Adaptive hp finite element method for fluorescence molecular tomography with simplified spherical harmonics approximation

Recently,the simplified spherical harmonics equations(SP)model has at tracted much att entionin modeling the light propagation in small tissue ggeometriesat visible and near-infrared wave-leng ths.In this paper,we report an eficient numerical method for fluorescence moleeular tom-ography(FMT)that combines the advantage of SP model and adaptive hp finite elementmethod(hp-FEM).For purposes of comparison,hp-FEM and h-FEM are,respectively applied tothe reconstruction pro cess with diffusion approximation and SPs model.Simulation experiments on a 3D digital mouse atlas and physical experiments on a phantom are designed to evaluate thereconstruction methods in terms of the location and the reconstructed fluorescent yield.Theexperimental results demonstrate that hp-FEM with SPy model,yield more accurate results thanh-FEM with difusion approximation model does.The phantom experiments show the potentialand feasibility of the proposed approach in FMT applications.

Performance evaluation of the simpli¯ed spherical harmonics approximation for cone-beam X-ray luminescence computed tomography imaging

As an emerging molecular imaging modality,cone-beam X-ray luminescence computed tomog-raphy(CB-XLCT)uses X-ray-excitable probes to produce near-infrared(NIR)luminescence and then reconst ructs three-dimensional(3D)distribution of the probes from surface measurements.A proper photon-transportation model is critical to accuracy of XLCT.Here,we presented a systematic comparison between the common-used Monte Carlo model and simplified spherical harmonics(SPN).The performance of the two methods was evaluated over several main spec-trums using a known XLCT material.We designed both a global measurement based on the cosine similarity and a locally-averaged relative error,to quantitatively assess these methods.The results show that the SP_(3) could reach a good balance between the modeling accuracy and computational efficiency for all of the tested emission spectrums.Besides,the SP_(1)(which is equivalent to the difusion equation(DE))can be a reasonable alternative model for emission wavelength over 692nm.In vivo experiment further demonstrates the reconstruction perfor-mance of the SP:and DE.This study would provide a valuable guidance for modeling the photon-transportation in CB-XLCT.

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 x 10-s, 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.

Terahertz quasi-perfect vortex beam with integer-order and fractional-order generated by spiral spherical harmonic axicon

We propose a new method to generate terahertz perfect vortex beam with integer-order and fractional-order. A new optical diffractive element composed of the phase combination of a spherical harmonic axicon and a spiral phase plate is designed and called spiral spherical harmonic axicon. A terahertz Gaussian beam passes through the spiral spherical harmonic axicon to generate a terahertz vortex beam. When only the topological charge number carried by spiral spherical harmonic axicon increases, the ring radius of terahertz vortex beam increases slightly, so the beam is shaped into a terahertz quasi-perfect vortex beam. Importantly, the terahertz quasi-perfect vortex beam can carry not only integer-order topological charge number but also fractional-order topological charge number. This is the first time that vortex beam and quasi-perfect vortex beam with fractional-order have been successfully realized in terahertz domain and experiment.

On Approximation by Reciprocals of Spherical Harmonics in L^p Norm

Let S^1-1,q≥2,be the surface of the unit sphere in the Euclidean space R^1,f（x）∈L^p（S^q-1）,f（x）≥0,f absohutely unegual to 0,1≤p≤＋∞,Then,it is proved in the present paper that there is a spherical harmonics PN（x） of order≤N and a constant C〉0 such that where ω（f,δ）L^p=sup 0〈t≤δ‖St（f）-f‖L^p is a kind of moduli of continuity and ^‖f-1/PN‖L^p≤Cω（f,N^-1）L^p,St（f,μ）=1/｜S^q-2｜Sin^2λt ∫-μμ’=t f（μ＇）dμ＇ is a translation operator.

CLEBSCH-GORDAN COEFFICIENTS,SPHERICAL HARMONICS,AND d FUNCTIONS

Note： A square-root sign is to be understood over every coefficient, e.g., for -8/15 read - √8√15.

Molecular Simulations using Spherical Harmonics

Computer aided drug design is to develop a chemical that binds to a target macromolecule known to play a key role in a disease state. In recognition of ligands by their protein receptors, molecular surfaces are often used because they represent the interacting part of molecules and they should reflex the complementarity between ligand and receptor. However, assessing the surface complementarity by searching all relative position of two surfaces is often computationally expensive. The complementarity of lobe hole is very important in protein ligand interactions. Spherical harmonic models based on expansions of spherical harmonic functions were used as a fingerprint to approximate the binding cavity and the ligand, respectively. This defines a new way to identify the complementarity between lobes and holes. The advantage of this method is that two spherical harmonic surfaces to be compared can be defined separately. This method can be used as a filter to eliminate candidates among a large number of conformations, and it will speed up the docking procedure. Therefore, it is possible to select complementary ligands or complementary conformations of a ligand and the macromolecules, by comparing their fingerprints previously stored in a database.

Estimating Illumination Parameters Using Spherical Harmonics Coefficients in Frequency Space

An algorithm is presented for estimating the direction and strength of point light with the strength of ambient illumination. Existing approaches evaluate these illumination parameters directly in the high dimensional image space, while we estimate the parameters in two steps： first by projecting the image to an orthogonal linear subspace based on spherical harmonic basis functions and then by calculating the parameters in the low dimensional subspace. The test results using the CMU PIE database and Yale Database B show the stability and effectiveness of the method. The resulting illumination information can be used to synthesize more realistic relighting images and to recognize objects under variable illumination.

Accurate estimation of Li/Ni mixing degree of lithium nickel oxide cathode materials

Li/Ni mixing negatively influences the discharge capacity of lithium nickel oxide and high-nickel ternary cathode materials.However,accurately measuring the Li/Ni mixing degree is difficult due to the preferred orientation of labbased XRD measurements using Bragg–Brentano geometry.Here,we find that employing spherical harmonics in Rietveld refinement to eliminate the preferred orientation can significantly decrease the measurement error of the Li/Ni mixing ratio.The Li/Ni mixing ratio obtained from Rietveld refinement with spherical harmonics shows a strong correlation with discharge capacity,which means the electrochemical capacity of lithium nickel oxide and high-nickel ternary cathode can be estimated by the Li/Ni mixing degree.Our findings provide a simple and accurate method to estimate the Li/Ni mixing degree,which is valuable to the structural analysis and screening of the synthesis conditions of lithium nickel oxide and high-nickel ternary cathode materials.

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP -In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.

Analytical Solutions to Definite Integrals for Combinations of Legendre, Bessel and Trigonometric Functions Encountered in Propagation and Scattering Problems in Spherical Coordinates

Mie theory is a rigorous solution to scattering problems in spherical coordinate system. The first step in applying Mie theory is expansion of some arbitrary incident field in terms of spherical harmonics fields in terms of spherical which in turn requires evaluation of certain definite integrals whose integrands are products of Bessel functions, associated Legendre functions and periodic functions. Here we present analytical results for two specific integrals that are encountered in expansion of arbitrary fields in terms of summation of spherical waves. The analytical results are in terms of finite summations which include Lommel functions. A concise analytical expression is also derived for the special case of Lommel functions that arise, rendering expensive numerical integration or other iterative techniques unnecessary.

Modeling and Interpreting CHAMP Magnetic Anomaly Field over China Continent Using Spherical Cap Harmonic Analysis

Based on the CHAMP Magsat data set, spherical cap harmonic analysis was used to model the magnetic fields over China continent. The data set used in the analysis includes the 15′×15′ gridded values of the CHAMP anomaly fields (latitude φ=25°N to 50°N and longitude λ=78°E to 135°E). The pole of the cap is located at φ=35°N and λ=110°E with half-angle of 30°. The maximum index (K max) of the model is 30 and the total number of model coefficients is 961, which corresponds to the minimum wavelength at the earth's surface about 400 km. The root mean square (RMS) deviations between the calculated and observed values are ～ 4 nT for ΔX, ～ 3 nT for ΔY and ～ 3.5 nT for ΔZ, respectively. Results show that positive anomalies are found mainly at the Tarim basin with ～6- 8 nT, the Yangtze platform and North China platform with ～4 nT, and the Songliao basin with ～4-6 nT. In contrast, negative anomaly is mainly located in the Tibet orogenic belt with the amplitude ～ (-6)-(-8) nT. Upward continuation of magnetic anomalies was used to semi-quantitatively separate the magnetic anomalies in different depths of crust. The magnetic anomalies at the earth's surface are from -6 to 10 nT for upper crust, middle crust -27 to 42 nT and lower crust -12 to 18 nT, respectively. The strikes of the magnetic anomalies for the upper crust are consistent with those for the middle crust, but not for the lower crust. The high positive magnetic anomalies mainly result from the old continental nucleus and diastrophic block (e.g. middle Sichuan continental nucleus, middle Tarim basin continental nucleus, Junggar diastrophic block and Qaidam diastrophic block). The amplitudes of the magnetic anomalies of the old continental nucleus and diastrophic block are related to evolution of deep crust. These results improve our understanding of the crustal structure over China continent.

Establishment of European Regional Ionosphere Model Based on Spherical Harmonic Functions

In order to study the temporal and spatial variation characteristics of the regional ionosphere and the modeling accuracy,the experiment is based on the spherical harmonic function model,using the GPS,Glonass,and Galileo dual-frequency observation data from the 305th-334th day of the European CORS network in 2019 to establish a global ionospheric model.By analyzing and evaluating the accuracy of the global ionospheric puncture points,VTEC,and comparing code products,the test results showed that the GPS system has the most dense puncture electricity distribution,the Glonass system is the second,and the Galileo system is the weakest.The values of ionospheric VTEC calculated by GPS,Glonass and Galileo are slightly different,but in terms of trends,they are the same as those of ESA,JPL and UPC.GPS data has the highest accuracy in global ionospheric modeling.GPS,Glonass and Galileo have the same trend,but Glonass data is unstable and fluctuates greatly.

ALMOST EVERYWHERE CONVERGENCE AND APPROXIMATION OF SPHERICAL HARMONIC EXPANSIONS

This paper deals with the order of magnitude of the partial sums of the spherical harmonic series and its convergence rate in Bessel potential spaces. The partial results obtained in the paper are the analogue of those on the circle.

ON APPROXIMATION BY SPHERICAL REPRODUCING KERNEL HILBERT SPACES

The spherical approximation between two nested reproducing kernels Hilbert spaces generated from different smooth kernels is investigated. It is shown that the functions of a space can be approximated by that of the subspace with better smoothness. Furthermore, the upper bound of approximation error is given.

IMAGING OF EEG BY SPHERICAL HARMONIC ANALYSIS

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Simulation for MSS-2 low-perigee elliptical orbit satellites:an example of lithospheric magnetic field modelling

A future constellation of at least four geomagnetic satellites(designated Macao Scientific Satellite-1(MSS-1)and Macao Scientific Satellite-2(MSS-2))was recently proposed,to continue high-quality geomagnetic observations in the post-Swarm period,focusing especially on collecting data that will provide a global,three-dimensional survey of the geomagnetic field.In this paper,we present a simulation of two years of orbits(2020.01.01-2022.01.01)of two satellites(tentatively denoted as MSS-2)that are constellated in elliptical(200×5,300 km)low-perigee orbits.By comparing error variances of Gauss coefficients,we investigate the sensitivity of lithospheric magnetic field modelling to data collected from various satellite orbits,including a near circular reference orbit of 300×350km,and elliptical orbit of 180×5,300 km,220×5,300 km,200×3,000 km and 200×1,500 km.We find that in two years the two MSS-2 satellites can collect 35,000 observations at altitude below 250 km,data that will be useful in advancing the quality of lithospheric magnetic field modelling;this number of observations reflects the fact that only 4.5%of the flight time of these satellites will be below250 km(just 6.4%of their flight time below 300 km).By combining observations from the MSS-2 satellites’elliptical orbits of 200×5,300km with observations from a circular reference orbit,the variance of the geomagnetic model can be reduced by a factor of 285 at spherical harmonic degree n=200 and by a factor of 1,300 at n=250.The planned lower perigee of their orbits allows the new satellites to collect data at unprecedentedly lower altitudes,thus dramatically improving the spatial resolution of satellite-derived lithospheric field models,(up to 80%at n=150).In addition,lowering the apogee increases the time interval during which the satellites fly at near-Earth altitudes,thus improving the model predictions at all spherical harmonic degrees(around 52%-62%at n=150).The upper limit of the expected improvement to the field model at the orbital apogee is not as good as at the perigee.However,data from the MSS-1 orbit can help fill the gap between data from the MSS-2 orbits and from the circular reference orbit for the low-degree part of the model.The feasibility of even lower-altitude flight requires further discussion with satellite engineers.

Evaluation of regional ionospheric grid model over China from dense GPS observations

The current global or regional ionospheric models have been established for monitoring the ionospheric variations. However, the spatial and temporal resolutions are not enough to describe total electron content（TEC） variations in small scales for China. In this paper, a regional ionospheric grid model（RIGM） with high spatial-temporal resolution（0.5 0.5 and 10-min interval） in China and surrounding areas is established based on spherical harmonics expansion from dense GPS measurements provided by Crustal Movement Observation Network of China（CMONOC） and the International GNSS Service（IGS）. The correlation coefficient between the estimated TEC from GPS and the ionosonde measurements is 0.97, and the root mean square（RMS） with respect to Center for Orbit Determination in Europe（CODE） Global Ionosphere Maps（GIMs） is 4.87 TECU. In addition, the impact of different spherical harmonics orders and degrees on TEC estimations are evaluated and the degree/order 6 is better. Moreover, effective ionospheric shell heights from300 km to 700 km are further assessed and the result indicates that 550 km is the most suitable for regional ionospheric modeling in China at solar maximum.

WHU-Grace01s:A new temporal gravity field model recovered from GRACE KBRR data alone

A new temporal gravity field model called WHU-Grace01s solely recovered from Gravity Recovery and Climate Experiment （GRACE） K-Band Range Rate （KBRR） data based on dynamic integral approach is presented in this paper. After meticulously preprocessing of the GRACE KBRR data, the root mean square of its post residuals is about 0.2 micrometers per second, and seventy-two monthly temporal solutions truncated to degree and order 60 are computed for the period from January 2003 to December 2008. After applying the combi- nation filter in WHU-Grace01s, the global temporal signals show obvious periodical change rules in the large-scale fiver basins. In terms of the degree variance, our solution is smaller at high degrees, and shows a good consistency at the rest of degrees with the Release 05 models from Center for Space Research （CSR）, GeoForschungsZentrum Potsdam （GFZ） and Jet Pro- pulsion Laboratory 0PL）. Compared with other published models in terms of equivalent water height distribution, our solution is consistent with those published by CSR, GFZ, JPL, Delft institute of Earth Observation and Space system （DEOS）, Tongji University （Tongji）, Institute of Theoretical Geodesy （ITG）, Astronomical Institute in University of Bern （AIUB） and Groupe de Recherche de Geodesie Spatiale （GRGS}, which indicates that the accuracy of WHU-Grace01s has a good consistency with the previously published GRACE solutions.

A Virtual Puncture Surgery System Based on Multi-Layer Soft Tissue and Force Mesh

Puncture is a common operation in surgery,which involves all kinds of tissue materials with different geometry and mechanical properties.As a new cross-disciplinary research area,Virtual Surgery(VS)makes simulation of soft tissue in puncture operation possible in virtual environment.In this paper,we introduce a VS-based puncture system composed by three-layer soft tissue,simulated with spherical harmonic function(SHF),which is covered with a force mesh,constructed by mass spring model(MSM).The two models are combined together with a parameter of SHF named surface radius,which provides MSM with real-time deformation data needed in force calculation.Meanwhile,force calculation,divided into the surface spring force and the puncture damping force,makes the force presentation better accord to the corresponding tissue characteristics.Moreover,a deformation resumption algorithm is leveraged to simulate the resumption phenomenon of the broken tissue surface.In evaluation experiment,several residents are invited to grades our model along with other four mainstream soft tissue models in terms of 7 different indicators.After the evaluation,the scores are analyzed by a comprehensive weighted grading method.Experiment results show that the proposed model has better performance during puncture operation than other models,and can well simulate surface resumption phenomenon when tissue surface is broken.