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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.

The Mohorovičić discontinuity (Moho) marks the boundary between Earth's crust and the underlying mantle, serving as a critical interface for understanding Earth's structure, composition, and geodynamic processes. This study introduces a novel iterative and stable algorithm for global Moho depth inversion. We first derive the gravity disturbance of the Moho interface in the spherical harmonic domain, expressed as a series of spherical harmonic coefficients. These forward expressions are then reformulated into an iterative scheme for Moho depth estimation. To ensure convergence, a damping factor is applied to suppress high‐frequency noise, and the process is constrained by observed gravity data to minimize residuals. The algorithm is validated using a synthetic Airy–Heiskanen interface in a closed‐loop test. Results show stable convergence within approximately three iterations, yielding minimal gravity residuals (∼0.05 mGal) and small depth errors (standard deviation: 0.07 km), demonstrating the method's high accuracy. A sensitivity analysis of constant and variable Moho density contrasts further shows that when density varies from 450 to 600, the mean difference is less than 1.0 km and the standard deviation is only 1.1 km, indicating that the solution is largely insensitive to density changes. Importantly, incorporating a variable density contrast significantly improves Moho depth recovery along mid‐ocean ridges. Finally, the method is applied to refined gravity disturbances that are maximally correlated with Moho depth, successfully recovering global Moho topography. Comparison with the CRUST1.0 seismic Moho model shows strong consistency in both spatial distribution and statistical measures, with depth residuals (standard deviation: 4.23 km) and gravity residuals (∼1.89 mGal), further confirming the robustness of the method. Notably, the use of variable Moho density contrast again provides substantial improvements along mid‐ocean ridges. Key Points We introduces a novel iterative stable algorithm for global Moho depth inversion To ensure convergence, a damping factor is applied to suppress high‐frequency noise

The Multiple Signal Classification (MUSIC) algorithm has become one of the most popular algorithms for estimating the direction-of-arrival (DOA) of multiple sources due to its simplicity and ease of implementation. Spherical microphone arrays can capture more sound field information than planar arrays. The collected multichannel speech signals can be transformed from the space domain to the spherical harmonic domain (SHD) for processing through spherical modal decomposition. The spherical harmonic domain MUSIC (SHD-MUSIC) algorithm reduces the dimensionality of the covariance matrix and achieves better DOA estimation performance than the conventional MUSIC algorithm. However, the SHD-MUSIC algorithm is prone to failure in low signal-to-noise ratio (SNR), high reverberation time (RT), and other multi-source environments. To address these challenges, we propose a novel joint spherical harmonic domain beam-space MUSIC (SHD-BMUSIC) algorithm in this paper. The advantage of decoupling the signal frequency and angle information in the SHD is exploited to improve the anti-reverberation property of the DOA estimation. In the SHD, the broadband beamforming matrix converts the SHD sound pressure to the beam domain output. Beamforming enhances the incoming signal in the desired direction and reduces the SNR threshold as well as the dimension of the signal covariance matrix. In addition, the 3D beam of the spherical array has rotational symmetry and its beam steering is decoupled from the beam shape. Therefore, the broadband beamforming constructed in this paper allows for the arbitrary adjustment of beam steering without the need to redesign the beam shape. Both simulation experiments and practical tests are conducted to verify that the proposed SHD-BMUSIC algorithm has a more robust adjacent source discrimination capability than the SHD-MUSIC algorithm.

This paper proposes a novel approach for improving the speech of a single speaker in noisy reverberant environments. The proposed approach is based on using a beamformer with a large number of virtual microphones with the suggested arrangement on an open sphere. Our method takes into account virtual microphone signal synthesizing using the non-parametric sound field reproduction in the spherical harmonics domain and the popular weighted prediction error method. We obtain entirely accurate beam steering towards a known source location with more directivity. The suggested approach is proven to perform effectively not just in boosting the directivity factor but also in terms of improving speech quality as measured by subjective metrics like the PESQ. In comparison to current research in the area of speech enhancement by beamformer, our experiments reveal more noise and reverberation suppression as well as improved quality in the enhanced speech samples due to the usage of virtual beam rotation in the fixed beamformer. Text for this section.

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.

In this paper, we propose and analyze an efficient spectral-Galerkin method based on a mixed formulation with dimension reduction for the Helmholtz transmission eigenvalue problem in spherical domains. By introducing an auxiliary function, we rewrite the original problem as an equivalent fourth-order coupled form in spherical coordinates. Using the properties of spherical harmonic and Laplace–Beltrami operator, we further decompose the original problem into a series of decoupled one-dimensional fourth-order linear eigenvalue problems, for which a new mixed variational formulation and its discretization is developed. For error estimates of numerical eigenvalues and eigenfunctions, we recall the spectral theory of compact operators. Towards this end, we derive the essential polar conditions, define a class of weighted Sobolev spaces, and most importantly, prove a sequence of two compact embedding properties for the weighted Sobolev spaces, based on which the spectral theory of compact operators for the variational formulation and discrete system can be established. Finally, some numerical examples are presented to confirm the theoretical error analysis and the efficiency of our algorithm.

A bstract We consider a non-supersymmetric domain-wall version of N = 4 SYM theory where five out of the six scalar fields have non-zero classical values on one side of a wall of codimension one. The classical fields have commutators which constitute an irreducible representation of the Lie algebra so (5) leading to a highly non-trivial mixing between color and flavor components of the quantum fields. Making use of fuzzy spherical harmonics on S 4 , we explicitly solve the mixing problem and derive not only the spectrum of excitations at the quantum level but also the propagators of the original fields needed for perturbative quantum computations. As an application, we derive the one-loop one-point function of a chiral primary and find complete agreement with a supergravity prediction of the same quantity in a double-scaling limit which involves a limit of large instanton number in the dual D3-D7 probe-brane setup.

Oral cancer surgery demands precise margin delineation to ensure complete tumor resection (healthy tissue margin ) while preserving postoperative functionality. Inadequate margins most frequently occur at the deep surgical margins, where tumors are located beneath the tissue surface; however, current fluorescent optical imaging systems are limited by their inability to quantify subsurface structures. Combining structured light techniques with deep learning may enable intraoperative margin assessment of 3D surgical specimens. A deep learning (DL)-enabled spatial frequency domain imaging (SFDI) system is investigated to provide subsurface depth quantification of fluorescent inclusions. A diffusion theory-based numerical simulation of SFDI was used to generate synthetic images for DL training. ResNet and U-Net convolutional neural networks were developed to predict margin distance (subsurface depth) and fluorophore concentration from fluorescence images and optical property maps. Validation was conducted using SFDI images of composite spherical harmonics, as well as simulated and phantom datasets of patient-derived tongue tumor shapes. Further testing was done in animal tissue with fluorescent inclusions. For oral cancer optical properties, the U-Net DL model predicted the overall depth, concentration, and closest depth with errors of , , and , respectively, using patient-derived tongue shapes with closest depths below 10 mm. In PpIX fluorescent phantoms of inclusion depths up to 8 mm, the closest subsurface depth was predicted with an error of . For tissue, the closest distance to the fluorescent inclusions with depths up to 6 mm was predicted with an error of . A DL-enabled SFDI system trained with images demonstrates promise in providing margin assessment of oral cancer tumors.

For a given plane domain, we add a constant multiple of the Dirac measure at a point in the domain and make a new domain called a quadrature domain. The quadrature domain is characterized as a domain such that the integral of a harmonic and integrable function over the domain equals the integral of the function over the given domain plus the integral of the function with respect to the added measure. The family of quadrature domains can be modeled as the Hele-Shaw flow with a free-boundary problem. We regard the given domain as the initial domain and the support point of the Dirac measure as the injection point of the flow. We treat the case in which the initial domain has a corner on the boundary and discuss the shape of the time-dependent domain around the corner immediately after the initial time. If the interior angle of the corner is less than

We present a method to recover absorption and reduced scattering spectra for each layer of a two-layer turbid media from spatial frequency-domain spectroscopy data. We focus on systems in which the thickness of the top layer is less than the transport mean free path   (  0.1  −  0.8l  *    )  . We utilize an analytic forward solver, based upon the N’th-order spherical harmonic expansion with Fourier decomposition   (  SHEFN  )   method in conjunction with a multistage inverse solver. We test our method with data obtained using spatial frequency-domain spectroscopy with 32 evenly spaced wavelengths within λ  =  450 to 1000 nm on six-layered tissue phantoms with distinct optical properties. We demonstrate that this approach can recover absorption and reduced scattering coefficient spectra for both layers with accuracy comparable with current Monte Carlo methods but with lower computational cost and potential flexibility to easily handle variations in parameters such as the scattering phase function or material refractive index. To our knowledge, this approach utilizes the most accurate deterministic forward solver used in such problems and can successfully recover properties from a two-layer media with superficial layer thicknesses.