Theory Of Spherical Harmonics Research Articles

Spin-Weighted Spherical Harmonics for Polarized Light Transport

The objective of polarization rendering is to simulate the interaction of light with materials exhibiting polarization-dependent behavior. However, integrating polarization into rendering is challenging and increases computational costs significantly. The primary difficulty lies in efficiently modeling and computing the complex reflection phenomena associated with polarized light. Specifically, frequency-domain analysis, essential for efficient environment lighting and storage of complex light interactions, is lacking. To efficiently simulate and reproduce polarized light interactions using frequency-domain techniques, we address the challenge of maintaining continuity in polarized light transport represented by Stokes vectors within angular domains. The conventional spherical harmonics method cannot effectively handle continuity and rotation invariance for Stokes vectors. To overcome this, we develop a new method called polarized spherical harmonics (PSH) based on the spin-weighted spherical harmonics theory. Our method provides a rotation-invariant representation of Stokes vector fields. Furthermore, we introduce frequency domain formulations of polarized rendering equations and spherical convolution based on PSH. We first define spherical convolution on Stokes vector fields in the angular domain, and it also provides efficient computation of polarized light transport, nearly on an entry-wise product in the frequency domain. Our frequency domain formulation, including spherical convolution, led to the development of the first real-time polarization rendering technique under polarized environmental illumination, named precomputed polarized radiance transfer, using our polarized spherical harmonics. Results demonstrate that our method can effectively and accurately simulate and reproduce polarized light interactions in complex reflection phenomena, including polarized environmental illumination and soft shadows.

Development of an individual 3D particle reconstruction method for discrete mechanical modeling: Interpolation by Fourier composition

Reconstruction methods for granular materials are an essential step in achieving an accurate geometrical basis in the use of particle methods such as the well-known discrete element method. In this study, a novel continuous analytical method will be proposed to achieve individual 3D reconstructions of granular materials. For this purpose, the 2D Fourier Descriptor theory is used and a generalization of it to the 3D case is obtained, which is quite different from the spherical harmonics theory. Some limitations existing in the original model will be addressed by developing and calibrating a generalized approach that results in a total of four main models. Particles from a study sample are reconstructed, and the models are evaluated based on four generalized error metrics, as well as the presence of some specific limitations detected in the models. Finally, the models are compared using a decision-making method in different decisional scenarios.

3D meso-scale simulation of chloride ion transportation in cracked concrete considering aggregate morphology

Many experimental and numerical studies have been conducted to explain chloride ion penetration in concrete including aggregates, ITZ and mortar. To my best knowledge, there are few devotion on the relationship between three-dimensional real cracks and chloride ion transportation. Based on spherical harmonic theory, reconstruct the particle generated from smoothing Voronoi polyhedron cell by Loop subdivision to simulate the morphology of real aggregate, and this process performs relatively good efficiency in generating sets of concrete model with different features (mainly aggregate volume fraction, fractal dimension). Consequently, a novel method of representing crack characteristics based on spherical harmonic theory is proposed for analysis, construct a bridge between aggregate morphology and crack characteristics. In view of chloride penetration depth, crack changes the profiles significantly, which accounts for the phenomenon that the existence of crack makes statistical results of chloride concentration on different cross-sections of models fit well with bimodal Gaussian distribution. Volume fraction and morphology of aggregates are mainly correlated to decline rate of gap between chloride concentration of region near crack and the remaining. Both are meaningful for further predict the time to corrosion initiation in reinforced concrete structures in marine environment.

A Sturm‐Liouville equation on the crossroads of continuous and discrete hypercomplex analysis

The paper studies discrete structural properties of polynomials that play an important role in the theory of spherical harmonics in any dimensions. These polynomials have their origin in the research on problems of harmonic analysis by means of generalized holomorphic (monogenic) functions of hypercomplex analysis. The Sturm‐Liouville equation that occurs in this context supplements the knowledge about generalized Vietoris number sequences , first encountered as a special sequence (corresponding to ) by Vietoris in connection with positivity of trigonometric sums. Using methods of the calculus of holonomic differential equations, we obtain a general recurrence relation for , and we derive an exponential generating function of expressed by Kummer's confluent hypergeometric function.

Contributions to Modified Spherical Harmonics in Four Dimensions

A modification of the classical theory of spherical harmonics in four dimensions is presented. The space $$\mathbb {R}^4 = \{(x,y,t,s)\}$$ is replaced by the upper half space $${\mathbb {R}}_{+}^{4}=\left\{ (x,y,t,s), s > 0 \right\} $$ , and the unit sphere S in $$\mathbb {R}^4$$ by the unit half sphere $$S_{+}=\left\{ (x,y,t,s): x^2 +y^2+ t^2+ s^2 =1, s > 0 \right\} $$ . Instead of the Laplace equation $$\Delta h = 0$$ we shall consider the Weinstein equation $$s\Delta u + k \frac{\partial u }{\partial s}= 0$$ , for $$k \in \mathbb {N}$$ . The Euclidean scalar product for functions on S will be replaced by a non-Euclidean one for functions on $$S_{+}$$ . It will be shown that in this modified setting all major results from the theory of spherical harmonics stay valid. In addition we shall deduct—with respect to this non-Euclidean scalar product—an orthonormal system of homogeneous polynomials, which satisfies the above Weinstein equation.

Finite element solution of the time-dependent SP3 equations using an implicit integration scheme

Abstract A coupled thermal-hydraulics and reactor physics code system is being developed at the Institute of Nuclear Techniques of the Budapest University of Technology and Economics based on a higher-order transport approximation, the simplified spherical harmonics theory. The advantage of this method is that with a small increase in computational demand, it provides additional accuracy compared to diffusion theory. Besides - due to the fact that the multi-group SP3 and diffusion equations have a mathematically similar form - it requires minimal effort to implement an SP3 solution algorithm to an existing diffusion code. This paper focuses on an algorithm developed by the authors which applies Galerkin weighted residual method for spatial and theta method for time discretization. Results of two-group kinetic SP3 calculations performed with the SPNDYN code are also presented for various one-dimensional perturbations taking into account the delayed neutron precursor balance equations as well. The flexible nature of the SP3 equations makes the developed code a good starting point for more realistic dynamic calculations in the future.

Modified Spherical Harmonics in Several Dimensions

A modification of the classical theory of spherical harmonics is presented. The space $${\mathbb {R}}^d = \{(x_1,\ldots ,x_d)\}$$ is replaced by the upper half space $${{\mathbb {R}}}_{+}^{d}=\left\{ (x_1,\ldots ,x_d), x_d > 0 \right\} $$, and the unit sphere $$S^{d-1}$$ in $${\mathbb {R}}^d$$ by the unit half sphere $$S_{+}^{d-1}=\left\{ (x_1,\ldots ,x_d): x_1^2 + \cdots + x_d^2 =1, x_d > 0 \right\} $$. Instead of the Laplace equation $$\Delta h = 0$$ we shall consider the Weinstein equation $$x_d\Delta u + (d-2)\frac{\partial u }{\partial x_d}= 0$$. The Euclidean scalar product for functions on $$S^{d-1}$$ will be replaced by a non-Euclidean one for functions on $$S_{+}^{d-1}$$. It will be shown that in this modified setting all major results from the theory of spherical harmonics stay valid. In case $$d=3$$ and $$d=4$$ the modified theory has already been treated.

Torque Calculation of Permanent Magnet Spherical Motor Based on Virtual Work Method

A double-layer permanent magnet spherical motor (PMSM) with same spatial distribution of stator poles and rotor poles is designed based on spherical harmonic theory in this article. The stator coils are equivalent to permanent magnets to simplify the calculation, and the torque model based on virtual work method is derived. First, the torque model of single pair of poles is derived on the premise of considering the fundamental component of the magnetic field. Then, the relative positions of stator poles and rotor poles, the direction angles of torque components are discussed. Finally, the torque model is obtained by space vector theory and superposition theorem. Moreover, the simulation and experiment are implemented to verify the feasibility and correctness of the proposed torque model. The obtained torque model in this article has similar expression with that of traditional motor, which lays a theoretical foundation for the structure optimization and control strategy of PMSM.

More on Modified Spherical Harmonics

A modification of the classical theory of spherical harmonics is presented. The space $${\mathbb {R}}^d = \{(x_1,\ldots ,x_d)\}$$ is replaced by the upper half space $${{\mathbb {R}}}_{+}^{d}=\left\{ (x_1,\ldots ,x_d), x_d > 0 \right\} $$ , and the unit sphere $$S^{d-1}$$ in $${\mathbb {R}}^d$$ by the unit half sphere $$S_{+}^{d-1}=\left\{ (x_1,\ldots ,x_d): x_1^2 + \cdots + x_d^2 =1, x_d > 0 \right\} $$ . Instead of the Laplace equation $$\Delta h = 0$$ we shall consider the Weinstein equation $$x_d\Delta u + k \frac{\partial u }{\partial x_d}= 0$$ , for $$k \in {\mathbb {N}}$$ . The Euclidean scalar product for functions on $$S^{d-1}$$ will be replaced by a non-Euclidean one for functions on $$S_{+}^{d-1}$$ . It will be shown that in this modified setting all major results from the theory of spherical harmonics stay valid. In case $$k=d-2$$ the modified theory has already been treated by the author.

Modified Spherical Harmonics in Four Dimensions

The classical theory of spherical harmonics on the unit sphere S is well-known. In an earlier paper, entitled “Modified Spherical Harmonics”, we dealt with a modification of this theory, adapted to the half-sphere $$S_{+}$$ , in case of three dimensions. In the present paper we extend these results to the four-dimensional case. Although the results look quite similar, their proofs are not. In $$\mathbb {R}^4 =\left\{ (x,y,t,s) \right\} $$ the Laplace equation $$\Delta h=0$$ will be replaced by the equation $$s\Delta u+2\,\frac{\partial u}{\partial s}=0$$ . Homogeneous polynomial solutions of this equation, if restricted to the half-sphere $$S_{+}=\left\{ (x,y,t,s): x^2 + y^2 + t^2 + s^2=1, s > 0 \right\} $$ are called modified spherical harmonics. Endowed with a non-Euclidean scalar product on $$S_{+}$$ , these functions behave like the classical spherical harmonics on the full sphere in $$\mathbb {R}^4$$ . We shall give an explicit expression for the corresponding zonal harmonics and dwell on their connection with the Poisson-type kernel, adapted to the above differential equation. Finally we shall give an explicit orthonormal system of modified spherical harmonics.

Modified Spherical Harmonics

A modification of the classical theory of spherical harmonics is presented. The unit sphere S in $${\mathbb{R}^3 = \{(x,y,t)\}}$$ is replaced by the half-sphere S + in the upper half space, the Euclidean scalar product on S by a non-Euclidean one on S +, and the Laplace equation $${\Delta h = 0}$$ by the equation $${t\Delta v + \frac{\partial v }{\partial t}= 0}$$ . It will be shown that most results from the theory of spherical harmonics in $${\mathbb{R}^3}$$ stay valid in this modified setting.

Uncertainty reevaluation in determining the volume of a silicon sphere by spherical harmonics in an Avogadro project

To determine the Avogadro constant with a target relative uncertainty of 2 × 10−8, the uncertainty component of the silicon sphere's volume introduced by the spherical harmonics method, which is usually used in determining the sphere's volume, is reevaluated. By means of representing the shape of the silicon sphere by an ellipsoid with Gaussian white noise in its diameters, the uncertainty of the current mapping methods based on the spherical harmonics theory can be estimated theoretically. It is evidenced that the uncertainty component attributed to the current mapping method is underestimated. To eliminate this effect as much as possible, the number of mapping points should be increased to more than before. Moreover, a new mapping method is proposed to accomplish the equal-area mapping with large number points on the silicon sphere.

Binding Balls: Fast Detection of Binding Sites Using a Property of Spherical Fourier Transform

The functional prediction of proteins is one of the most challenging problems in modern biology. An established computational technique involves the identification of three-dimensional local similarities in proteins. In this article, we present a novel method to quickly identify promising binding sites. Our aim is to efficiently detect putative binding sites without explicitly aligning them. Using the theory of Spherical Harmonics, a candidate binding site is modeled as a Binding Ball. The Binding Ball signature, offered by the Spherical Fourier coefficients, can be efficiently used for a fast detection of putative regions. Our contribution includes the Binding Ball modeling and the definition of a scoring function that does not require aligning candidate regions. Our scoring function can be computed efficiently using a property of Spherical Fourier transform (SFT) that avoids the evaluation of all alignments. Experiments on different ligands show good discrimination power when searching for known binding sites. Moreover, we prove that this method can save up to 40% in time compared with traditional approaches.

A generalization of the fundamental theorem of spherical harmonic theory

We consider a boundary-value problem of the first kind for a self-adjoint differential operator with constant coefficients on a domain in ℝn bounded by an ellipsoid; boundary conditions are defined by an arbitrary polynomial of degree N. It is proved that the solution of the problem is again a polynomial of degree ≤N.

Spherical harmonics and integration in superspace

In this paper, the classical theory of spherical harmonics in is extended to superspace using techniques from Clifford analysis. After defining a super-Laplace operator and studying some basic properties of polynomial null-solutions of this operator, a new type of integration over the supersphere is introduced by exploiting the formal equivalence with an old result of Pizzetti. This integral is then used to prove orthogonality of spherical harmonics of different degree, Green-like theorems and also an extension of the important Funk–Hecke theorem to superspace. Finally, this integration over the supersphere is used to define an integral over the whole superspace, and it is proven that this is equivalent with the Berezin integral, thus providing a more sound definition of the Berezin integral.

Efficient 3D shape matching and retrieval using a concrete radialized spherical projection representation

We present a 3D shape retrieval methodology based on the theory of spherical harmonics. Using properties of spherical harmonics, scaling and axial flipping invariance is achieved. Rotation normalization is performed by employing the continuous principal component analysis along with a novel approach which applies PCA on the face normals of the model. The 3D model is decomposed into a set of spherical functions which represents not only the intersections of the corresponding surface with rays emanating from the origin but also points in the direction of each ray which are closer to the origin than the furthest intersection point. The superior performance of the proposed methodology is demonstrated through a comparison against state-of-the-art approaches on standard databases.

Regional gravity modeling in terms of spherical base functions

This article provides a survey on modern methods of regional gravity field modeling on the sphere. Starting with the classical theory of spherical harmonics, we outline the transition towards space-localizing methods such as spherical splines and wavelets. Special emphasis is given to the relations among these methods, which all involve radial base functions. Moreover, we provide extensive applications of these methods and numerical results from real space-borne data of recent satellite gravity missions, namely the Challenging Minisatellite Payload (CHAMP) and the Gravity Recovery and Climate Experiment (GRACE). We also derive high-resolution gravity field models by effectively combining space-borne and surface measurements using a new weighted level-combination concept. In addition, we outline and apply a strategy for constructing spatio-temporal fields from regional data sets spanning different observation periods.

Holomorphic horospherical duality “sphere-cone”

We describe a construction of complex geometrical analysis which corresponds to the classical theory of spherical harmonics.

The compatibility conditions in altimetry–gravimetry boundary value problems

The role of compatibility conditions for altimetry–gravimetry boundary value problems (AGBVPs) is quite complicated by way of: guaranteeing the existence of the solution; smoothing the data (boundary conditions) at the boundary; and providing a higher level of regularization and smoothness to the solution. Pseudo-differential operators (PDOs) can be applied not only for the reformulation of AGBVPs in a simple way, but also for imposing compatibility conditions at the coastline. It is shown here that spherical PDOs can be combined with the theory of spherical harmonics and spherical wavelets to obtain a numerical solution for AGBVPs with compatibility conditions along the coastline. The role of compatibility conditions, is summarized and emphasized and compatibility conditions are given in an explicit form based on the combined representation of functionals of the disturbing potential in terms of spherical harmonics, spherical PDOs and wavelets. Numerical aspects of the application of the compatibility conditions to a case-study area in Canada are discussed.

SPHERICAL HARMONICS ASSOCIATED TO THE LAPLACE–BESSEL OPERATOR AND GENERALIZED SPHERICAL CONVOLUTIONS

A theory of spherical harmonics associated to the Laplace–Bessel differential operator is developed. Natural analogs of the Plancherel theory, the Laplace formula, the Funk–Hecke formula, the product formula, and the addition theorem are obtained. Symmetry properties of the Fourier–Bessel transform, decompositions of smooth functions, and convolution operators are studied.