Spherical Harmonic Polynomials Research Articles

Comparison of polynomial and rational function cornea models for effective dimensionality reduction

This study aims to examine geometric models of the corneal surface that can be used to reduce in reasonable time the dimensionality of datasets of normal anterior corneas. Polynomial models (P) like Zernike polynomials (ZP) and spherical harmonic polynomials (SHP) were obvious candidates along with their rational function (R) counterparts, namely Zernike rational functions (ZR) and spherical harmonic rational functions (SHR, new model). Knowing that both SHP and ZR were more accurate than ZP for the modeling of normal and keratoconus corneas, it was expected that both spherical harmonic (SH) models (SHP and SHR) would be more accurate than their Zernike (Z) counterparts (ZP and ZR, respectively), and both rational (R) models (SHR and ZR) more accurate than their polynomial counterparts (SHP and ZP, respectively) for a low dimensional space (coefficient number J < 30). This was the case. The SH factor contributed more to accuracy than the R factor. Considering the corneal processing time as a function of J, P models were processed in quasi-linear time with a quasi-null slope and rational models in polynomial time. Z models were faster than SH models, and increasingly so in their R version. In sum, for corneal dimensionality reduction, SHR is the most accurate model, but its processing time is increasingly prohibitive unless the best coefficient combination is identified beforehand. ZP is the fastest model and is reasonably accurate with normal corneas for exploratory tasks. SHP is the best compromise between accuracy and speed.

The accuracy of geostatistics for regional geomagnetic modeling in an archipelago setting.

Indonesia as an archipelago country relies on a limited number and clustered distributed repeat station networks. This paper explores the use of geostatistical modeling to overcome this data limitation. The model data set consisted of repeat station data from 1985 to 2015 epoch. The geostatistical methods utilized included ordinary kriging (OK), collocated cokriging (CC), and kriging with external drift (KED). The model generated using these geostatistical methods was then compared to spherical cap harmonic analyses (SCHA) and polynomial models. The geostatistical model was shown to perform better, with greater accuracy in declination, inclination, and total intensity, as indicated by the root mean square error (RMSE). We have demonstrated that the geostatistical method is a promising approach in the modeling of regional geomagnetic field, especially in areas with limited and clustered distributed data.

Non‐Singular Calculation of Geomagnetic Vectors and Geomagnetic Gradient Tensors

AbstractSome spherical harmonic expressions of gravitational and geomagnetic field elements will become infinite when computation point approaches polar regions, as the sine function of the geocentric co‐latitude contained in the denominator tends to be zero. Currently, this singularity problem has been solved for gravitational field case, however, it remains unsolved for geomagnetic vectors (GVs) and geomagnetic gradient tensors (GGTs). Because the latter use Schmidt semi‐normalized associated Legendre function (SNALF), which is different from fully‐normalized associated Legendre function used in the former. To overcome this singularity problem, we derive new non‐singular expressions of the first‐ and second‐order derivatives of Schmidt SNALF (order m equals to 0 or 1), and the corresponding two kinds of spherical harmonic polynomials. When the non‐singular expressions are applied to the traditional formulae of GVs and GGTs, more practical expressions of GVs and GGTs with non‐singularity are formulated by refining the cases that the order m equals to 0, 1, 2 and other values. Furthermore, to provide flexible calculation strategies for Schmidt SNALF, we derive four kinds of recursive formulae, including the standard forward row recursion, the standard forward column recursion, the cross degree and order recursion, and the Belikov recursion. The standard formula to check the calculation accuracy of various recursive formulae of SNALF are also given. Besides, we demonstrate the effectiveness and reliability of the new derived non‐singular expressions of GVs and GGTs and analyze the computation speed and stability of the four recursive formulae by extensive numerical experiments.

A mesh-free treatment for even parity neutron transport equation

Common family of numerical methods such as finite element and finite difference involve the discretization of domain into regular grids or meshes. Mesh generation and re-meshing process, however, for obtaining required accuracy are complicated and time-consuming. Recently, Mesh Free (MFree) methods have been developed to overcome the drawbacks of such conventional numerical methods. In the present work, a novel approach is developed for solving multi-group neutron transport equation in two-dimensional X-Y geometry based on a mesh free scheme using the Radial Point Interpolation Method (RPIM). The K+ principle which is an extremum variational principle for solving even parity neutron transport equation is our choice to implement RPIM. The directional dependence of even-parity angular flux is expanded by the spherical harmonic polynomials which leads to PN method. The multi-quadrics radial basis function is used to construct the RPIM shape functions for spatial approximation of angular flux to obtain the discretized MFree weak-form of neutron transport equation. As the RPIM shape functions possess the Kronecker delta function property, essential boundary conditions is enforced as efficiently as in the finite element method (FEM). The obtained results are compared with some mesh-based methods such as conventional finite element method to evaluate the performance of the presented approach. It is demonstrated that the proposed method is robust, stable, reliable and efficient for treatment of neutron transport.

Propagation and estimation of the dynamical behaviour of gravitationally interacting rigid bodies

Next-generation planetary tracking methods, such as interplanetary laser ranging (ILR) and same-beam interferometry (SBI) promise an orders-of-magnitude increase in the accuracy of measurements of solar system dynamics. This requires a reconsideration of modelling strategies for the translational and rotational dynamics of natural bodies, to ensure that model errors are well below the measurement uncertainties.The influence of the gravitational interaction of the full mass distributions of celestial bodies, the so-called figure-figure effects, will need to be included for selected future missions. The mathematical formulation of this problem to arbitrary degree is often provided in an elegant and compact manner that is not trivially relatable to the formulation used in space geodesy and ephemeris generation. This complicates the robust implementation of such a model in operational software packages. We formulate the problem in a manner that is directly compatible with the implementation used in typical dynamical modelling codes: in terms of spherical harmonic coefficients and Legendre polynomials. An analytical formulation for the associated variational equations for both translational and rotational motion is derived.We apply our methodology to both Phobos and the KW4 binary asteroid system, to analyze the influence of figure-figure effects during estimation from next-generation tracking data. For the case of Phobos, omitting these effects during estimation results in relative errors of 0.42% and 0.065% for the bar{C}_{20} and bar{C}_{22} spherical harmonic gravity field coefficients, respectively. These values are below current uncertainties, but orders of magnitude larger than those obtained from past simulations for accurate tracking of a future Phobos lander, showing the need to apply the methodology outlined in this manuscript for selected future missions.

Mean Value Property and Harmonicity on Carnot-Carath\xe9odory Groups

We study strongly harmonic functions in Carnot–Carathéodory groups defined via the mean value property with respect to the Lebesgue measure. For such functions we show their Sobolev regularity and smoothness. Moreover, we prove that strongly harmonic functions satisfy the sub-Laplace equation for the appropriate gauge norm and that the inclusion is sharp. We observe that appropriate spherical harmonic polynomials in ℍ1 are both strongly harmonic and satisfy the sub-Laplace equation. Our presentation is illustrated by examples.

Kriging with Unknown Variance Components for Regional Ionospheric Reconstruction.

Ionospheric delay effect is a critical issue that limits the accuracy of precise Global Navigation Satellite System (GNSS) positioning and navigation for single-frequency users, especially in mid- and low-latitude regions where variations in the ionosphere are larger. Kriging spatial interpolation techniques have been recently introduced to model the spatial correlation and variability of ionosphere, which intrinsically assume that the ionosphere field is stochastically stationary but does not take the random observational errors into account. In this paper, by treating the spatial statistical information on ionosphere as prior knowledge and based on Total Electron Content (TEC) semivariogram analysis, we use Kriging techniques to spatially interpolate TEC values. By assuming that the stochastic models of both the ionospheric signals and measurement errors are only known up to some unknown factors, we propose a new Kriging spatial interpolation method with unknown variance components for both the signals of ionosphere and TEC measurements. Variance component estimation has been integrated with Kriging to reconstruct regional ionospheric delays. The method has been applied to data from the Crustal Movement Observation Network of China (CMONOC) and compared with the ordinary Kriging and polynomial interpolations with spherical cap harmonic functions, polynomial functions and low-degree spherical harmonic functions. The statistics of results indicate that the daily ionospheric variations during the experimental period characterized by the proposed approach have good agreement with the other methods, ranging from 10 to 80 TEC Unit (TECU, 1 TECU = 1 × 1016 electrons/m2) with an overall mean of 28.2 TECU. The proposed method can produce more appropriate estimations whose general TEC level is as smooth as the ordinary Kriging but with a smaller standard deviation around 3 TECU than others. The residual results show that the interpolation precision of the new proposed method is better than the ordinary Kriging and polynomial interpolation by about 1.2 TECU and 0.7 TECU, respectively. The root mean squared error of the proposed new Kriging with variance components is within 1.5 TECU and is smaller than those from other methods under comparison by about 1 TECU. When compared with ionospheric grid points, the mean squared error of the proposed method is within 6 TECU and smaller than Kriging, indicating that the proposed method can produce more accurate ionospheric delays and better estimation accuracy over China regional area.

Gradient nonlinearity calibration and correction for a compact, asymmetric magnetic resonance imaging gradient system

Due to engineering limitations, the spatial encoding gradient fields in conventional magnetic resonance imaging cannot be perfectly linear and always contain higher-order, nonlinear components. If ignored during image reconstruction, gradient nonlinearity (GNL) manifests as image geometric distortion. Given an estimate of the GNL field, this distortion can be corrected to a degree proportional to the accuracy of the field estimate. The GNL of a gradient system is typically characterized using a spherical harmonic polynomial model with model coefficients obtained from electromagnetic simulation. Conventional whole-body gradient systems are symmetric in design; typically, only odd-order terms up to the 5th-order are required for GNL modeling. Recently, a high-performance, asymmetric gradient system was developed, which exhibits more complex GNL that requires higher-order terms including both odd- and even-orders for accurate modeling. This work characterizes the GNL of this system using an iterative calibration method and a fiducial phantom used in ADNI (Alzheimer’s Disease Neuroimaging Initiative). The phantom was scanned at different locations inside the 26 cm diameter-spherical-volume of this gradient, and the positions of fiducials in the phantom were estimated. An iterative calibration procedure was utilized to identify the model coefficients that minimize the mean-squared-error between the true fiducial positions and the positions estimated from images corrected using these coefficients. To examine the effect of higher-order and even-order terms, this calibration was performed using spherical harmonic polynomial of different orders up to the 10th-order including even- and odd-order terms, or odd-order only. The results showed that the model coefficients of this gradient can be successfully estimated. The residual root-mean-squared-error after correction using up to the 10th-order coefficients was reduced to 0.36 mm, yielding spatial accuracy comparable to conventional whole-body gradients. The even-order terms were necessary for accurate GNL modeling. In addition, the calibrated coefficients improved image geometric accuracy compared with the simulation-based coefficients.

Relating coefficients of expansion of a function to its norm

Relations of the coefficients of the expansion of a function by orthogonal sequence with the norm of that function are given. The results contain Hausdorff–Young and Hardy–Littlewood type inequalities. The many applications that show the use of the results and their values are crucial. Many of the applications are for expansion by orthogonal polynomials on domain D and weight w. For w=1 the results are given for polytopes and other bounded convex domains. For the ball, simplex and the cube the weights dealt with are Jacobi-type. For D=R, R+, and Rd(d>1) we give results using Freud (including Hermite) weights, Laguerre weights and Hermite weights respectively. Expansions by trigonometric polynomials on the torus and by spherical harmonic polynomials on the sphere are also dealt with.

MO-FG-CAMPUS-IeP3-02: Gradient Nonlinearity Calibration and Correction for Full-Volume Imaging of a Compact Asymmetric MRI Gradient System

Purpose: Due to engineering limitations, the spatial encoding gradient fields generated by an MRI scanner are inherently nonlinear and images must undergo geometric correction prior to display. The gradient nonlinearity (GNL) of a conventional symmetric whole-body gradient is typically characterized using a spherical harmonic polynomial (SHP) expansion of magnetic gradient fields with up to fifth-order terms. Only odd-order terms are required due to design symmetry (i.e., 3rd/5th-order). Recently, a high-performance asymmetric gradient system was developed which exhibits more complex GNL. Here, the GNL across the 26cm diameter-spherical-volume (DSV) of this system is characterized using a phantom-based iterative calibration procedure. Methods: The Alzheimer’s Disease Neuroimaging Initiative (ADNI) phantom was scanned with a 3D IR-FSPGR sequence (sagittal acquisition, Nx=Ny=256, Nz=196, Δx=Δy=1.05mm, Δz=1.3mm, BW=±125kHz) on the asymmetric gradient system. The phantom (20cm diameter with 160 spherical fiducials of 1.0/1.5cm diameter) was imaged at isocenter and at spatial offsets of ±3cm along the superior/inferior, right/left, anterior/posterior axes to cover the 26cm DSV. The spatial positions of fiducials were measured from MR images and compared with the design positions to extract distortion field, which is then used to fit the SHP model using an iterative calibration procedure. The effect of higher-order terms was tested by incrementally increasing the model order from 5 to 10. The calibrated coefficients were then used to geometrically correct the images, and for each case the residual root-mean-square-error (RMSE) was determined. For comparison, RMSE was also determined for images corrected using coefficients obtained from electromagnetic simulation. Results: The calibrated coefficients (10th-order) reduce RMSE from 0.96mm (using a 10th-order correction from electromagnetic simulation) to 0.36mm. Conclusion: The model parameters for GNL correction over the 26cm DSV of an asymmetric gradient system were successfully determined. The use of calibrated coefficients up to 10thorder coefficients including even-order terms yielded improved geometric accuracy (RMSE) comparable to conventional symmetric gradients. Funding support: NIH R01EB010065

Local random phase approximation with projected oscillator orbitals

An approximation to the many-body London dispersion energy in molecular systems is expressed as a functional of the occupied orbitals only. The method is based on the local-RPA theory. The occupied orbitals are localized molecular orbitals and the virtual space is described by projected oscillator orbitals, i.e. functions obtained by multiplying occupied localized orbitals with solid spherical harmonic polynomials having their origin at the orbital centroids. Since we are interested in the long-range part of the correlation energy, responsible for dispersion forces, the electron repulsion is approximated by its multipolar expansion. This procedure leads to a fully non-empirical long-range correlation energy expression. Molecular dispersion coefficients calculated from determinant wave functions obtained by a range-separated hybrid method reproduce experimental values with less than 15% error.

Rotational invariants constructed by the products of three spherical harmonic polynomials**Supported by National Natural Science Foundation of China (11174099, 11075014) and NSERC of Canada

The rotational invariants constructed by the products of three spherical harmonic polynomials are expressed generally as homogeneous polynomials with respect to the three coordinate vectors in the compact form, where the coefficients are calculated explicitly in this paper.

TH-CD-207-11: Gradient Nonlinearity Calibration and Correction for Head-Only Asymmetric Gradient System

Purpose: Due to engineering limitations, the gradient fields in clinical MRI inevitably contain high-order, nonlinear components. The presence of gradient nonlinearity (GNL) causes image geometrical distortion. Standard correction methods are based on parameterization of the simulated gradient fields with spherical harmonic polynomials. Conventional whole-body gradient systems typically employ symmetric designs, and the GNL for such systems usually contains only odd-order terms (up to 5th-order). Recently, a high-performance, head-only gradient system was developed. Due to asymmetric design, this new system exhibits more complex GNL profiles. Here, we demonstrate measurement-based high-order(N>5) GNL-correction on this new system using a fiducial phantom and iterative model-fitting procedure. Methods: The Alzheimer’s Disease Neuroimaging Initiative (ADNI) phantom was scanned with a 3D IR-FSPGR sequence (sagittal acquisition plane, Nx=Ny=256, Nz=196, x= y=1.05mm, z=1.3mm, BW=125kHz) on the head-only gradient system operating at 80mT/m, 500T/m/s. This phantom contains 160 fiducial spheres (diameter=1.0 or 1.5cm) that are distributed within a diameter=20cm spherical shell. The spatial positions of the fiducials were then measured from the distorted images and fit using a spherical harmonic polynomial model and iterative fitting procedure. The model order was increased from 1 to 10 step-by-step to test the effects of including high-order terms (N>5). Then, the coefficients were used to correct distortion. The residual root-mean-square-error (RMSE) was calculated from fiducial positions tracked from GNL-corrected images. Results: The RMSE analysis shows that GNL-distortion is reduced from 3.93 to 0.39mm by using correction terms up to 7th-order including both even-and odd-order terms. The addition of higher orders (N>7) provides negligible benefit. Conclusion: The GNL of a high-performance, head-only gradient system was successfully measured. The use of up to 7th-order correction terms was found to be sufficient for correcting GNL-distortion in a typical brain scan. These techniques could also improve geometric accuracy for whole-body MRI systems used for radiation therapy planning.

Relating smoothness to expressions involving Fourier coefficients or to a Fourier transform

Coefficients of expansion of a function by trigonometric, algebraic and spherical harmonic orthogonal polynomials are related to the smoothness of that function. Hausdorff–Young type and Hardy–Littlewood type inequalities will be utilized. Expressions involving the Fourier transform of a function are also related to the measure of smoothness of that function.

Absolutely convergent expansions

The classical Bernstein estimate for absolutely convergent Fourier series and its multivariate extension are improved. Analogous results for absolutely convergent expansion by Legendre polynomials and by spherical harmonic polynomials are given. For expansion by a Fourier series it follows that the present result implies the classical result. Moreover, there are many functions that are shown to have an absolutely convergent Fourier series using the present estimate while not satisfying the conditions of the previous one.

On Siegel-Eisenstein series attached to certain cohomological representations

We introduce a Siegel-Eisenstein series of degree 2 which generates a cohomological representation of Saito-Kurokawa type at the real place. We study its Fourier expansion in detail, which is based on an investigation of the confluent hypergeometric functions with spherical harmonic polynomials. We will also consider certain Mellin transforms of the Eisenstein series, which are twisted by cuspidal Maass wave forms, and show their holomorphic continuations to the whole plane.

Smoothness of a function and the growth of its Fourier transform or its Fourier coefficients

In a recent paper Bray and Pinsky [1] estimated the growth of f ̂ ( ξ ) , the Fourier transform of f ( x ) where x ∈ R d , by some moduli of smoothness. We show here that noticeably better results can be derived as an immediate corollary of previous theorems in [2]. The improvements include dealing with higher levels of smoothness and using the fact that for higher dimensions (when d ≥ 2 ) the description of smoothness requires less information. Using a similar technique, we also deduce relations between the smoothness of f ( x ) for x ∈ S d − 1 or x ∈ T d and the growth of the coefficients of the expansion by spherical harmonic polynomials or trigonometric polynomials.

Jackson theorem in [formula omitted], for functions on the sphere

The best approximation of functions in L p ( S d − 1 ) , 0 < p < 1 by spherical harmonic polynomials is shown to be bounded by a modulus of smoothness recently introduced by the second author.

Triangulated spherical splines for geopotential reconstruction

We present an alternate mathematical technique than contemporary spherical harmonics to approximate the geopotential based on triangulated spherical spline functions, which are smooth piecewise spherical harmonic polynomials over spherical triangulations. The new method is capable of multi-spatial resolution modeling and could thus enhance spatial resolutions for regional gravity field inversion using data from space gravimetry missions such as CHAMP, GRACE or GOCE. First, we propose to use the minimal energy spherical spline interpolation to find a good approximation of the geopotential at the orbital altitude of the satellite. Then we explain how to solve Laplace’s equation on the Earth’s exterior to compute a spherical spline to approximate the geopotential at the Earth’s surface. We propose a domain decomposition technique, which can compute an approximation of the minimal energy spherical spline interpolation on the orbital altitude and a multiple star technique to compute the spherical spline approximation by the collocation method. We prove that the spherical spline constructed by means of the domain decomposition technique converges to the minimal energy spline interpolation. We also prove that the modeled spline geopotential is continuous from the satellite altitude down to the Earth’s surface. We have implemented the two computational algorithms and applied them in a numerical experiment using simulated CHAMP geopotential observations computed at satellite altitude (450 km) assuming EGM96 (nmax = 90) is the truth model. We then validate our approach by comparing the computed geopotential values using the resulting spherical spline model down to the Earth’s surface, with the truth EGM96 values over several study regions. Our numerical evidence demonstrates that the algorithms produce a viable alternative of regional gravity field solution potentially exploiting the full accuracy of data from space gravimetry missions. The major advantage of our method is that it allows us to compute the geopotential over the regions of interest as well as enhancing the spatial resolution commensurable with the characteristics of satellite coverage, which could not be done using a global spherical harmonic representation.

SHEF: a vHTS geometrical filter using coefficients of spherical harmonic molecular surfaces

SHEF (spherical harmonic coefficient filter), a geometrical matching procedure constituting a preliminary step in the virtual high throughput screening of large databases of small drug-like molecules, is demonstrated. This filter uses a description of both the binding site of the target and the ligand surfaces using spherical harmonic polynomial expansions. Using this representation, which is based on limited sets of spherical harmonic coefficients, considerably reduces the complexity of surface complementarity calculation. As a first test, 188 known protein-ligand complexes were used, and the results of docking the abstracted ligands into the bare proteins using SHEF were compared to the original X-ray structures. The ability of SHEF to retrieve known ligands "hidden" in a virtual library of 1,000 randomly selected drug-like compounds is also demonstrated.