Spherical Harmonic Transform Research Articles

CunuSHT: GPU Accelerated Spherical Harmonic Transforms on Arbitrary Pixelizations

Abstract We present cunuSHT https://github.com/Sebastian-Belkner/cunuSHT, a general-purpose Python package that wraps a highly efficient CUDA implementation of the nonuniform spin-0 spherical harmonic transform. The method is applicable to arbitrary pixelization schemes, including schemes constructed from equally-spaced iso-latitude rings as well as completely nonuniform ones. The algorithm has an asymptotic scaling of $\mathcal {O}{(\ell _{\rm max}^3)}$ for maximum multipole ℓmax and can be made to achieve machine precision accuracy, considering band-limited transforms for which $N\approx \ell _{\rm max}^2$ (where N is the number of pixels in the map). While cunuSHT is developed for applications in cosmology in mind, it is applicable to various other interpolation problems on the sphere. We outperform the fastest available CPU algorithm at problem sizes ℓmax ∼ 4 · 102 and larger. The speed-up increases with the problem size and reaches a factor of up to 5 for problems with a nonuniform pixelization and ℓmax > 4 · 103 when comparing a single modern GPU to a modern 32-core CPU. This performance is achieved by utilizing the double Fourier sphere method in combination with the nonuniform fast Fourier transform and by avoiding transfers between the host and device. For scenarios without GPU availability, cunuSHT wraps existing CPU libraries. cunuSHT is publicly available and includes tests, documentation, and demonstrations.

FSH3D: 3D Representation via Fibonacci Spherical Harmonics

AbstractSpherical harmonics are a favorable technique for 3D representation, employing a frequency‐based approach through the spherical harmonic transform (SHT). Typically, SHT is performed using equiangular sampling grids. However, these grids are non‐uniform on spherical surfaces and exhibit local anisotropy, a common limitation in existing spherical harmonic decomposition methods. This paper proposes a 3D representation method using Fibonacci Spherical Harmonics (FSH3D). We introduce a spherical Fibonacci grid (SFG), which is more uniform than equiangular grids for SHT in the frequency domain. Our method employs analytical weights for SHT on SFG, effectively assigning sampling errors to spherical harmonic degrees higher than the recovered band‐limited function. This provides a novel solution for spherical harmonic transformation on non‐equiangular grids. The key advantages of our FSH3D method include: 1) With the same number of sampling points, SFG captures more features without bias compared to equiangular grids; 2) The root mean square error of 32‐degree spherical harmonic coefficients is reduced by approximately 34.6% for SFG compared to equiangular grids; and 3) FSH3D offers more stable frequency domain representations, especially for rotating functions. FSH3D enhances the stability of frequency domain representations under rotational transformations. Its application in 3D shape reconstruction and 3D shape classification results in more accurate and robust representations. Our code is publicly available at https://github.com/Miraclelzk/Fibonacci-Spherical-Harmonics.

Fast and Accurate Spherical Harmonic Transform for Spatio-Temporal Regular Grid Data

Fast and Accurate Spherical Harmonic Transform for Spatio-Temporal Regular Grid Data

Analyzing Earth's Position based on the Anisotropic Characteristics of Cosmic Microwave Background Radiation

This paper elaborates on the results of an exhaustive study regarding the Earth's position in the universe based on the Cosmic Microwave Background (CMB) radiation map, which is the latest discovery in modern astronomy. CMB radiation provides crucial insights into the early distribution of mass and energy in the universe. The aim of this research is to understand the theory and mechanisms behind the formation of polarity structures in the CMB and analyze their correlation with Earth's position in the overall structure of the universe. Intensity measurement data of CMB radiation published by COBE, WMAP, and Planck present temperature distribution data in coordinates in FITS (Flexible Image Transport System) format files. Subsequently, a spherical harmonic transformation is performed to obtain spherical harmonic coefficients a_lm as equations that represent dipole, quadrupole, octupole models, and various other multipole models. The analysis of the correlation in the temperature distribution of CMB radiation involves detailing various patterns found in the dipole, quadrupole, and octopole models, demonstrating quasi-symmetry characteristics with Earth at its center. An analysis of anisotropic CMB data yields an interesting hypothesis that the Earth's position plays a role in shaping the structure of the universe on a certain scale. Even more extremely, it can be said that Earth is at the center of the universe. This finding prompts profound contemplation about Earth's position in the structure of the cosmos, opening the door for further research in this field.

The Spherical Fast Multipole Method (sFMM) for Gravitational Lensing Simulation

In this paper, we present a spherical Fast Multipole Method (sFMM) for ray-tracing simulation of gravitational lensing on a curved sky. The sFMM is a nontrivial extension of the Fast Multiple Method to sphere , and it can accurately solve the Poisson equation with time complexity of , where N is the number of particles. It is found that the time complexity of the sFMM is near O(N) and the computational accuracy can reach 10−10 in our test. In addition, compared with the Fast Spherical Harmonic Transform, the sFMM is not only faster but also more accurate, as it has the ability to reserve high-frequency components of the density field. These merits make the sFMM an optimum method to simulate the gravitational lensing on a curved sky, which is the case for upcoming large-area sky surveys, such as the Vera Rubin Observatory and the China Space Station Telescope.

An End-to-End Scheme for Learning Over Compressed Data Transmitted Through a Noisy Channel

Within the emerging area of goal-oriented communications, this paper introduces a novel end-to-end transmission scheme dedicated to learning over a noisy channel, under the constraint that no prior training dataset is available. In this scheme, the transmitter makes use of powerful Spherical Harmonic Transform and Irregular Hexagonal Quadratic Amplitude Modulation techniques, while the receiver relies on a Complex-Valued Neural Network (CVNN) so as to realize the learning task onto the received noisy data. As a main feature of the proposed scheme, the transmitter is fixed and does not depend on the source statistics, while the receiver is trained from a first data transmission phase, thus providing an efficient transmission-versus-learning approach under the considered constraint. The proposed transmission scheme may be adapted to a variety of learning problems, and the paper specifically investigates clustering and classification, two very common learning tasks. In the last part of the paper, the source/channel coding rate of the proposed transmission scheme is evaluated theoretically and from numerical simulations. This analysis shows a clear advantage in terms of coding rate of our scheme compared to conventional coding approaches, when targeting the same learning performance level.

Accelerating Spherical Harmonic Transforms for a Large Number of Sky Maps

The spherical harmonic transform is a powerful tool in the analysis of spherical data sets, such as cosmic microwave background data. In this work, we present a new scheme for spherical harmonic transforms that supports both CPU and GPU computations, which is especially efficient on a large number of sky maps. By comparing our implementation with the standard Libsharp-HEALPix program, we demonstrate a 2–10 times speedup for the CPU implementation, and an up to 30 times speedup when a state-of-the-art GPU is employed. This new scheme’s software package is available via an open-source GitHub repository.

Quantification of Crystal Packing Similarity from Spherical Harmonic Transform

In this work, we present a new computational approach to characterize and classify molecular packing in the solid states. The key idea is to project each neighboring molecule (or short contact) from the centered molecule into a unit sphere according to the interaction energy. Consequently, the similarity between two spherical images can be evaluated from the spherical harmonics expansion based on the maximum cross-correlation. We apply this approach to successfully reproduce the previous packing assignment on a small amount of data with an improved categorization. Furthermore, we conduct a packing similarity analysis over 2000 hydrocarbon crystal data sets and uncover a set of abundant packing motifs. Unlike the previous approaches based on the subjective visual comparison at the real space, our approach provides a more robust way to measure the packing similarity, thus paving the way for a rapid classification of large scale crystal data.

Magnitude Modeling of Personalized HRTF Based on Ear Images and Anthropometric Measurements

In this paper, we propose a global personalized head-related transfer function (HRTF) method based on anthropometric measurements and ear images. The model consists of two sub-networks. The first is the VGG-Ear Model, which extracts features from the ear images. The second sub-network uses anthropometric measurements, ear features, and frequency information to predict the spherical harmonic (SH) coefficients. Finally, the personalized HRTF is obtained through inverse spherical harmonic transform (SHT) reconstruction. With only one training, the HRTF in all directions can be obtained, which greatly reduces the parameters and training cost of the model. To objectively evaluate the proposed method, we calculate the spectral distance (SD) between the predicted HRTF and the actual HRTF. The results show that the SD provided by this method is 5.31 dB, which is better than the average HRTF of 7.61 dB. In particular, the SD value is only increased by 0.09 dB compared to directly using the pinna measurements.

Multiple Scattering by Two PEC Spheres Using Translation Addition Theorem

An analysis of multiple scattering by two Perfect Electric Conducting (PEC) spheres using translation Addition Theorem (AT) for spherical vector wave functions is presented. Specifically, the Cruzan formalism is used to represent the AT for spherical harmonics, which introduces the translation coefficients for transformation of spherical harmonics from one coordinate to another. The adoption of these coefficients with the use of two PEC spheres in a near zone region makes the calculation of multiple scattering electric fields very efficient. As an illustration, the mathematical formation using advanced computational approaches was inspected. Then, the generic truncation criteria in the scattered electric field by two PEC spheres was deeply investigated using translation AT. However, the numerical validation was obtained using Comsol simulation software. This approach will allow to evaluate the scattering from macro-structures composed of spherical particles, i.e., biological molecules, clouds of airborne particles, etc. An original and fully general solution to the problem using vector quantities is introduced, and the convergence of the solution in several numerical examples is also demonstrated. This approach takes into account the effect of multiple scattering by two PEC spheres for spherical vector function.

Three-dimensional reconstruction of realistic stone-based materials with controllable stone inclusion geometries

Stone-based materials are heterogeneous construction materials that mainly consist of coarse stones and fine matrices. When using the numerical simulation tools (FEM, DEM, FDEM) to study the mechanical properties of the tone-based materials, the irregular stone-inclusion geometries cannot be neglected. The main objective of this study is to generate three-dimensional models of realistic stone-based materials with controllable stone-inclusion geometries. To achieve this goal, two major methodologies are adopted: (1) the Spherical Harmonic Transform (SHT) method and Inverse Monte-Carlo (IMC) algorithm are employed to randomly generate the 3D particle model of irregular stones with controllable geometries at three different shape scales, including form, roundness and roughness, and (2) based on the Spherical Harmonic function, an overlapping detection algorithm is proposed to facilitate the rapid allocation of the 3D SHT-based particles. The proposed algorithm can quantitatively control several geometrical features of the generated stone-based materials in an efficient and precise manner. Finally, the application of the proposed approach is demonstrated through the discrete modelling of stone-based materials with different stone contents and stone shapes. The proposed study has the significance to pave a viable pathway for stochastic modelling of stone-based materials pertaining to various construction and manufacture processes.

An efficient algorithm to compute the RCS of UAV swarmbased on spherical harmonic transform

ABSTRACT It is time-consuming to simulate the unmanned aerial vehicle (UAV) swarm by the Multi-Level Fast Multipole Algorithm (MLFMA). This paper proposes a method to accelerate the simulation by neglecting the coupling among UAVs and computing the scattered field by Spherical Harmonic Transform (SHT). The key point is evaluating the scattered electric field of a UAV on an eccentric sphere by changing the order of summation of the spherical harmonics and then accelerating the summation by Fast Fourier Transform (FFT). The example of a swarm of 6 UAVs shows that the bistatic Radar Cross Section (RCS) obtained by the proposed method is in good agreement with the RCS obtained by MLFMA, and the proposed method requires 4.76 times less memory and is 10.93 times faster than MLFMA.

An implementation of single‐precision fast spherical harmonic transform in Yin–He global spectral model

AbstractIn this article, we describe an implementation of single‐precision fast spherical harmonic transform (SHT) in the Yin–He global spectral model (YHGSM). The potential of single‐precision arithmetic is explored for accelerating fast SHT. In particular, we assess the impact of using single‐precision fast SHT on the meteorological skill of YHGSM. Compared to double‐precision SHT, single‐precision fast SHT against is accelerated by about 59.83%, whereas the run‐time reduction achieved 25.28% in model integration for TL2047L137. The simulation results indicate that single‐precision fast SHT can improve computational efficiency without a noticeable impact on the forecast skill.

Analysis of Synoptic Disturbance in Maritim Continent Using Spherical Harmonics Transformation Method

This study aims to modify the idea of ​​WK99-analyzing the existence of signature in wave-number and frequency spectrum when the analyzed Outgoing Longwave Radiation (OLR) data is associated with a unique phenomenon in Maritime Continent (MC), Borneo vortex (BV). Although BV is often related to easterly equatorial wave disturbances, there was no specific study to examine its behavior in the spectral domain. The purpose of this study is to develop a method to diagnose certain signatures of BV through spectrum analysis of OLR data when BV occurs.
 In contrast to previous studies, to present the unique phenomenon in MC as a target for diagnostics, spectrum analysis is performed by Spherical Harmonics (SH). The results of the OLR data spectrum comparison when BV occurs with the OLR data spectrum when BV does not occur that in the anti-symmetrical component when BV occurs, the tropical depression-mixed Rossby gravity (TD-MRG) wave spectrum is stronger, especially in zonal wave-numbers 5 to 10 and frequencies 0.12 to 0.2. Similar to the anti-symmetrical component, the TD-MRG spectrum of the symmetrical component is also stronger at zonal wave-numbers 5 to 10 and frequencies 0.12 to 0.2. Moreover, the westward-propagating inertio-gravity (WIG) wave spectrum in the symmetrical component is also stronger at the time of BV than when there is no BV, especially at zonal wave-numbers 7 to 13 and frequencies 0.38 to 0.5. It can be concluded that when BV occurs, equatorial waves that propagate to the west, especially the types of TD-MRG and WIG waves are stronger than when BV did not occur. The results of durational grouped spectrum analysis of OLR data when BV occurs indicate that the longer the duration of BV occurs, the stronger the spectrum of TD-MRG and WIG wave types gets.

On the statistical performance of spherical harmonics MUSIC

In this paper, we present a theoretical analysis of the statistical performance of spherical harmonics multiple signal classification (SH-MUSIC), which is a representative localization algorithm for spherical microphone arrays. What we are concerned about is the influence of spherical harmonics transformation on localization performance, which is mainly manifested in the following aspects: how the spatial aliasing affects the estimation of direction of arrivals (DOAs) in the large aperture case; and how to characterize the resolution capacity of SH-MUSIC in the small aperture case. We advance the research through the detailed examination of the null spectrum of SH-MUSIC. First, by applying the Taylor series expansion of the null spectrum, we give a bias expression to expound the impact of spatial aliasing on DOA estimation. The post-processing for bias reduction is then suggested. Second, utilizing the statistical distribution of the sample covariance matrix and also using the addition theorem of spherical harmonics, we derive an expression for the threshold at which SH-MUSIC can resolve two closely spaced sources to clarify SH-MUSIC’s advantages in the small aperture case. Besides, numerical simulations are provided to verify the analytical results.

A spherical harmonic transform approach to the indexing of electron back-scattered diffraction patterns

A new approach is proposed for the indexing of electron back-scattered diffraction (EBSD) patterns. The algorithm employs a spherical master EBSD pattern and computes its cross-correlation with a back-projected experimental pattern using the spherical harmonic transform (SHT). This approach is significantly faster than the recent dictionary indexing algorithm, but shares the latter’s robustness against noise. The underlying theory is presented, followed by example applications, one on a series of Ni EBSD data sets recorded with decreasing signal-to-noise ratio, the other on a large shot-peened Al data set. The dependence of indexing speed and memory usage on the SHT bandwidth is explored. The speed gains of the new algorithm are achieved by executing real-valued Fast Fourier Transforms, explicitly incorporating crystallographic symmetry in the cross-correlation computation, and using efficient loop ordering to improve the caching behavior. The algorithm produces a cross-correlation array in the zyz Euler space; an orientation refinement procedure is proposed based on analytical derivatives of the Wigner d functions. The new approach can be applied to any diffraction modality for which the scattered intensity can be represented on a spherical surface.

Computing Mutual Impedance of Antennas by Spherical Harmonic Transform

The coupling of two antennas can be described with mutual impedance. This paper proposes computing the mutual impedance by reaction theorem, which involves integrating vector products of near fields over a sphere surrounding the second antenna. We express the near fields of both antennas by the vector spherical harmonic transform (SHT), evaluate the near fields of the first antenna on the eccentric sphere by a new fast Fourier transform (FFT)/interpolation method efficiently, and then obtain the mutual impedance by the reaction integrals. This new method has a computation complexity of $O(N^{3})$ , where N represents the truncation order in SHT. A comparison with the higher order moment method (HOMOM) indicates that the proposed method is a fast, accurate, stable, and universal method to analyze the coupling of any two antennas.

Fast algorithms for integral formulations of steady-state radiative transfer equation

We investigate integral formulations and fast algorithms for the steady-state radiative transfer equation with isotropic and anisotropic scattering. When the scattering term is a smooth convolution on the unit sphere, a model reduction step in the angular domain using the Fourier transformation in 2D and the spherical harmonic transformation in 3D significantly reduces the number of degrees of freedoms. The resulting Fourier coefficients or spherical harmonic coefficients satisfy a Fredholm integral equation of the second kind. We study the uniqueness of the equation and proved an a priori estimate. For a homogeneous medium, the integral equation can be solved efficiently using the FFT and iterative methods. For an inhomogeneous medium, the recursive skeletonization factorization method is applied instead. Numerical simulations demonstrate the efficiency of the proposed algorithms in both homogeneous and inhomogeneous cases and for both transport and diffusion regimes.

Equations of Interaction of Weakly Non-Spherical Gas Bubbles in Liquid

Equations of spatial hydrodynamic interaction of gas bubbles in an ideal incompressible liquid are derived with allowing for small deformations of their surfaces. The derivation is carried out by the method of spherical functions using the Bernulli integral, the kinematic and dynamic boundary conditions on the bubble surfaces. At that, an original expression of transformation of irregular solid spherical harmonics at parallel translation of the coordinate system is applied. The result of the derivation is a system of ordinary differential equations of the second order in the radii of the bubbles, the position-vectors of their centers and the vectors characterizing small deviations of the bubble surfaces from the spherical one. Due to their compactness, these equations are much more convenient for analysis and numerical solution than their known analogs.

Performance Evaluation of the Fast Spherical Harmonic Transform Algorithm in the Yin–He Global Spectral Model

Abstract In this paper, we describe an implementation of the fast spherical harmonic transform (SHT) algorithm in the Yin–He global spectral model (YHGSM). A new analysis method is proposed to study the potential instability of interpolative decomposition and to evaluate the performance of fast SHT on the MilkyWay-2 supercomputer. The novel aspect of the proposed method is the use of the computational complexity analysis method by studying the properties of interpolative decompositions. Furthermore, three test cases are conducted to verify fast SHT in YHGSM. The results demonstrate that fast SHT is feasible and efficient for reducing the computational complexity and memory cost of SHT while still keeping enough accuracy.