Planar Approximations Research Articles

Об одном методе построения однородных плоскостных аппроксимаций фильтрующего генератора

Planar approximations of filter generators can be used to restore its initial state from a segment of the output sequence. The results of the study of one method for constructing plane approximations of a special form are presented.

Planar fiber orientation: Jeffery, non-orthotropic closures, and reconstructing distribution functions

Many models for fiber suspensions use the second-order orientation tensor to represent the orientation state. This requires that the fourth-order orientation tensor be approximated using a closure formula. The prediction of non-linear properties also requires reconstructing an approximate orientation distribution function. All existing fourth-order closures and most reconstructions have orthotropic symmetry, which limits their accuracy. In this paper, closures and reconstruction are explored for planar fiber orientation, using a recent parameterization of fourth-order orientation tensors by Bauer and Böhlke (2021, 2022a, 2022b). We show that the orientation distribution for fibers that follow Jeffery’s equation, together with the associated natural closure, form the central axis of the space of fourth-order orientation tensors. A new family of non-orthotropic planar closure approximations is created by including the rate-of-deformation tensor as an additional independent variable. These closures are shown to be substantially more accurate than any planar orthotropic closure. We also explore the Jeffery, Bingham, ellipse radius, and fourth-order maximum entropy (ME4) functions for reconstructing the orientation distribution. When compared to distributions from the Folgar-Tucker model, ME4 using the exact fourth-order orientation tensor gives the most accurate reconstruction, while ME4 with the fourth-order tensor from the new non-orthotropic closure is nearly as good. • The Jeffery distribution is central to planar fourth-order orientation tensors. • The first-ever non-orthotropic closure approximations are developed. • Non-orthotropic closures are more accurate than any previous planar closure. • Fourth-order maximum entropy distributions give the most accurate reconstruction.

Fast and Accurate Registration of Terrestrial Point Clouds Using a Planar Approximation of Roof Features

3D reconstructed models are becoming more diffused daily, especially in the Cultural Heritage field. These geometric models are typically obtained from elaborating a 3D point cloud. A significant limit in using these methods is the realignment of different point clouds acquired from different acquisitions, particularly for those whose dimensions are millions of points. Although several methodologies have tried to propose a solution for this necessity, none of these seems to solve definitively the problems related to the realignment of large point clouds. This paper presents a new and innovative procedure for the fine registration of large point clouds. The method performs an alignment by using planar approximations of roof features, taking the roof’s extension into account. It looks particularly suitable for the alignment of large point clouds acquired in urban and archaeological environments. The proposed methodology is compared in terms of accuracy and time with a standard photogrammetric reconstruction based on Ground Control Points (GCPs) and other ones, aligned by the Iterative Closest Point method (ICP) and markers. The results evidence the excellent performance of the methodology, which could represent an alternative for aligning extensive photogrammetric reconstructions without the use of GCPs.

Bend 3d mixed virtual element method for Darcy problems

In this study, we propose a virtual element scheme to solve the Darcy problem in three physical dimensions. The main novelty is that curved elements are naturally handled without any degradation of the solution accuracy. Indeed, in presence of curved boundaries, or internal interfaces, the geometrical error introduced by planar approximations may dominate the convergence rate limiting the benefit of high-order approximations. We consider the Darcy problem in its mixed form to directly obtain accurate and mass conservative fluxes without any post-processing. An important step to derive the proposed scheme is the integration over curved polyhedrons, here presented and discussed. Finally, we show the theoretical analysis of the scheme as well as several numerical examples to support our findings.

Mapping dark matter on the celestial sphere with weak gravitational lensing

ABSTRACT Convergence maps of the integrated matter distribution are a key science result from weak gravitational lensing surveys. To date, recovering convergence maps has been performed using a planar approximation of the celestial sphere. However, with the increasing area of sky covered by dark energy experiments, such as Euclid, the Vera Rubin Observatory’s Legacy Survey of Space and Time (LSST), and the Nancy Grace Roman Space Telescope, this assumption will no longer be valid. We recover convergence fields on the celestial sphere using an extension of the Kaiser–Squires estimator to the spherical setting. Through simulations, we study the error introduced by planar approximations. Moreover, we examine how best to recover convergence maps in the planar setting, considering a variety of different projections and defining the local rotations that are required when projecting spin fields such as cosmic shear. For the sky coverages typical of future surveys, errors introduced by projection effects can be of the order of tens of percent, exceeding 50 per cent in some cases. The stereographic projection, which is conformal and so preserves local angles, is the most effective planar projection. In any case, these errors can be avoided entirely by recovering convergence fields directly on the celestial sphere. We apply the spherical Kaiser–Squires mass-mapping method presented to the public Dark Energy Survey science verification data to recover convergence maps directly on the celestial sphere.

Edge correction for intensity estimation of 3D heterogeneous point processes from replicated data

A common task in the analysis of point patterns is to estimate the intensity of the underlying process, i.e., the expected number of points per unit area or volume at each position over space. In biological studies, a specific feature of point patterns and processes is their confinement within bounded domains of Rd because of the organization of biological systems into various compartments (tissues, cells, nuclei, etc.). This induces a systematic negative bias in intensity estimates at the boundary of the confinement domain. Here, we address this edge effect issue in the context of intensity estimation from multiple realizations, which is particularly relevant to biological studies because of experimental replications. We introduce an edge correction method for a recently proposed maximum likelihood intensity estimator based on distance statistics that requires no additional parameter in the estimation method. We describe corrections relying on locally planar or spherical shape approximations of the domain boundary. Based on corrected estimators, we propose a strategy for the statistical mapping of a 3D bounded process that adapts to the shape of the domain boundary. We demonstrate quantitatively the robustness of the correction method using point patterns simulated over domains of Rd with various shapes. We illustrate the practical usefulness of the method by analyzing the 3D spatial organization of compartments within plant cells. The method should be useful for the statistical analysis of biological structures at different scales.

Precise geoid computation using Stokes-Helmert's scheme and strict integrals of topographic effects

The high-precision local geoid model was computed based on the improved Stokes-Helmert′s boundary value problem and strict integrals of topographic effects. This proposed method involves three steps. First, the mathematical form of Stokes-Helmert′s boundary value problem was derived, and strict computational formulas regarding topographic effects were provided to overcome the disadvantage of planar approximations. Second, a gravimetric geoid model was constructed using the proposed Stokes-Helmert′s scheme with a heterogeneous data set. Third, a least squares adjustment method combined with a multi-surface function model was employed to remove the bias between the gravimetric geoid model and the GNSS/leveling data and to refine the final local geoid model. The accuracy of the final geoid model was evaluated using independent GNSS/leveling data. Numerical results show that an external precision of 1.45 cm is achievable.

Numerical study on adjusting parameters to improve gaze estimation using planar approximations from electro-oculogram signal voltage ratios.

Gaze or eye movements are used as a communication interface in daily life. Herein, we developed a simple method for gaze estimations based on planar approximations of voltage ratios calculated from multiple electro-oculogram signals not affected by drift phenomena, which decrease accuracy. Subsequently, we conducted simulations using an eyeball battery model and investigated the effects of adjusting electrode arrangements, determination coefficients for planar approximations of voltage ratios, and threshold values for angles between simultaneous linear equations, to improve the estimation accuracy and decreased the number of required electrodes. Numerical experiments were used to identify arrangements of six electrodes with errors that were approximately 5° less than those of nine-electrode L-shaped arrangements, indicating improved estimation accuracy with fewer electrodes.

Modelling the view factor of a ‘grain-like’ observer near a tilted pool fire via planar approximation approach

• Planar approximation to view factor, F, expressed in terms of two sensitive parameters: α and β . • The larger the values of β , the closer the view factor predictions of PHAST 7.2 and planar approximations. • Large enough values of β would guarantee the observer receives a maximum view factor and hence maximum radiation. • F decreases (decays polynomially) with increasing observer distances away from the flame surface. • Both approximations were found to coincide for: β = 48 . 961 ∘ , X = 30 m, α = 0.01 , γ = 0.77 , L = 12.14 m, Θ = 55 . 17 ∘ , D = 5 m. Modelling the view factor, F , of a ‘grain-like’ observer near a tilted pool fire assumed to be cylindrical in shape, requires the observer (on the same ground level as the pool fire) to be as close as possible to the flame surface so that the flame surface is viewed as a plane. The orientation of the observer is to receive a maximum view, with the view factor integrated over the flame area seen by the observer. The derived expression for F was expressed in terms of standard functions and sensitive parameters: α , defined to ensure the observer receives a maximum view of the flame surface and β , the angle of inclination of the differential observer plane. Results from a planar approximation to F are compared with those of PHAST 7.2 simulated results in MATLAB . The values of the planar approximation to F was found to increase with increasing β . This suggests that the larger the value of β , the more the radiation received by a near observer. For β = 48 . 961 ∘ , horizontal distance, X = 30 m, α = 0.01 , flame length, L = 12.14 m, tilt angle, θ = 55 . 17 ∘ , and pool diameter, D = 5 m: planar and PHAST 7.2 approximations to F were found to coincide up to 6 significant digits and differ by 3.2 × 10 − 8 . The close agreement between both approximations depend heavily on the choice of flame properties and sensitive parameters. Interestingly, this result may depend sensitively on the choice of parameters, leading to prediction of planar approximation to F much higher or lower than those of PHAST 7.2 approximations. The advantage of this approach is that a knowledge of the plane where the largest concentration of radiation is located will help minimise loss of lives to fire hazards and improve the efficiency of risk safety assessment/management.

An optimal flattening algorithm for ship hull plate fabricated by thermal forming

Normally, ship hull plates contain compound curvature distributions and they can only be fabricated by integrating mechanical rolling with bending and following them with regional heating to achieve required configuration. This paper mainly proposes a comprehensive flattening process to determine optimal planar shape, shrinkage strain distribution and mapping relation for three typical ship hull plates by thermal forming, which practically benefit for automatic forming. An innovative geometric method is first presented to obtain initial planar approximation and strain square sum subject to negative principal strain constraints within each element from planar to desired surface is formulated based on nonlinear kinetic analysis. Then, we employ area equivalent method through transforming spatial quadrilateral into planar quadrilateral to simplify optimization model. Finally, initial planar approximations are regarded as iteration initial points to solve the model based on sequential quadratic programming. The results indicate that the proposed process can effectively achieve optimal planar shape, mapping relation and negative principal strain to help achieve all heating paths and heat source parameters at one time, thus automation.

Theory of corticothalamic brain activity in a spherical geometry: Spectra, coherence, and correlation.

Corticothalamic neural field theory is applied to a spherical geometry to better model neural activity in the human brain and is also compared with planar approximations. The frequency power spectrum, correlation, and coherence functions are computed analytically and numerically. The effects of cortical boundary conditions and resulting modal aspects of spherical corticothalamic dynamics are explored, showing that the results of spherical and finite planar geometries converge to those for the infinite planar geometry in the limit of large brain size. Estimates are made of the point at which modal series can be truncated and it is found that for physiologically plausible parameters only the lowest few spatial eigenmodes are needed for an accurate representation of macroscopic brain activity. A difference between the geometries is that there is a low-frequency 1/f spectrum in the infinite planar geometry, whereas in the spherical geometry it is 1/f^{2}. Another difference is that the alpha peak in the spherical geometry is sharper and stronger than in the planar geometry. Cortical modal effects can lead to a double alpha peak structure in the power spectrum, although the main determinant of the alpha peak is corticothalamic feedback. In the spherical geometry, the cross spectrum between two points is found to only depend on their relative distance apart. At small spatial separations the low-frequency cross spectrum is stronger than for an infinite planar geometry and the alpha peak is sharper and stronger due to the partitioning of the energy into discrete modes. In the spherical geometry, the coherence function between points decays monotonically as their separation increases at a fixed frequency, but persists further at resonant frequencies. The correlation between two points is found to be positive, regardless of the time lag and spatial separation, but decays monotonically as the separation increases at fixed time lag. At fixed distance the correlation has peaks at multiples of the period of the dominant frequency of system activity.

2D/3D image charge for modeling field emission

Analytic image charge approximations exist for planar and spherical metal surfaces but approximations for more complex geometries, such as the conical and wirelike structures characteristic of field emitters, are lacking. Such models are the basis for the evaluation of Schottky lowering factors in equations for current density. The development of a multidimensional image charge approximation, useful for a general thermal-field emission equation used in space charge studies, is given and based on an analytical model using a prolate spheroidal geometry. A description of how the model may be adapted to be used with a line charge model appropriate for carbon nanotube and carbon fiber field emitters is discussed.

Development of a Method for Gaze Estimation Based on Planar Approximations of Voltage Ratios Calculated from Multiple Electro-oculogram Signals

Development of a Method for Gaze Estimation Based on Planar Approximations of Voltage Ratios Calculated from Multiple Electro-oculogram Signals

3D planar representation of stereo depth images for 3DTV applications.

The depth modality of the multiview video plus depth (MVD) format is an active research area, whose main objective is to develop depth image based rendering friendly efficient compression methods. As a part of this research, a novel 3D planar-based depth representation is proposed. The planar approximation of multiple depth images are formulated as an energy-based co-segmentation problem by a Markov random field model. The energy terms of this problem are designed to mimic the rate-distortion tradeoff for a depth compression application. A novel algorithm is developed for practical utilization of the proposed planar approximations in stereo depth compression. The co-segmented regions are also represented as layered planar structures forming a novel single-reference MVD format. The ability of the proposed layered planar MVD representation in decoupling the texture and geometric distortions make it a promising approach. Proposed 3D planar depth compression approaches are compared against the state-of-the-art image/video coding standards by objective and visual evaluation and yielded competitive performance.

Slip distributions on active normal faults measured from LiDAR and field mapping of geomorphic offsets: an example from L'Aquila, Italy, and implications for modelling seismic moment release

Surface slip distributions for an active normal fault in central Italy have been measured using terrestrial laser scanning (TLS), in order to assess the impact of changes in fault orientation and kinematics when modelling subsurface slip distributions that control seismic moment release. The southeastern segment of the surface trace of the Campo Felice active normal fault near the city of L'Aquila was mapped and surveyed using techniques from structural geology and using TLS to define the vertical and horizontal offsets of geomorphic slopes since the last glacial maximum (15±3ka). The fault geometry and kinematics measured from 43 sites and throw/heave measurements from geomorphic offsets seen on 250 scarp profiles were analysed using a modification of the Kostrov equations to calculate the magnitudes and directions of horizontal principal strain-rates. The map trace of the studied fault is linear, except where a prominent bend has formed to link across a former left-stepping relay-zone. The dip of the fault and slip direction are constant across the bend. Throw-rates since 15±3ka decrease linearly from the fault centre to the tip, except in the location of the prominent bend where higher throw rates are recorded. Vertical coseismic offsets for two palaeo earthquake ruptures seen as fresh strips of rock at the base of the bedrock scarp also increase within the prominent bend. The principal strain-rate, calculated by combining strike, dip, slip-direction and post 15±3ka throw rate, decreases linearly from the fault centre towards the tip; the strain-rate does not increase across the prominent fault bend. The above shows that changes in fault strike, whilst having no effect on the principal horizontal strain-rate, can produce local maxima in throw-rates during single earthquakes that persist over the timescale of multiple earthquakes (15±3ka). Detailed geomorphological and structural characterisation of active faults is therefore a critical requirement in order to properly define fault activity for the purpose of accurate seismic hazard assessment. We discuss the implications of modelling subsurface slip distributions for earthquake ruptures through inversion of GPS, InSAR and strong motion data using planar fault approximations, referring to recent examples on the nearby Paganica fault that ruptured in the Mw 6.3 2009L'Aquila Earthquake.

A 3-D Volume-of-Fluid advection method based on cell-vertex velocities for unstructured meshes

A new geometrical Volume-of-Fluid (VOF) method for capturing interfaces on three-dimensional (3-D) Cartesian and unstructured meshes is introduced. The method reconstructs interfaces as first- and second-order piecewise planar approximations (PLIC), and advects volumes in a single unsplit Lagrangian–Eulerian (LE) geometrical algorithm based on constructing flux polyhedrons by tracing back the Lagrangian trajectories of the cell-vertex velocities. In this way, the situations of overlapping between flux polyhedrons are minimized, consequently, the accuracy in the solution of the advection equation is improved by minimizing the creation of overshoots (volume fractions over one), undershoots (volume fractions below zero) and wisps (fluid in void regions or vice versa). However, if not treated carefully, the use of cell-vertex velocities may result in the construction of flux polyhedrons that contain nonplanar faces and that do not conserve volume. Therefore, this work explains in detail a set of geometric algorithms necessary to overcome these two drawbacks. In addition, the new VOF method is analyzed numerically on 3-D Cartesian and unstructured meshes, first, by reconstructing the interface of spherical geometries and, second, by evaluating the final advection result of a sphere placed in a rotation, shear and deformation field.

Watertight Scenes from Urban LiDAR and Planar Surfaces

AbstractThe demand for large geometric models is increasing, especially of urban environments. This has resulted in production of massive point cloud data from images or LiDAR. Visualization and further processing generally require a detailed, yet concise representation of the scene's surfaces. Related work generally either approximates the data with the risk of over‐smoothing, or interpolates the data with excessive detail. Many surfaces in urban scenes can be modeled more concisely by planar approximations. We present a method that combines these polygons into a watertight model. The polygon‐based shape is closed with free‐form meshes based on visibility information. To achieve this, we divide 3‐space into inside and outside volumes by combining a constrained Delaunay tetrahedralization with a graph‐cut. We compare our method with related work on several large urban LiDAR data sets. We construct similar shapes with a third fewer triangles to model the scenes. Additionally, our results are more visually pleasing and closer to a human modeler's description of urban scenes using simple boxes.

Planar, spherical and ellipsoidal approximations of Poisson's integral in near zone

Planar, spherical and ellipsoidal approximations of Poisson's integral in near zonePlanar, spherical, and ellipsoidal approximations of Poisson's integral for downward continuation (DWC) of gravity anomalies are discussed in this study. The planar approximation of Poisson integral is assessed versus the spherical and ellipsoidal approximations by examining the outcomes of DWC and finally the geoidal heights. We present the analytical solution of Poisson's kernel in the point-mean discretization model that speed up computation time 500 times faster than spherical Poisson kernel while preserving a good numerical accuracy. The new formulas are very simple and stable even for regions with very low height. It is shown that the maximum differences between spherical and planar DWC as well as planar and ellipsoidal DWC are about 6 mm and 18 mm respectively in the geoidal heights for a rough mountainous area such as Iran.

Technical Note: The heterogeneous Zeldovich factor

Abstract. In this technical note we present the exact form of the Zeldovich factor for heterogeneous nucleation on spherical pre-existing particles. We study the error caused by planar pre-existing surface approximations, which have been used in our earlier heterogeneous nucleation model and elsewhere in the literature. We also test the significance of widely used approximations for cluster surface area and circumference. We conclude that the approximations do not affect the predicted onset saturation. Especially for small pre-existing particles the nucleation rates calculated with the exact and approximative models differ significantly.

Analysis of Finite Arrays of Axially Directed Printed Dipoles on Electrically Large Circular Cylinders

Various arrays consisting of finite number of printed dipoles on electrically large dielectric coated circular cylinders are investigated using a hybrid method of moments/Green's function technique in the spatial domain. This is basically an "element by element" approach in which the mutual coupling between dipoles through space as well as surface waves is incorporated. The efficiency of the method comes from the computation of the Green's function, where three types of spatial domain Green's function representations are used interchangeably, based on their computational efficiency and regions where they remain accurate. Numerical results are presented in the form of array current distributions, active reflection coefficient and far-field pattern to indicate the efficiency and accuracy of the method. Furthermore, these results are compared with similar results obtained from finite arrays of printed dipoles on grounded planar dielectric slabs. It is shown that planar approximations, except for small separations, can not be used due to the mutual coupling between the array elements. Consequently, basic performance metrics of printed dipole arrays on coated cylinders show significant discrepancies when compared to their planar counterparts.