Azimuthal Projection Research Articles

An Area Preserving Projection from the Regular Octahedron to the Sphere

In this paper, we propose an area preserving bijective map from the regular octahedron to the unit sphere $${\mathbb{S}^2}$$ , both centered at the origin. The construction scheme consists of two steps. First, each face F i of the octahedron is mapped to a curved planar triangle $${\mathcal{T}_i}$$ of the same area. Afterwards, each $${\mathcal{T}_i}$$ is mapped onto the sphere using the inverse Lambert azimuthal equal area projection with respect to a certain point of $${\mathbb{S}^2}$$ . The proposed map is then used to construct uniform and refinable grids on a sphere, starting from any triangular uniform and refinable grid on the triangular faces of the octahedron.

SU-E-l-01: Investigating the Dependence of 2D and 3D Scatter-To-Primary Ratios on Breast Density in Clinical Breast CT.

To characterize 2D and 3D scatter-to-primary ratios (SPR) and investigate the dependence of SPR on breast density based on clinicalpatient imaging on our dedicated SPECT-CT mammotomography system. As a part of an on-going IRB approved protocol, 7 consented women underwent a breast imaging study with our SPECT-CT system. Using a quasi-monochromatic x-ray cone beam and flat panel detector, 240 projections were obtained. 6 beam stop array (BSA) projections were also obtained over 40 degree intervals. The CT data were scatter corrected using a phantom-validated algorithm based on the BSA technique. For each projection, 2D scatter fluence was determined behind each beam stop shadow through the patient's breast, and cubic spline interpolated throughout the thresholded breast region. 2D SPRs were calculated as the average of 6 measured ratios of scatter projections to scatter corrected (primary) projections. Angular cubic spline interpolation was performed to obtain the remaining 234 azimuthal BSA projections for reconstruction correction. Corrected and uncorrected projection images were individually reconstructed and 3D SPRs calculated as the ratio of difference between primary and un-corrected volumes to primary volumes. Histograms of the reconstructed patient data-sets yielded distinct peaks representing linear attenuation coefficients of glandular+skin and fatty tissue. Percentage glandular and fatty tissue was estimated by fitting a double Gaussian to the histogram and integrating area under the curve. Only 4 CT data sets devoid of motion and truncation artifacts were suitable for scatter correction. Preliminary results show that 2D SPR values peak at the center of breast volumes and were greatest (∼50%) for the most glandular breast, whereas 3D SPR values remained fairly constant throughout the breast and did not obviously correlate with density, consistent with prior phantom results. 2D SPR values are greater in breasts with higher glandular tissue composition, whereas 3D SPR values appear independent of breast composition. This work is funded by the National Cancer Institute of the National Institutes of Health grant RO1-CA096821, with partial support (JHP) from NIH T32-EB001040. MPT is the inventor of this CT technology, and is named as an inventor on the patent for this technology awarded to Duke University. If this technology becomes commercially successful, MPT and Duke could benefit financially.

A general rule for disk and hemisphere partition into equal-area cells

A new general rule is presented to define procedures in order to cut a disk or a hemisphere into an imposed number of equal-area cells. The system has several degrees of freedom that can be fixed, for instance, by enforcing the cells aspect ratios. Therefore, the cells can have very comparable forms, i.e. close to the square. This kind of method is effectively useful because it is not possible to build exact dense uniform distributions of points on the sphere. However, it will be shown that it is easy to cover the sphere, the hemisphere or the disk with equal-area cells. This capability makes easy, for instance, the implementation of stratified sampling in Monte Carlo methods. Moreover, the use of different azimuthal projections allows to link problems initially stated either on the hemisphere or within the circle.

Attitude Parameterizations as Higher-Dimensional Map Projections

A class of vectorial attitude parameterizations that are formulated as a product of the unit rotation vector and various functions of the rotation angle is examined. When related to a four-dimensional unit quaternion, these vectorial parameterizations are shown to be analogous to higher-dimensional azimuthal projections from a threedimensional unit hypersphere. Several types of these projections are examined. Singularities are identified and numerical accuracy is evaluated based on the singular value decomposition of the attitude kinematics. It is shown how shadow parameterizations can be constructed in order to alleviate the kinematical singularities. It is also shown that the kinematical passivity and optimality of the Rodrigues and modified Rodrigues parameters are special cases of themore general result that holds for awider range of vectorial parameterizations. This result is used to formulate and compare passivity-based control laws using various parameterizations.

Mapping system for coregistration of cardiac mri and ex vivo tissue sampling

To design a method suitable for obtaining tissue samples from regions of different function as ascertained by magnetic resonance imaging (MRI). In vivo MRI was used to create azimuthal projections of the heart from dilated cardiomyopathy transplant patients with the cardiac valves in the center and four concentric rings representing the septum and free wall. Tagged MRI could identify regions of different contractile strength that were then transferred onto the map projection. The resulting tissue sampling map was used to guide dissection of tissue samples from the explanted heart for analysis by electron microscopy (EM) as well as provide samples for subsequent mRNA analysis. Accuracy of the sampling was determined in a sheep heart using 17 fiduciary markers glued to the epicardial surface. Tagged MRI identified areas of "normal" (%S(c) -11), "poor" (%S(c) -4) and "failed" contraction (%S(c) +3). The mapping method we developed enabled straightforward sampling of these regions after surgical excision. EM showed good tissue preservation while the test of accuracy using the fiduciary markers showed a sampling accuracy of 0.3 ± 3.7 mm. This was similar to the resolution of tagged MRI images themselves. The methods we have developed can accurately guide tissue sampling for ex vivo tissue analysis.

Uniform spherical grids via equal area projection from the cube to the sphere

We construct an area preserving map from a cube to the unit sphere S2, both centered at the origin. More precisely, each face Fi of the cube is first projected to a curved square Si of the same area, and then each Si is projected onto the sphere by inverse Lambert azimuthal equal area projection, with respect to the points situated at the intersection of the coordinate axes with S2. This map is then used to construct uniform and refinable grids on a sphere, starting from any grid on a square.

New uniform grids on the sphere

Aims. A method for constructing new uniform grids on the sphere is given. Methods. We define a bijection in R 2 , which maps squares onto discs and preserves areas. Then we use this bijection, combined with Lambert azimuthal projection, for lifting uniform grids from the square to the sphere. Results. We can obtain uniform spherical grids that allow a hierarchical data manipulation and have an isolatitudinal distribution of cells. Compared with HEALPix grids, nowadays the most used in astronomy and astrophysics, our grids have the advantage of allowing easier implementation, and in addition one can move approximating functions from the square to the sphere by a simple technique.

Mapping technique of climate fields between GCM's and ice models

Abstract. Here, we present a mapping method OBLIMAP, which projects and interpolates fields like surface temperature, surface mass balance, and surface height between a geographical based coordinate system of a General Circulation Model (GCM) and a rectangular based Ice Model (IM). We derive an oblique stereographic projection and its inverse, which holds for any area at the Earth's surface, and which can be combined with two different interpolation methods. The first one is suited to interpolate the projected fields of a coarse GCM grid on a fine meshed IM grid. The second one is appropriate for the opposite case. Both grids are allowed to be arbitrary and irregularly spaced. Therefore the OBLIMAP technique is suitable for any GCM-IM combination. After a first scan of the GCM grid coordinates and the specification of the IM grid, fast mapping of various fields is possible. To and fro (GCM-IM-GCM) mapping tests with the Climate Community System Model (CCSM) at T42 resolution (~313 km) and the Regional Atmospheric Climate Model (RACMO) at ~11 km and ~55 km, show average temperature differences of less than 0.1 K with small standard deviations. OBLIMAP, available at GMD, is an accurate, robust and well-documented mapping method for coupling an IM with a GCM or to map state of the art initial and forcing fields available at geographical coordinates to any local IM grid with an optimal centered oblique projection. Currently, the oblique stereographic and the oblique Lambert azimuthal equal-area projections for both the sphere and the ellipsoid are implemented in OBLIMAP.

Reconstruction from a single diffraction pattern of azimuthally projected electron density of molecules aligned parallel to a single axis

Diffraction from the individual molecules of a molecular beam, aligned parallel to a single axis by a strong electric field or other means, has been proposed as a means of structure determination of individual molecules. As in fiber diffraction, all the information extractable is contained in a diffraction pattern from incidence of the diffracting beam normal to the molecular alignment axis. The limited size of the object results in continuous diffraction patterns characterized by neither Bragg spots nor layer lines. Equations relating the scattered amplitudes to the molecular electron density may be conveniently formulated in terms of cylindrical harmonics. For simulated diffraction patterns from short C nanotubes aligned along their axes, iterative solution of the equation for the zeroth-order cylindrical harmonic and its inverse with appropriate constraints in real and reciprocal space enables the phasing of the measured amplitudes, and hence a reconstruction of the azimuthal projection of the molecule.

Molecular structure determination from x-ray scattering patterns of laser-aligned symmetric-top molecules

We investigate the molecular structure information contained in the x-ray diffraction patterns of an ensemble of rigid CF(3)Br molecules aligned by an intense laser pulse at finite rotational temperature. The diffraction patterns are calculated at an x-ray photon energy of 20 keV to probe molecular structure at angstrom-scale resolution. We find that a structural reconstruction algorithm based on iterative phase retrieval fails to extract a reliable structure. However, the high atomic number of Br compared with C or F allows each diffraction pattern to be treated as a hologram. Using this approach, the azimuthal projection of the molecular electron density about the alignment axis may be retrieved.

THE DUNHUANG CHINESE SKY: A COMPREHENSIVE STUDY OF THE OLDEST KNOWN STAR ATLAS

This paper presents an analysis of the star atlas included in the medieval Chinese manuscript Or.8210/S.3326 discovered in 1907 by the archaeologist Aurel Stein at the Silk Road town of Dunhuang and now housed in the British Library. Although partially studied by a few Chinese scholars, it has never been fully displayed and discussed in the Western world. This set of sky maps (12 hour-angle maps in quasi-cylindrical projection and a circumpolar map in azimuthal projection), displaying the full sky visible from the Northern Hemisphere, is up to now the oldest complete preserved star atlas known from any civilisation. It is also the earliest known pictorial representation of the quasi-totality of Chinese constellations. This paper describes the history of the physical object—a roll of thin paper drawn with ink. We analyse the stellar content of each map (1,339 stars, 257 asterisms) and the texts associated with the maps. We establish the precision with which the maps were drawn (1.5-4° for the brightest stars) and examine the type of projections used. We conclude that precise mathematical methods were used to produce the Atlas. We also discuss the dating of the manuscript and its possible author, and we confirm the date +649-684 (early Tang Dynasty) as most probable based on the available evidence. This is at variance with a prior estimate of around +940. Finally, we present a brief comparison with later sky maps, both from China and Europe.

VIPERdb2: an enhanced and web API enabled relational database for structural virology

VIPERdb (http://viperdb.scripps.edu) is a relational database and a web portal for icosahedral virus capsid structures. Our aim is to provide a comprehensive resource specific to the needs of the virology community, with an emphasis on the description and comparison of derived data from structural and computational analyses of the virus capsids. In the current release, VIPERdb2, we implemented a useful and novel method to represent capsid protein residues in the icosahedral asymmetric unit (IAU) using azimuthal polar orthographic projections, otherwise known as Φ–Ψ (Phi–Psi) diagrams. In conjunction with a new Application Programming Interface (API), these diagrams can be used as a dynamic interface to the database to map residues (categorized as surface, interface and core residues) and identify family wide conserved residues including hotspots at the interfaces. Additionally, we enhanced the interactivity with the database by interfacing with web-based tools. In particular, the applications Jmol and STRAP were implemented to visualize and interact with the virus molecular structures and provide sequence–structure alignment capabilities. Together with extended curation practices that maintain data uniformity, a relational database implementation based on a schema for macromolecular structures and the APIs provided will greatly enhance the ability to do structural bioinformatics analysis of virus capsids.

Comparison of geometrical mapping for ring diagram analysis

Mapping the solar surface is a crucial step in any local helioseismology technique. Because the acoustic waves propagate along great circles at the solar surface, it has been shown that these circles need to be used in the geometrical construction of the plane grid. We study different types of projections based on great circles for the calculation of sub-surface flows from ring diagram analysis of GONG data. Azimuthal equidistant projection, transverse cylindrical projection, gnomonic projection and stereographic projection produce almost the same velocity fields with standard patch sizes (15°x 15°). The difference between the four projections is more noticeable when larger patches (30° x 30°) are used.

Sexual Dimorphism in the Hoverfly Motion Vision Pathway

Many insects perform high-speed aerial maneuvers in which they navigate through visually complex surrounds. Among insects, hoverflies stand out, with males switching from stationary hovering to high-speed pursuit at extreme angular velocities [1]. In dipterans, 50-60 large interneurons -- the lobula-plate tangential cells (LPTCs) -- detect changes in optic flow experienced during flight [2-5]. It has been predicted that large LPTC receptive fields are a requirement of accurate "matched filters" of optic flow [6]. Whereas many fly taxa have three horizontal system (HS) LPTC neurons in each hemisphere, hoverflies have four [7], possibly reflecting the more sophisticated flight behavior. We here show that the most dorsal hoverfly neuron (HS north [HSN]) is sexually dimorphic, with the male receptive field substantially smaller than in females or in either sex of blowflies. The (hoverfly-specific) HSN equatorial (HSNE) is, however, sexually isomorphic. Using complex optic flow, we show that HSN, despite its smaller receptive field, codes yaw velocity as well as HSNE. Responses to a target moving against a plain or textured background suggest that the male HSN could potentially play a role in target pursuit under some conditions.

Unusual Map Projections

In this paper I look at azimuthal projections, equal area projections, and spherical and map distances, with emphasis on less known variants.

Quasinormal Spectrum and Quantization of the Kerr–Newman Black Hole

The intermediate asymptotic quasinormal mode spectrum of the charged scalar and Dirac fields in the near extremal Kerr–Newman black hole is studied analytically. It is found that the quasinormal mode spectrum can be expressed in terms of the Hawking temperature Thb, the electric potential ϕ+ and the horizon's angular velocity ΩH for the case of (eϕ_ + mΩ H) > (1 - 4πThb)Re(ω) (where e is the charge and m is the azimuthal projection number), whereas it is only relevant to the charge and the mass parameter for another case. It is also shown that by using the Bohr's correspondence principle, the fundamental change in the black-hole surface area induced by the emission of a rotating charged quantum from the Kerr–Newman black hole is in accord with the Bekenstein–Mukhanov general prediction.

Land cover dynamic changes in northern China: 1989–2003

The 13 provinces (autonomous regions and municipalities) in northern China are located in latitude 31°–54°N and longitude 73°–136°E including Beijing, Tianjin, Hebei, Shanxi, Inner Mongolia, Jilin, Liaoning, Heilongjiang, Shaanxi, Gansu, Ningxia, Qinghai, and Xinjiang, where environments are fragile. In recent years, the eco-environmental problems such as vegetation destruction, desertification and soil erosion etc. become serious because of climate change and unreasonable human activities. In this paper, landscape pattern and its evolution in northern China from 1989 to 2003 was investigated by the combined use of RS and GIS based on the basic theory and method of landscape ecology. Land use/cover maps of the study area in 1989, 1999 and 2003 were produced by using 1 km monthly NOAA Advanced Very High Resolution Radiometer (AVHRR) and SPOT/VGT Normalized Difference Vegetation Index (NDVI) dataset from national climate bureau of China which were geo-registered to Lambert azimuthal equal-area map projection and were used in the paper. Landscape evolution in the area over the study period was investigated by two methods: (a) the changes of various landscape metrics were analyzed using the landscape structure analysis program FRAGSTATS; (b) the transition matrix of landscape patch types was calculated with the help of the RS and GIS software. The results showed that from 1989 to 2003, the landscape within the study area had undertaken a complicated evolution in landscape structure and composition. The diversity index and evenness index increased during the period, which means that the landscape pattern tended to be diversified and even. The fragmentation index of grassland, forestland and water areas also increased significantly. This showed that the distribution and structure of forestland, grassland and water areas had been changed greatly during the period, especially grassland which became more and more fragmentized, and its fragmentation index increased from 19.23% to 88.72%. The transitions of the landscape types were mainly shown by the changes among forestland, grassland and farmland, and grassland changing into unable land. Over the study period, grassland and water areas had decreased remarkably, accounting for 15% and 37% from 1989 to 1999 and 24.79% and 49.25% from 1999 to 2003 respectively. The grassland and water resources play an important role in the eco-environment and economic development of the region. So, they must be protected carefully. According to the analysis, we can conclude that the eco-environment in the study area is obviously degenerated due to unreasonable human activities and climate changes and some measures should be taken to combat the environ-mental degradation.

Modeling the Potential Swath Coverage of Nadir and Off-Nadir Pointable Remote Sensing Satellite-Sensor Systems

Pointable sensor systems onboard many earth resources satellites today, particularly the higher spatial resolution sensors, provide for a near infinite set of collection opportunities. Satellite orbits of these systems are not systematic repetitive tracks. Predicting future collection opportunities requires predicting where the satellite will be and then computing the potential swath coverage from a pointable sensor along these orbits. While each agency or company models its own satellite-sensor systems, few publicly available sources exist for mapping future satellite ground tracks. Evaluating collection opportunities from multiple satellite-sensor systems from different agencies/companies is problematic. The purpose of the research described in this article was to develop a generic approach for modeling future satellite-sensor collection opportunities. In this article, formulae are developed for computing the potential swath coverage, and an algorithm is designed for constructing the potential swath coverage area. The solution to the swath coverage problem is based on spherical trigonometry, a well known map projection (i.e., azimuthal equidistant map projection) used in an unconventional dynamic form, and a satellite orbital propagation model. We demonstrate how the computation of the swath coverage area can be accomplished using a temporal series of re-centered map projections.

Map Projections Minimizing Distance Errors

Maps convey important information about distances between pairs of points. It is therefore desirable to minimize the errors made in representing distances between pairs of points on maps. Since it is just as bad to have two points on the map at twice their proper separation as to have them at half their proper separation, it is the root-mean-square (rms) logarithmic distance between random points in the mapped region that we will minimize. The best previously known projection of the entire sphere for distances is the Lambert equal-area azimuthal, with an rms logarithmic distance error of σ = 0.343. By way of comparison, the Mercator projection has σ = 0.444 and the Mollweide, σ = 0.390. We present three new projections – the Gott equal-area elliptical, with perfect shapes on the central meridian; the Gott-Mugnolo equal-area elliptical; and the Gott-Mugnolo azimuthal, with rms logarithmic distance errors of σ = 0.365, σ = 0.348, and σ = 0.341 respectively – that improve on previous projections of their type. The Gott-Mugnolo azimuthal projection has the lowest distance errors of any map and is produced by a new technique using “forces” between pairs of points on a map, which make the points move so as to minimize σ. The Gott equal-area elliptical projection produces a particularly attractive map of Mars, and the Gott-Mugnolo azimuthal projection produces an interesting map of the Moon, both of which we also show.

STAR results on high transverse momentum, heavy flavour and electromagnetic probes

We summarize here recent results from the STAR collaboration focusing on processes involving large momentum transfers. Measurements of angular correlations of di-hadrons are explored in both the pseudorapidity (η) and azimuthal (ϕ) projections. In central Au+Au, an elongated structure is found in the η projection which persists up to the highest measured p⊥. After quantifying the particle yield in this structure and subtracting it from the near-side yield, we observe that the remainder exhibits a behaviour strikingly similar to that of the near-side yield in d+Au. For heavy flavour production, using electron–hadron correlations in p + p collisions, we obtain an estimate of the b-quark contribution to the non-photonic electrons in the p⊥ region 3–6 GeV/c, and find it consistent with FONLL calculations. Together with the observed suppression of non-photonic electrons in Au+Au, this strongly suggests suppression of b-quark production in Au+Au collisions. We discuss results on the mid-rapidity ϒ cross-section in p + p collisions. Finally, we present a proof-of-principle measurement of photon–hadron (γ–h) correlations in p + p collisions, paving the way for the tomographic study of the matter produced in central Au+Au via γ-jet measurements.