active or dominant within the mixed recordings, benefits many acoustic signal processing techniques, while with very little research focusing on this topic. This article uses a higher-order sensor array and proposes a spherical harmonics (SH) domain detector that is applicable under diverse acoustic scenarios. The detector utilizes multi-dimensional scaling (MDS) by projecting a recently introduced SH-domain feature denoted relative harmonic coefficients (RHC) into a low-dimensional space, where the single-source zones can be easily recognized as they locate at distinctive positions compared to the overlapped and source-absent zones. The method eases limitations suffered by the baseline detectors, such as direct-path wave propagation and far-field requirement, and is additionally capable of counting the unknown source number and automatically clustering the detected single-source zones into each source they originally belong. Extensive experimental results in both simulated and real-life environments have confirmed the proposed detector outperforms the baselines, especially in strongly noisy, reverberant and near-field soundfield.

Sound Source Localization in Spherical Harmonics Domain Based on High-Order Ambisonics Signals Enhancement Neural Network

Sound Source Localization in Spherical Harmonics Domain Based on High-Order Ambisonics Signals Enhancement Neural Network

A proposed method to improve the WER of an ASR system in the noisy reverberant room

This paper proposes a novel approach to reducing the word error rate (WER) of an automatic speech recognition (ASR) system in a noisy reverberant room. This research utilizes the integration of beamforming, dereverberation, and ambisonic. Based on the demonstrated formula, the proposed system synthesizes the signal of desired points on the sphere surface from a combination of 32 signals of a uniform spherical microphone array (USMA). This method uses the non-parametric sound field reproduction technique in the spherical harmonics domain (SHD). Also, the suggested new geometry determines the place of the desired points. In addition to improving the dereverberation performance, the proposed method also improves the performance of the beamformer in terms of directivity factor (DF) and white noise gain (WNG). The results show that objective metrics such as PESQ are significantly improved, and the WER of the Kaldi and the WeNet ASR systems is reduced considerably.

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.

A diffuseness of a sound field in the domains of spherical harmonics and plane waves

Sound field diffuseness plays a key role in room acoustics. Although several attempts have been made to characterize diffuseness quantitatively, how to quantify the extent of diffuseness has yet to be understood. This paper considers a directional diffuseness of a sound field from a viewpoint of statistics. Using an existing approach to the evaluation of diffuseness that employs a covariance of electric acoustic signals in the spherical-harmonics domain, a limited-degree directional diffuseness is formulated. In addition, a spherical design is used to model the diffuse sound field composed of finite number of plane waves. The formulation and the model can contribute to a better understanding of the evaluation of the diffuseness. Results also suggest that Gaussianity of a sound field in the spherical-harmonics domain can be interpreted as Gaussianity of plane waves in the diffuse field model.

A study on constructing arbitrary directivity of sound sources by fitting the spherical harmonics coefficients

We attempt to obtain room impulse responses by sound sources with arbitrary directional characteristics by directivity synthesis in the spherical harmonic domain. For this purpose, we use directional room impulse responses measured with a general loudspeaker pointed in multiple directions. Specifically, we attempt to synthesize the directivity from measurements in 24 directions, including the horizontal and elevation angles of 45 degrees each and the case where the loudspeaker was pointed directly above and below. This enables us to construct an omnidirectional sound source that strictly meets ISO standards or a more flexible directional that reflects musical instruments' directivity. For the omnidirectional case, we are also considering how much we can reduce the number of measurement directions and propose a practical index. In addition, because the proposed method uses a commercially available coaxial loudspeaker with enough sound quality and performs frequency spectrum correction at the same time as directional synthesis, it is possible to synthesize room impulse responses that can be used for auralization.

Modeling and reproduction of sound field by moving complex sound source

Previously, Wave Field Synthesis (WFS), Near-field Compensated Higher-Order Ambisonics (NFC-HOA), Boundary Surface Control (BoSC), and the Spectral Division Method (SDM) have been proposed as sound field synthesis methods. These methods are based on matching the acoustic physical quantities at the boundary of the reproduced sound field with those of the desired sound field. Sound field synthesis techniques can be classified into data-based and model-based methods. The former aim at reproducing sound fields captured by microphone arrays, while the latter aim at synthesizing sound fields based on mathematical models. Most existing sound field reproduction methods only consider simple sound source models, such as static point sources, instead of sound sources with complex radiation properties. To generate a more realistic sound image, this study investigated the formulation of a sound field generated by a sound source with complex radiation properties moving along arbitrary trajectories in a spherical harmonic domain, and reproduced the sound field using loudspeaker arrays.

Multi-Objective NSGA-II Optimization for Broadband Beamforming with Spherical Harmonic Domain Assistance.

Sidelobe suppression is a major challenge in wideband beamforming for acoustic research, especially in high noise and reverberation environments. In this paper, we propose a multi-objective NSGA-II wideband beamforming method based on a spherical harmonic domain for spherical microphone arrays topology. The method takes white noise gain, directional index and maximum sidelobe level as the optimization objectives of broadband beamforming, adopts the NSGA-II optimization strategy with constraints to estimate the Pareto optimal solution, and provides three-dimensional broadband beamforming capability. Our method provides superior sidelobe suppression across different spherical harmonic orders compared to commonly used multi-constrained single-objective optimal beamforming methods. We also validate the effectiveness of our proposed method in a conference room setting. The proposed method achieves a white noise gain of 8.28 dB and a maximum sidelobe level of -23.42 dB at low frequency, while at high frequency it yields comparable directivity index results to both DolphChebyshev and SOCP methods, but outperforms them in terms of white noise gain and maximum sidelobe level, measuring 16.14 dB and -25.18 dB, respectively.

Closed-form multiple source direction-of-arrival estimator under reverberant environments.

Accurate direction-of-arrival (DOA) estimation of multiple sources, simultaneously active in a reverberant environment, remains a challenge, as the multi-path acoustic reflections and overlapped periods dramatically distort the direct-path wave propagation. This article proposes a prominent solution localizing multiple sources in a reverberant environment using closed-form estimates, circumventing any exhaustive search over the two-dimensional directional space. Apart from a low complexity cost, the algorithm has robustness to reverberant, inactive, and overlapped periods and an ease of operation in practice, achieving sufficient accuracy compared to state-of-the-art approaches. Specifically, this algorithm localizes an unknown number of sources through four steps: (i) decomposing the frequency domain signals on a spherical array to the spherical harmonics domain; (ii) extracting the first-order relative harmonic coefficients as the input features; (iii) achieving direct-path dominance detection and localization using closed-form estimation; and (iv) estimating the number of sources and their DOAs based on those pass the direct-path detection. Experimental results, using extensive simulated and real-life recordings, confirm the algorithm with a significantly reduced computational complexity, while preserving competitive localization accuracy as compared to the baseline approaches. Additional tests confirm this low-complexity algorithm even with a potential capacity for online DOA tracking of multiple moving sources.

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.

An Improved Two-Stage Spherical Harmonic ESPRIT-Type Algorithm

Sensor arrays are gradually becoming a current research hotspot due to their flexible beam control, high signal gain, robustness against extreme interference, and high spatial resolution. Among them, spherical microphone arrays with complex rotational symmetry can capture more sound field information than planar arrays and can convert the collected multiple speech signals into the spherical harmonic domain for processing through spherical modal decomposition. The subspace class direction of arrival (DOA) estimation algorithm is sensitive to noise and reverberation, and its performance can be improved by introducing relative sound pressure and frequency-smoothing techniques. The introduction of the relative sound pressure can increase the difference between the eigenvalues corresponding to the signal subspace and the noise subspace, which is helpful to estimate the number of active sound sources. The eigenbeam estimation of signal parameters via the rotational invariance technique (EB-ESPRIT) is a well-known subspace-based algorithm for a spherical microphone array. The EB-ESPRIT cannot estimate the DOA when the elevation angle approaches 90°. Huang et al. proposed a two-step ESPRIT (TS-ESPRIT) algorithm to solve this problem. The TS-ESPRIT algorithm estimates the elevation and azimuth angles of the signal independently, so there is a problem with DOA parameter pairing. In this paper, the DOA parameter pairing problem of the TS-ESPRIT algorithm is solved by introducing generalized eigenvalue decomposition without increasing the computation of the algorithm. At the same time, the estimation of the elevation angle is given by the arctan function, which increases the estimation accuracy of the elevation angle of the algorithm. The robustness of the algorithm in a noisy environment is also enhanced by introducing the relative sound pressure into the algorithm. Finally, the simulation and field-testing results show that the proposed method not only solves the problem of DOA parameter pairing, but also outperforms the traditional methods in DOA estimation accuracy.

Efficient representation of head-related transfer functions in continuous space–frequency domains

Utilizing spherical harmonic (SH) domain has been established as the default method of obtaining continuity over space in head-related transfer functions (HRTFs). This paper concerns different variants of extending this solution by replacing SHs with four-dimensional (4D) continuous functional models in which frequency is imagined as another physical dimension. Recently developed hyperspherical harmonic (HSH) representation is compared with models defined in spherindrical coordinate system by merging SHs with one-dimensional basis functions. The efficiency of both approaches is evaluated based on the reproduction errors for individual HRTFs from HUTUBS database, including detailed analysis of its dependency on chosen orders of approximation in frequency and space. Employing continuous functional models defined in 4D coordinate systems allows HRTF magnitude spectra to be expressed as a small set of coefficients which can be decoded back into values at any direction and frequency. The best performance was noted for HSHs and SHs merged with reverse Fourier–Bessel series, with the former featuring better compression abilities, achieving slightly higher accuracy for low number of coefficients. The presented models can serve multiple purposes, such as interpolation, compression or parametrization for machine learning applications, and can be applied not only to HRTFs but also to other types of directivity functions, e.g. sound source directivity.

Particle-Velocity-Based Mixed-Source Sound Field Translation for Binaural Reproduction

Following the rise of virtual reality is a demand for sound field reproduction techniques that allow the user to interact and move within acoustic reproductions with six-degrees-of-freedom. To this end, a mixed-source model of near-field and far-field virtual sources has been introduced to improve the performance of sound field translation in binaural reproductions of spatial audio recordings. The previous works, however, expand the sound field in terms of the mixed sources based on sound pressure. In this paper, we develop a new mixed-source expansion based on particle velocity, which contributes to more precise reconstruction of the interaural phase difference and, therefore, contributes to improved human perception of sound localization. We represent particle velocity over space using velocity coefficients in the spherical harmonic domain, and the driving signals of the virtual mixed-sources are estimated by constructing cost functions to optimize the velocity coefficients. Compared to the state-of-the-art method, sound-pressure-based mixed-source expansion, we show through numerical simulations that the proposed particle-velocity-based mixed-source expansion has better reconstruction performance in sparse solutions, allowing for sound field translation with better perceptual immersion over a larger space. Finally, we perceptually validate the proposed method through a Multiple Stimulus with Hidden Reference and Anchor (MUSHRA) experiment for a single source scenario. The experimental results support the better perceptual immersion of the proposed method.

Spherical Atomic Norm-Inspired Approach for Direction-of-Arrival Estimation of EM Waves Impinging on Spherical Antenna Array with Undefined Mutual Coupling

A spherical antenna array (SAA) is an array-designed arrangement capable of scanning in almost all the radiation sphere with constant directivity. It finds recent applications in aerospace, spacecraft, vehicular and satellite communications. Therefore, estimation of the direction-of-arrival (DoA) of electromagnetic (EM) waves that impinge on an SAA with unknown mutual coupling called for research attention. This paper proposed a spherical harmonic atomic norm minimization (SHANM) for DoA estimation using an SAA configuration. The gridless sparse signal recovery problem is considered in the spherical harmonic (SH) domain in conjunction with the atomic norm minimization (ANM). Because of the unavailability of the Vandermonde structure in the SH domain, the theorem of Vandermonde decomposition that is the mathematical basis of the traditional ANM methods finds no application in SH. Addressing this challenge, a low-dimensional semidefinite programming (SDP) approach implementing the SHANM method is developed. This approach is independent of Vandermonde decomposition, and directly recovers the atomic decomposition in SH. The numerical experimental results show the superior performance of the proposed method against the previous methods. In addition, accounting for the impacts of mutual coupling, an experimental measured data, which is the generally accepted ground of testing any method, is employed to illustrate the efficacy and robustness of the proposed methods. Finally, for achieving DoA estimation with sufficient localization accuracy using a SAA, the proposed SHANM-based method is a better option.