Harmonic Basis Research Articles

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

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

Distribution amplitude and decay constant of 1S and 2S state light mesons in the light-front quark model

Studying meson structures is essential for gaining insights into the nonperturbative nature of Quantum Chromodynamics (QCD). This study will focus on calculating the decay constant and distribution amplitudes (DAs) of unflavored light mesons (π and ρ) in the 1S and 2S states using the Light Front Quark Model. This study utilizes the QCD-motivated Hamiltonian, taking into account both contact and smeared spin-spin interactions. The two lowest harmonic oscillator bases are employed in this work to achieve improved results for the 2S states. The study found the optimal mixing parameter in basis expansion coefficients to be θ = 10°. Light meson properties, including the mass spectrum, decay constant, and twist-2 DAs, are then predicted using model parameters fixed through the variational principle. While the contact spin-spin interaction yields poor accuracy, the results from the smeared interaction generally agree well with experimental data and other theoretical models with $ f_{\pi}^{1S}=130 $ MeV and $ f_{\rho}^{1S}=210 $ MeV in the mixed state. Unlike the case for the 1S state, it should be noted that the properties of the 2S state are sensitive to the mixing parameter θ. In addition, we observe that the decay constant for ρ(2S) is $ f_{\rho}^{2S}=116 $ MeV. While for π(2S) the decay constant is extremely small with the value of $ f_{\pi}^{2S}=0.9 $ MeV, which is mainly due to the dynamical chiral symmetry breaking.

Estimating the connectional brain template based on multi-view networks with bi-channel graph neural network

Multi-view networks constructed from multiple brain functional or structural measures (e.g., fractional anisotropy, fiber number, and fiber length, etc.) describe brain connectivity from different views. The connectional brain template (CBT) of multi-view networks from a given population provides a standard graph template incorporating complementary information for analyzing intergroup differences. However, preserving the complex topology of multi-view networks and building the CBT end-to-end remains challenging. Hence, we proposed a bi-channel convolution framework, including a local connection channel (LCC) with edge-conditioned convolution layers and a global network channel (GNC) with graph convolution network, to estimate the CBT in an end-to-end manner. The LCC captured the local connection topology across individual networks, and the GNC learned the global network topology through the high-order population network built by the similarity between the node strength distributions of individual networks for each view. Additionally, we improved topological similarity constraint in the loss function by minimizing the KL divergence of the node strength and degree distributions between the CBT and multi-view networks. Compared with four existing methods, our CBT exhibited optimal centrality, topological structure retention ability and differentiation between the younger and older age groups. Using the harmonic bases of the CBT, we found that the total harmonic energy of cortical thickness presented significant differences among different age groups. Our method provided a standard common template for examining the structural connection changes in the spatial domain and the anatomical features alterations in the harmonic domain, thus distinguishing the different population groups.

A single-image human body reconstruction method combining normal recovery and frequency domain completion

<p><span lang="EN-US">Reconstructing a 3D human body from a single image is convenient and efficient, but it faces challenges when the face is heavily occluded or the person is wearing loose clothing. In this paper, we propose a frequency domain-based method for completing missing parts of the human body using manifold harmonics and frequency domain analysis. Our approach involves linear interpolation of the incomplete 3D human body obtained through normal integration. The interpolated points are then projected onto the appropriate dimension of the frequency domain space using manifold harmonic bases. Through Laplacian mesh editing, the interpolated points are replaced, resulting in a refined and complete 3D human body. Our method surpasses the limitations of template human bodies and marching cubes algorithms, enabling more detailed feature reconstruction. By locally completing the body in a low-dimensional frequency domain space, our method avoids over smoothing and bulging issues, effectively filling the missing regions while maintaining mesh smoothness consistency. Experimental results demonstrate the effectiveness and superiority of our frequency domain-based completion method for accurate and detailed 3D human body reconstruction from single images.</span></p>

Spectral representation of EEG data using learned graphs with application to motor imagery decoding

Electroencephalography (EEG) data entail a complex spatiotemporal structure that reflects ongoing organization of brain activity. Characterization of the spatial patterns is an indispensable step in numerous EEG processing pipelines. We present a novel method for transforming EEG data into a spectral representation. First, we learn subject-specific graphs from each subject’s EEG data. Second, by eigendecomposition of the normalized Laplacian matrix of each subject’s graph, an orthonormal basis is obtained using which any given EEG map of the subject can be decomposed, providing a spectral representation of the data. We show that energy of EEG maps is strongly associated with low frequency components of the learned basis, reflecting the smooth topography of EEG maps. As a proof-of-concept for this alternative view of EEG data, we consider the task of decoding two-class motor imagery (MI) data. To this aim, the spectral representations are first mapped into a discriminative subspace for differentiating two-class data using a projection matrix obtained by the Fukunaga–Koontz transform (FKT). An SVM classifier is then trained and tested on the resulting features to differentiate MI classes. The method is benchmarked against features extracted from a subject-specific functional connectivity matrix as well as four alternative MI-decoding methods on Dataset IVa of BCI Competition III. Experimental results show the superiority of the proposed method over alternative approaches in differentiating MI classes, reflecting the added benefit of (i) decomposing EEG data using data-driven, subject-specific harmonic bases, and (ii) accounting for class-specific temporal variations in spectral profiles.

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

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

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

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

Doubly-charmed pentaquark in a quark model with a complete set of harmonic oscillator bases

Doubly-charmed pentaquark in a quark model with a complete set of harmonic oscillator bases

Bandura in the Late 1990s and Early 2000s Compositions by Maryna Denysenko: Genre and Stylistic Context of Ideas

The article summarizes the use of bandura in the compositional ideas of Maryna Denysenko, a famous Ukrainian composer (1962–2022). The nature of the bandura use in the artist’s works suggests the process of the academicization of the instrument. Engaging bandura in the Winter and Spring play and in the August the Sickle cycle, Maryna Denysenko makes this instrument a mediator between the musical material of a non-folklore nature and the folklore themes declared in the programmatic titles of the works. In general, Denysenko’s results demonstrate high mastery of compositional technique and artistic perfection, corresponding to contemporary musical trends. This once again confirms the urgent need to revive and explore her work. The analysis of Maryna Denysenko’s works featuring bandura revealed the following stylistic features of the composer’s creative method: a tendency to a programmatic approach, to the realization of folklore themes without explicit references to the characteristics of traditional genres of Ukrainian folklore in the musical language, and to a pronounced integrity of the composition. Her ideas in the field of instrumental bandura are characterized by the use of decentralized harmonic systems and modality in the principles of the harmonic basis of her compositions. These are the regularities of constructing colorful intonational material based on movements ofseconds asthe main structural microelements. The components of the author’sstyle of the instrumental chamber works with bandura were revealed, including the exchange ofroles between bandura and piano at certain moments of the work; elaborate composition with the dominating improvisation; a significant role of polyphonic techniques (use of equirhythmic counterpoint, various imitations, canons with two or more voices); numerous instances of inventive sound imagery (for example, the depiction of the sounds of drops and gurgling water in the third movement with the help of bells, a xylophone, or the layering of aleatoric techniques of bandura performance and pizzicato strings).

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

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

Matrix quantization of gravitational edge modes

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

Dirichlet‐type absorbing boundary conditions for peridynamic scalar waves in two‐dimensional viscous media

AbstractConstruction of absorbing boundary conditions (ABCs) for nonlocal models is generally challenging, primarily due to the fact that nonlocal operators are commonly associated with volume constrained boundary conditions. Moreover, application of Fourier and Laplace transforms, which are essential for the majority of available methods for ABCs, to nonlocal models is complicated. In this paper, we propose a simple method to construct accurate ABCs for peridynamic scalar wave‐type problems in viscous media. The proposed ABCs are constructed in the time and space domains and are of Dirichlet type. Consequently, their implementation is relatively simple, since no derivatives of the wave field are required. The proposed ABCs are derived at the continuum level, from a semi‐analytical solution of the exterior domain using harmonic exponential basis functions in space and time (plane‐wave modes). The numerical implementation is done using a meshfree collocation approach employed within a boundary layer adjacent to the interior domain boundary. The modes satisfy the peridynamic numerical dispersion relation, resulting in a compatible solution of the interior region (near‐field) with that of the exterior region (far‐field). The accuracy and stability of the proposed ABCs are demonstrated with several numerical examples in two‐dimensional unbounded domains.

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

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

Wave packet enriched finite elements for thermo-electro-elastic wave propagation problems with discontinuous wavefronts

In this article, we study the numerical properties of the recently developed local (element-domain) harmonic basis function enriched finite element (FE) for wave propagation problems. The study includes the effect of the enrichment harmonics on the conditioning of system matrices and how the diagonal scaling mass lumping technique influences the possible ill-conditioning and the solution accuracy. Subsequently, a wave packet enriched thermoelectromechanical FE formulation is presented for axisymmetric and planar wave propagation problems in piezoelastic media. The conventional Lagrange interpolations for the displacement, temperature, and electric potential fields are enriched with the element-domain sinusoidal functions which satisfy the partition of unity condition. The extended Hamilton’s principle and constitutive relations of the Lord–Shulman and Green–Lindsay generalized piezothermoelasticity are employed to derive the coupled system of equations of motion which is solved using Newmark- direct time integration scheme. The performance of the proposed elements is assessed and wave characteristics are studied for the problems of thermoelastic shock waves in an elastic annular disk and thermoelectric shock waves in a piezoelectric hollow cylinder. The element shows significant improvement in the computational efficiency and accuracy over the conventional FE for these problems involving sharp discontinuities in the fields at the wavefronts.

Solution method for time-frequency responds of servo control systems based on frequency domain characteristics

In view of that the tracking error originates from the amplitude attenuation and phase lag of the setpoints frequency component caused by the servo control system at each moment, this paper puts forward the idea of obtaining the setpoints time-frequency response (STFR) by solving the amplitude attenuation and phase lag of each frequency component of the setpoints at each moment. First, the feasibility of the proposed idea is proved theoretically; Then, the setpoints time-frequency transform is carried out and the solving method of STFR and setpoints time-frequency response error (STFRE) is established; Finally, taking the servo control system as the object, the STFR and STFRE are solved, and the corresponding relationship with the output time-frequency transformation and tracking error is analyzed to verify the proposed method. The contributions of the proposed method are: the method achieves the solution of STFR based on the amplitude attenuation and phase lag of the servo system; it avoids the limitation that only harmonic wavelet bases must be used in the existing method and can be applied to the solution of the time-frequency response of actual setpoints; it can separate the setpoints loss or tracking error caused by amplitude attenuation and phase lag; it can determine the redundancy or lack of strategies such as filters for certain setpoints and servo control system.

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

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

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

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

Hadronic structure on the light front. IV. Heavy and light baryons

This work is a continuation in our series of papers, that addresses quark models of hadronic structure on the light front. The chief focus of this paper is the quantum-mechanical solution of the three-quark model Hamiltonian, describing baryons. In Jacobi coordinates, we use a harmonic oscillator basis for the transverse directions. For the longitudinal momentum fractions $x_i$, the pertinent basis follows from quantum mechanics in a "triangular cup" potential, which we solve exactly. We calculate the masses and light front wave functions for the flavor symmetric $\frac 32^+$ baryons $bbb,ccc,sss,uuu$.

Machine learning light hypernuclei

We employ a feed-forward artificial neural network to extrapolate at large model spaces the results of ab-initio hypernuclear No-Core Shell Model calculations for the Λ separation energy BΛ of the lightest hypernuclei, Λ3H, Λ4H and Λ4He, obtained in computationally accessible harmonic oscillator basis spaces using chiral nucleon-nucleon, nucleon-nucleon-nucleon and hyperon-nucleon interactions. The overfitting problem is avoided by enlarging the size of the input dataset and by introducing a Gaussian noise during the training process of the neural network. We find that a network with a single hidden layer of eight neurons is sufficient to extrapolate correctly the value of the Λ separation energy to model spaces of size Nmax=100. The results obtained are in agreement with the experimental data in the case of Λ3H and the 0+ state of Λ4He, although they are off of the experiment by about 0.3 MeV for both the 0+ and 1+states of Λ4H and the 1+ state of Λ4He. We find that our results are in excellent agreement with those obtained using other extrapolation schemes of the No-Core Shell Model calculations, showing this that an ANN is a reliable method to extrapolate the results of hypernuclear No-Core Shell Model calculations to large model spaces.

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

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