Harmonic Representation Research Articles

Using Truncated Data Sets in Spherical-Scanning Antenna Measurements

We discuss the mitigation of truncation errors in spherical-scanning measurements by use of a constrained least-squares estimation method. The main emphasis is the spherical harmonic representation of probe transmitting and receiving functions; however, our method is applicable to near-field measurement of electrically small antennas for which full-sphere data are either unreliable or unavailable.

Spherical harmonic modelling to ultra-high degree of Bouguer and isostatic anomalies

The availability of high-resolution global digital elevation data sets has raised a growing interest in the feasibility of obtaining their spherical harmonic representation at matching resolution, and from there in the modelling of induced gravity perturbations. We have therefore estimated spherical Bouguer and Airy isostatic anomalies whose spherical harmonic models are derived from the Earth’s topography harmonic expansion. These spherical anomalies differ from the classical planar ones and may be used in the context of new applications. We succeeded in meeting a number of challenges to build spherical harmonic models with no theoretical limitation on the resolution. A specific algorithm was developed to enable the computation of associated Legendre functions to any degree and order. It was successfully tested up to degree 32,400. All analyses and syntheses were performed, in 64 bits arithmetic and with semi-empirical control of the significant terms to prevent from calculus underflows and overflows, according to IEEE limitations, also in preserving the speed of a specific regular grid processing scheme. Finally, the continuation from the reference ellipsoid’s surface to the Earth’s surface was performed by high-order Taylor expansion with all grids of required partial derivatives being computed in parallel. The main application was the production of a 1′ × 1′ equiangular global Bouguer anomaly grid which was computed by spherical harmonic analysis of the Earth’s topography–bathymetry ETOPO1 data set up to degree and order 10,800, taking into account the precise boundaries and densities of major lakes and inner seas, with their own altitude, polar caps with bedrock information, and land areas below sea level. The harmonic coefficients for each entity were derived by analyzing the corresponding ETOPO1 part, and free surface data when required, at one arc minute resolution. The following approximations were made: the land, ocean and ice cap gravity spherical harmonic coefficients were computed up to the third degree of the altitude, and the harmonics of the other, smaller parts up to the second degree. Their sum constitutes what we call ETOPG1, the Earth’s TOPography derived Gravity model at 1′ resolution (half-wavelength). The EGM2008 gravity field model and ETOPG1 were then used to rigorously compute 1′ × 1′ point values of surface gravity anomalies and disturbances, respectively, worldwide, at the real Earth’s surface, i.e. at the lower limit of the atmosphere. The disturbance grid is the most interesting product of this study and can be used in various contexts. The surface gravity anomaly grid is an accurate product associated with EGM2008 and ETOPO1, but its gravity information contents are those of EGM2008. Our method was validated by comparison with a direct numerical integration approach applied to a test area in Morocco–South of Spain (Kuhn, private communication 2011) and the agreement was satisfactory. Finally isostatic corrections according to the Airy model, but in spherical geometry, with harmonic coefficients derived from the sets of the ETOPO1 different parts, were computed with a uniform depth of compensation of 30 km. The new world Bouguer and isostatic gravity maps and grids here produced will be made available through the Commission for the Geological Map of the World. Since gravity values are those of the EGM2008 model, geophysical interpretation from these products should not be done for spatial scales below 5 arc minutes (half-wavelength).

Pion-proton correlations and asymmetry measurement in Au + Au collisions at √s NN = 200 GeV data

Correlations between non-identical particles at small relative velocity probe asymmetries in the average space-time emission points at freeze-out. The origin of such asymmetries may be from long-lived resonances, bulk collective effects, or differences in the freeze-out scenario for the different particle species. STAR has extracted pion-proton correlation functions from a dataset of Au+Au collisions at √sNN = 200 GeV. We present correlation functions in the spherical harmonic decomposition representation, for different centralities and for different combinations of pions and (anti-)protons.

Validation of international reference ionosphere models using in situ measurements from GRACE K-band ranging system and CHAMP planar Langmuir probe

The in situ measurements of electron contents from GRACE K-band (dual-frequency) ranging system and CHAMP planar Langmuir probe were used to validate the international reference ionosphere (IRI) models. The comparison using measurements from year 2003 to 2007 shows a general agreement between data and the model outputs. The improvement in the newer IRI model (IRI-2007) is evident with the measurements from the GRACE satellites orbiting at the higher altitude. We present the comparison between the models and data comprehensively for various cases in solar activity, local time, season, and latitude. The IRI models do not well predict the electron density in the years 2006 and later, when the solar activity is extremely low. The IRI models generally overestimate the electron density during local winter while they underestimate during local summer. In the equatorial region, the large difference at local sunrise lasts for all years and all seasons. The IRI models do not perform well in predicting the anomaly in the polar region such as the Weddell Sea Anomaly. These discrepancies are likely due to smoothed (12-month averaged) solar activity indices used in the IRI models and due to insufficient spherical harmonic representation not able to capture small spatial scales. In near future, further improvement on the IRI models is expected by assimilating those in situ satellite data by implementing higher resolution (spatial and temporal) parameterizations.

A Novel Haptic Deformation Modelling Method using Spherical Harmonic Representation and Radial Basis Function Neural Network

In the haptic modelling of deformable soft objects in virtual reality, due to the complicated relationship between the deformation and the applied force, it is a challenging task to achieve the balance between accuracy and efficiency. In this paper, a novel approach based on spherical harmonic (SH) representation and Radial Basis Function neural network(RBF-NN) to simulating the virtual deformation of soft objects was proposed. The object surface was represented by SH, which features of orthonormality, rotational invariance, and multi-resolution. Given the density, Young's modulus, Poisson ratio of the object and interactive position of the applied force, the haptic deformation model was estimated by a Radial Basis Function (RBF) based neural network model, which was trained with Least Error Measure method. The simulation results indicated that the haptic deformation model could be accurately calculated using the proposed approach and the computational time was acceptable for real-time applications.

Analysis of truncation and sampling errors in the spherical harmonic representation of soundfields

The use of microphone arrays to record soundfields for subsequent immersive reproduction, real time or off-line beamforming applications are becoming increasingly prevalent. The representation and analysis of such recorded soundfields using finite-order spherical harmonic coefficients is elegant and has a number of advantages. The focus of this paper is to quantify the accuracy of such a representation which is limited by errors introduced by several types of non-idealities: manufacturing non-idealities of the array such as microphone positional errors, coupling between microphones, etc.; physical non-idealities due to the discrete spatial sampling of the soundfield which introduces spatial aliasing and the inherent truncation of the modal representation of the soundfield to a finite order; and mathematical errors due to the computational technique used to derive the modal coefficients. This paper investigates the latter two classes of errors and quantifies the significant advantages in accuracy achievable when the physical structure of the microphone array allows the use of the orthogonal properties of the spherical harmonic basis functions to extract the modal coefficients. In addition, we extend previously published analysis of the truncation error, which was limited to incident plane waves, to spherical waves.

Multiterm solution of a generalized Boltzmann kinetic equation for electron and positron transport in structured and soft condensed matter

In this paper, we generalize the semiclassical Boltzmann kinetic equation for dilute gases to consider highly nonequilibrium electrons and positrons in soft condensed matter, accounting rigorously for all types of interactions, including positronium formation, and allowing for both coherent and incoherent scattering processes. The limitations inherent in the seminal paper of Cohen and Lekner [M. H. Cohen and J. Lekner, Phys. Rev. 158, 305 (1967); Y. Sakai, J. Phys. D 40, R441 (2007)] are avoided by solving the kinetic equation using a "multiterm" spherical harmonic representation of the velocity distribution function, as well as formulating a necessarily nonperturbative treatment of nonconservative collisional processes such as positronium formation. Numerical calculations of transport properties are carried out for a Percus-Yevick model of a hard-sphere system, and for positrons in liquid argon. New phenomena are predicted, including structure-induced negative conductivity and anisotropic diffusion.

A REMARK ON THE BRYLINSKI CONJECTURE FOR ORBIFOLDS

AbstractThe paper presents a proof of the Brylinski conjecture for compact Kähler orbifolds. The result is a corollary of the foliated version of the Mathieu theorem on symplectic harmonic representations of de Rham cohomology classes. The proofs are based on the idea of representing an orbifold as the leaf space of a Riemannian foliation and on the correspondence between foliated and holonomy invariant objects for foliated manifolds.

How to make a clean separation between CMB E and B modes with proper foreground masking

We investigate the E/B decomposition of CMB polarization on a masked sky. In real space, operators of E and B mode decomposition involve only differentials of CMB polarization. We may, therefore in principle, perform a clean E/B decomposition from incomplete sky data. Since it is impractical to apply second derivatives to observation data, we usually rely on spherical harmonic transformation and inverse transformation, instead of using real-space operators. In spherical harmonic representation, jump discontinuities in a cut sky produces Gibbs phenomenon, unless a spherical harmonic expansion is made up to an infinitely high multipole. By smoothing a foreground mask, we may suppress the Gibbs phenomenon effectively in a similar manner to apodization of a foreground mask discussed in other works. However, we incur foreground contamination by smoothing a foreground mask, because zero-value pixels in the original mask may be rendered non-zero by the smoothing process. In this work, we investigate an optimal foreground mask, which ensures proper foreground masking and suppresses Gibbs phenomenon. We apply our method to a simulated map of the pixel resolution comparable to the Planck satellite. The simulation shows that the leakage power is lower than unlensed CMB B mode power spectrum of tensor-to-scalar ratio r ~ 1 × 10-7. We compare the result with that of the original mask. We find that the leakage power is reduced by a factor of 106 ~ 109 at the cost of a sky fraction 0.07, and that that the enhancement is highest at lowest multipoles. We confirm that all the zero-value pixels in the original mask remain zero in our mask. The application of this method to the Planck data will improve the detectability of primordial tensor perturbation.

Linear filtration methods for statistical analysis of periodically correlated random processes—Part II: Harmonic series representation

The estimates of probability characteristics for periodically correlated signals that are based on harmonic series representation are analyzed. Two methods for stationary components estimations are discussed: Hilbert transform-based method and a frequency shift method. Application of frequency shift method to real and simulated data is shown.

Potential energy surfaces for van der waals complexes of rare gases with H2S and H2S2: Extension to xenon interactions and hyperspherical harmonics representation

AbstractThis article is an account and extension of a series of recent investigations, which using extensive quantum chemical methods provide analytical hyperspherical representations of the potential energy surfaces for the interactions of rare gases with H2S as a rigid molecule, and H2S2, considered as a floppy molecule with respect to torsional mode. For the H2S‐rare gas systems, the representation is based on a minimal model, here introduced and discussed. For H2S2, the study of the interaction with Xe, not considered previously, completes the series. The results are discussed with reference to the properties and trends expected for interactions of van der Waals type. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Numerical modelling of organic waste dispersion from fjord located fish farms

In this study, a three-dimensional particle tracking model coupled to a terrain following ocean model is used to investigate the dispersion and the deposition of fish farm particulate matter (uneaten food and fish faeces) on the seabed due to tidal currents. The particle tracking model uses the computed local flow field for advection of the particles and random movement to simulate the turbulent diffusion. Each particle is given a settling velocity which may be drawn from a probability distribution according to settling velocity measurements of faecal and feed pellets. The results show that the maximum concentration of organic waste for fast sinking particles is found under the fish cage and continue monotonically decreasing away from the cage area. The maximum can split into two maximum peaks located at both sides of the centre of the fish cage area in the current direction. This process depends on the sinking time (time needed for a particle to settle at the bottom), the tidal velocity and the fish cage size. If the sinking time is close to a multiple of the tidal period, the maximum concentration point will be under the fish cage irrespective of the tide strength. This is due to the nature of the tidal current first propagating the particles away and then bringing them back when the tide reverses. Increasing the cage size increases the likelihood for a maximum waste accumulation beneath the fish farm, and larger farms usually means larger biomasses which can make the local pollution even more severe. The model is validated by using an analytical model which uses an exact harmonic representation of the tidal current, and the results show an excellent agreement. This study shows that the coupled ocean and particle model can be used in more realistic applications to help estimating the local environmental impact due to fish farms.

Predicting drug polypharmacology using a novel surface property similarity-based approach

In recent years, polypharmacology is becoming anincreasingly important aspect in drug design. For exam-ple, pharmaceutical companies are discovering more andmore cases in which multiple drugs bind to a given tar-get (promiscuous targets) and in which a given drugbinds to more than one target (promiscuous ligands).Both of these phenomena are clearly of great impor-tance when considering drug side-effects. Given thatscreening drugs against all the proteins expressed by thehuman genome is infeasible, several computational tech-niques for predicting the pharmacological profiles ofdrugs have been developed, ranging from statistical ana-lyses of chemical fingerprints and biological activities [1]to 3D docking of ligand structures into protein pockets.Here we present a novel shape-based approach whichuses spherical harmonic (SH) representations [2,3] tocompare molecular surfaces and key surface propertiesvery efficiently. This approach compares targets by theSH similarity of their ligands and also of their bindingpockets. This allows promiscuous ligands and targets tobe identified and characterized.In this contribution, we present details of ourapproach applied to a subset of the MDL Drug DataReport (MDDR) database containing 65367 compoundsdistributed over 249 diverse pharmacological targets forwhich experimental binding information is known. Thesimilarity of each ligand to each target’s ligand set isquantified and used to predict promiscuity. To ourknowledge, this is the largest all-against-all polypharma-cological study to have been carried out using shape-based techniques. We compare our promiscuity predic-tions with computational and experimental resultsobtained by Keiser et al.[4]. We also analyse thecorrelation between binding pocket shapes and ligand-based promiscuity predictions using the ligand andpocket shape similarity matrices.

Synchrony‐capture filterbank: A novel cochlear signal processing model.

Examination of the representation of low harmonics of complex sounds in the auditory nerve shows a striking feature known as “synchrony capture.” Fibers of an entire cochlear region are driven almost exclusively by one local, dominant harmonic component [Delgutte and Kiang, 1984]. Sharp boundaries characteristic of such synchrony capture are also seen between the different CF regions driven by different dominant, formant‐region harmonics for multiformant vowels. Based on this observation, we propose a model for peripheral processing, which is not just a filter bank but behaves more like signal adaptive receivers. We call this model synchrony capture filterbank (SCFB). SCFB consists of a traditional gammatone filterbank, the individual filters of which are then cascaded with a bandpass filter (BPF) triplet. The BPF triplet is an adaptive tone follower and consists of three overlapping bandpass filters whose center frequencies can be changed using feedback. The amplitudes at the output of the the bandpass filters are feedback to tune the BPF triplet such that it centers itself on top of the dominant tone in the input signal. The SCFB exhibits the synchrony capture behavior that is observed in real auditory nerve fibers.

Some theory for anisotropic processes on the sphere

We investigate properties of an existing class of models for Gaussian processes on the sphere that are invariant to shifts in longitude. The class is obtained by applying first-order differential operators to an isotropic process and potentially adding an independent isotropic term. For a particular choice of the operators, we derive explicit forms for the spherical harmonic representation of these processes’ covariance functions. Because the spherical harmonic representation is a spectral one, these forms allow us to draw conclusions about the local properties of the processes. For one, the coefficients in the spherical harmonic representation relate to the equivalence and orthogonality of the measures induced by the models. It turns out that under certain conditions the models will lack consistent parameter estimability even when the process is observed everywhere on the sphere. We also consider the ability of the models to capture isotropic tendencies on the local level, a phenomenon observed in some data.

Validating and Improving Elastic Network Models Using Molecular Dynamics

Elastic network models (ENMs) offer the capability to study collective motions of large systems without the overwhelming computational cost of all-atom molecular dynamics (MD). They accomplish this by replacing the standard force field terms with simple harmonic connections, typically between neighboring alpha carbons; the collective motions can then be computed using normal mode analysis. We investigate new methods for performing ENM calculations to improve their sensitivity and fidelity. First, we have explored using a higher resolution representation for the protein, supplementing the standard 1-node per residue approach by adding a second node to represent the side chain. Our aim is to utilize this increase in spatial resolution, enabling the ENMs to predict the effects of subtler changes, such as the binding of small ligands. Second, we have explored the efficacy of different choices for the spring constants in the harmonic representation. Following our previous work (Romo and Grossfield. Proteins, 2010), we have implemented a new functional form, which distinguishes bonded and nonbonded interactions. In both cases, we have validated our approaches via quantitative comparison to several microsecond-scale all-atom simulations of G protein-coupled receptors.

3D Face Reconstruction from a Single Image Using a Single Reference Face Shape

Human faces are remarkably similar in global properties, including size, aspect ratio, and location of main features, but can vary considerably in details across individuals, gender, race, or due to facial expression. We propose a novel method for 3D shape recovery of faces that exploits the similarity of faces. Our method obtains as input a single image and uses a mere single 3D reference model of a different person's face. Classical reconstruction methods from single images, i.e., shape-from-shading, require knowledge of the reflectance properties and lighting as well as depth values for boundary conditions. Recent methods circumvent these requirements by representing input faces as combinations (of hundreds) of stored 3D models. We propose instead to use the input image as a guide to "mold" a single reference model to reach a reconstruction of the sought 3D shape. Our method assumes Lambertian reflectance and uses harmonic representations of lighting. It has been tested on images taken under controlled viewing conditions as well as on uncontrolled images downloaded from the Internet, demonstrating its accuracy and robustness under a variety of imaging conditions and overcoming significant differences in shape between the input and reference individuals including differences in facial expressions, gender, and race.

Synthesis of Room Impulse Responses for Variable Source Characteristics

Every acoustic source, e.g. a speaker, a musical instrument or a loudspeaker, generally has a frequency dependent characteristic radiation pattern, which is preeminent at higher frequencies. Room acoustic measurements nowadays only account for omnidirectional source characteristics. This motivates a measurement method that is capable of obtaining room impulse responses for these specific radiation patterns by using a superposition approach of several measurements with technically well-defined sound sources. We propose a method based on measurements with a 12-channel independentlydriven dodecahedron loudspeaker array rotated by an automatically controlled turntable.Radiation patterns can be efficiently described with the use of spherical harmonics representation. We propose a method that uses this representation for the spherical loudspeaker array used for the measurements and the target radiation pattern to be used for the synthesis.We show validating results for a deterministic test sound source inside in a small lecture hall.

Quality Estimates in Geoid Computation by EGM08

Quality Estimates in Geoid Computation by EGM08The high-degree Earth Gravitational Model EGM08 allows for geoid determination with a resolution of the order of 5'. Using this model for estimating the quasigeoid height, we estimate the global root mean square (rms) commission error to 5 and 11 cm, based on the assumptions that terrestrial gravity contributes to the model with an rms standard error of 5 mGal and correlation length 0:01° and 0:1°, respectively. The omission error is estimated to—0:7Δg [mm], where Δg is the regional mean gravity anomaly in units of mGal.In case of geoid determination by EGM08, the topographic bias must also be considered. This is because the Earth's gravitational potential, in contrast to its spherical harmonic representation by EGM08, is not a harmonic function at the geoid inside the topography. If a correction is applied for the bias, the main uncertainty that remains is that from the uncertainty in the topographic density, which will still contribute to the overall geoid error.

Using Spherical Harmonic Surface Property Representations for Ligand-Based Virtual Screening.

Ligand-based virtual screening (VS) techniques have become well established in the drug discovery process. However, despite their relative success, there still exists the problem of how to define the initial query compounds and which of their conformations should be used. Here, we propose a novel shape plus surface property approach using multiple local spherical harmonic (SH) functions. We also investigate the use of shape-based and shape plus property-based consensus SH queries calculated in several different ways. The utility of these approaches is compared using the 40 pharmaceutically relevant targets of the DUD database. Our results show that using a combination of SH-based properties often gives better VS performance than using simple shape-based queries. Shape-based consensus queries also perform well, but we find that explicit 3D shape-property conformations should be retained for highly flexible ligands.