Ray Geometry Research Articles

Geometry of fast magnetosonic rays, wavefronts and shock waves

Fast magnetosonic waves in a two-dimensional plasma are studied in the geometrical optics approximation. The geometry of rays and wavefronts influences decisively the formation and ulterior evolution of shock waves. It is shown that the curvature of the curve where rays start and the angle between rays and wavefronts are the main parameters governing a wide variety of possible outcomes.

Geometry and dynamics of fast magnetosonic wavefronts near magnetic null points

The behavior of two-dimensional fast magnetosonic waves in the vicinity of isolated points where the magnetic field vanishes is studied analytically. The geometry of rays and wavefronts is described, and the curvature of both is found using conformal mapping techniques. These results are applied to the formation of shock waves, obtaining that shock formation is guaranteed at a finite time for any initial condition of the perturbation when the wavefront is concave and the rays tend to focus, whereas otherwise shocks occur only for a certain range of initial conditions.

Camera array calibration for light field acquisition

Light field cameras are becoming popular in computer vision and graphics, with many research and commercial applications already having been proposed. Various types of cameras have been developed with the camera array being one of the ways of acquiring a 4D light field image using multiple cameras. Camera calibration is essential, since each application requires the correct projection and ray geometry of the light field. The calibrated parameters are used in the light field image rectified from the images captured by multiple cameras. Various camera calibration approaches have been proposed for a single camera, multiple cameras, and a moving camera. However, although these approaches can be applied to calibrating camera arrays, they are not effective in terms of accuracy and computational cost. Moreover, less attention has been paid to camera calibration of a light field camera. In this paper, we propose a calibration method for a camera array and a rectification method for generating a light field image from the captured images. We propose a two-step algorithm consisting of closed form initialization and nonlinear refinement, which extends Zhang's well-known method to the camera array. More importantly, we introduce a rigid camera constraint whereby the array of cameras is rigidly aligned in the camera array and utilize this constraint in our calibration. Using this constraint, we obtained much faster and more accurate calibration results in the experiments.

Time-domain Helmholtz-Kirchhoff integral for forward surface scattering in a refractive medium

Time-domain Helmholtz-Kirchhoff integral (H-K integral) for forward surface scattering is extended to a refractive medium. Ray theory is applied to obtain the Green’s function, while the normal derivative in the integral is evaluated analytically using a ray geometry along with the ray-based Green’s function. This approach allows for stationary phase approximation which reduces the surface integration to a line integration. For high-frequency signals with a narrow bandwidth, an asymptotic form of the time-domain H-K integral then can be derived using the Fourier transform. The pressure field scattered from a sinusoidal surface wave in a constant-gradient sound speed profile is evaluated and compared to the result based on a conventional ray model.

WIGWAM Reverberation Revisited

Operation WIGWAM was a test of a 30 kt nuclear depth charge conducted in deep water 500 miles southwest of San Diego on 14 May 1955. Its primary purpose was to determine the effectiveness of that device as an antisubmarine weapon. The acoustic pulse from the test, initially an intense shockwave, radiated throughout the North and South Pacific Oceans. Acoustic reflections from topographic features were recorded for several hours after the explosion by SOund Fixing And Ranging (SOFAR) hydrophones at Point Sur, California, and Kaneohe, Hawaii. Sheehy and Halley (1957) identified peaks of the recorded coda with reflections from specific topographic features at great distances (e.g., the Hawaiian Islands, French Polynesia, or Fiji). With modern data for seafloor topography and ocean sound speed, these coda were computed with surprising accuracy using simple geodesic rays reflected from islands and seamounts. The intensity variations of the coda are mostly determined by simple ray geometry, together with modest attenuation. Coda peaks are often obtained from rays arriving simultaneously from multiple, but disparate, topographic features. Online Material: Figures of computed and measured coda and associated geodesic paths.

SU-D-213-05: Design, Evaluation and First Applications of a Off-Site State-Of-The-Art 3D Dosimetry System

Purpose: To design, construct and commission a prototype in-house three dimensional (3D) dose verification system for stereotatic body radiotherapy (SBRT) verification at an off-site partner institution. To investigate the potential of this system to achieve sufficient performance (1mm resolution, 3% noise, within 3% of true dose reading) for SBRT verification. Methods: The system was designed utilizing a parallel ray geometry instigated by precision telecentric lenses and an LED 630nm light source. Using a radiochromic dosimeter, a 3D dosimetric comparison with our gold-standard system and treatment planning software (Eclipse) was done for a four-field box treatment, under gamma passing criteria of 3%/3mm/10% dose threshold. Post off-site installation, deviations in the system’s dose readout performance was assessed by rescanning the four-field box irradiated dosimeter and using line-profiles to compare on-site and off-site mean and noise levels in four distinct dose regions. As a final step, an end-to-end test of the system was completed at the off-site location, including CT-simulation, irradiation of the dosimeter and a 3D dosimetric comparison of the planned (Pinnacle{sup 3}) to delivered dose for a spinal SBRT treatment(12 Gy per fraction). Results: The noise level in the high and medium dose regions of the four field box treatment wasmore » relatively 5% pre and post installation. This reflects the reduction in positional uncertainty through the new design. This At 1mm dose voxels, the gamma pass rates(3%,3mm) for our in-house gold standard system and the off-site system were comparable at 95.8% and 93.2% respectively. Conclusion: This work will describe the end-to-end process and results of designing, installing, and commissioning a state-of-the-art 3D dosimetry system created for verification of advanced radiation treatments including spinal radiosurgery.« less

Singular value decomposition for the cone-beam transform in the ball

Abstract The X-ray computerized tomography is concerned to the reconstruction of a function f from line (ray) integrals of f. In general, we may have the parallel ray or divergent ray geometry. In this paper we develop an analytical singular value decomposition method for the cone-beam transform in divergent ray geometry. For the parallel ray transform a singular value decomposition has been derived by P. Maass in 1987.

How effective can optical-CT 3D dosimetry be without refractive fluid matching?

Achieving accurate optical CT 3D dosimetry without the use of viscous refractive index (RI) matching fluids would greatly increase convenience. Software has been developed to simulate optical CT 3D dosimetry for a range of scanning configurations including parallel-beam, point and converging light sources. For each configuration the efficacy of 3 refractive media were investigated: air, water, and a fluid closely matched to Presage (RI = 1.00, 1.33 and 1.49 respectively). The results revealed that the useable radius of the dosimeter (i.e. where data was within 2% of truth) reduced to 68% for water-matching, and 31% for dry-scanning in air. Point source incident ray geometry produced slightly more favourable results, although variation between the three geometries was relatively small. The required detector size however, increased by a factor six for dry-scanning, introducing cost penalties. For applications where dose information is not required in the periphery, some dry and low-viscous matching configurations may be feasible.

Ray geometry in non-pinhole cameras: a survey

A pinhole camera collects rays passing through a common 3D point and its image resembles what would be seen by human eyes. In contrast, a non-pinhole (multi-perspective) camera combines rays collected by different viewpoints. Despite their incongruity of view, their images are able to preserve spatial coherence and can depict, within a single context, details of a scene that are simultaneously inaccessible from a single view, yet easily interpretable by a viewer. In this paper, we thoroughly discuss the design, modeling, and implementation of a broad class of non-pinhole cameras and their applications in computer graphics and vision. These include mathematical (conceptual) camera models such as the General Linear Cameras and real non-pinhole cameras such as catadioptric cameras and projectors. A unique component of this paper is a ray geometry analysis that uniformly models these non-pinhole cameras as manifolds of rays and ray constraints. We also model the thin lens as a ray transform and study how ray geometry is changed by the thin lens for studying distortions and defocusing. We hope to provide mathematical fundamentals to satisfy computer vision researchers as well as tools and algorithms to aid computer graphics and optical engineering researchers.

Nelson Goodman’s Arguments Against Perspective: A Geometrical Analysis

Nelson Goodman’s highly influential book Languages of Art argued that that all visual experience is merely a cultural construct. In presenting his thesis Goodman relied on a series of geometrical arguments intended to show that the geometrical construction of perspective does not rely on the geometry of light rays. In this article I analyze the geometrical validity of Goodman’s arguments.

On the horoboundary and the geometry of rays of negatively curved manifolds

We study the Gromov compactification of quotients X= G of a Hadamard space X by a discrete group of isometries G, pointing out the main differences with the simply connected case. We prove a criterion for the Busemann equivalence of rays on these quotients and show that the “visual” description of the Gromov boundary breaks down, producing examples for the main pathologies that may occur in the nonsimply connected case, such as: divergent rays having the same Busemann functions, points on the Gromov boundary that are not Busemann functions of any ray, and discontinuity of the Busemann functions with respect to the initial conditions. Finally, for geometrically finite quotients X= G, we recover a simple description of the Gromov boundary, and prove that in this case the compactification is a singular manifold with boundary, with a finite number of conical singularities.

Quality factors of deformed dielectric cavities

An analysis is provided of the degradation that arises in the quality factor of a whispering gallery mode when a circular or spherical dielectric cavity is deformed. The large quality factors of such resonators are important to their use in applications such as sensors, wavelength filters or lasers. Yet a straightforward analysis of the effect of shape deformation on quality factors cannot be given because the underlying complex ray data demanded by a standard eikonal approximation frequently does not exist. In this paper we exploit an approach that has been successfully used elsewhere to describe the strong directional emission of such systems, based on a perturbative treatment of the relevant complex ray families. Applicable when the radial perturbation is formally of the order of a wavelength, the resulting approximation successfully describes changes to the quality factor using the ray geometry in a neighbourhood of a discrete set of escaping rays guiding the directions of maximum emitted intensity.

Prestack shot-gather depth migration by a rigid flow of Gaussian wave packets

We introduce a new method for prestack depth migration of seismic common-shot gathers. The computational procedure follows standard steps of the reverse-time migration, i.e., downward continuation of the source and the receiver wavefields, followed by application of an imaging condition (e.g. zero-lag cross-correlation of these fields). In our method we first find a sparse data representation with a small number of Gaussian wave packets. We then approximate the downward wavefield propagation (for the source and the receiver fields) by a rigid flow of these wave packets along seismic rays. In this case, the wave packets are simply translated and rotated according to the ray geometry. One advantage of using Gaussian wave packets is that analytic formulas can be used for translation, rotation, and the application of the cross-correlation imaging condition. Moreover, they allow more sparse representations than competing methods. Finally we formulate a computationally and memory efficient migration procedure, as only few rays have to be traced, and since it is cheap to compute the cross-correlation for the intersecting rays.

Characteristics of stereo images from detectors in focal plane array

The equivalent ray geometry of two horizontally aligned detectors at the focal plane of the main antenna in a millimeter wave imaging system is analyzed to reveal the reason why the images from the detectors are fused as an image with a depth sense. Scanning the main antenna in both horizontal and vertical directions makes each detector perform as a camera, and the two detectors can work like a stereo camera in the millimeter wave range. However, the stereo camera geometry is different from that of the stereo camera used in the visual spectral range because the detectors' viewing directions are diverging to each other and they are a certain distance apart. The depth sense is mainly induced by the distance between detectors. The images obtained from the detectors in the millimeter imaging system are perceived with a good depth sense. The disparities responsible for the depth sense are identified in the images.

The geometry of sound rays in a wind

We survey the close relationship between sound and light rays and geometry. In the case where the medium is at rest, the geometry is the classical geometry of Riemann. In the case where the medium is moving, the more general geometry known as Finsler geometry is needed. We develop these geometries ab initio, with examples, and in particular show how sound rays in a stratified atmosphere with a wind can be mapped to a problem of circles and straight lines.

Optical metrics and projective equivalence

Trajectories of light rays in a static spacetime are described by unparametrized geodesics of the Riemannian optical metric associated with the Lorentzian spacetime metric. We investigate the uniqueness of this structure and demonstrate that two different observers, moving relative to one another, who both see the Universe as static may determine the geometry of the light rays differently. More specifically, we classify Lorentzian metrics admitting more than one hyper-surface orthogonal timelike Killing vector and analyze the projective equivalence of the resulting optical metrics. These metrics are shown to be projectively equivalent up to diffeomorphism if the static Killing vectors generate a group $SL(2,\mathbb{R})$, but not projectively equivalent in general. We also consider the cosmological $C$ metrics in Einstein-Maxwell theory and demonstrate that optical metrics corresponding to different values of the cosmological constant are projectively equivalent.

Global-scalePwave tomography optimized for prediction of teleseismic and regional travel times for Middle East events: 1. Data set development

[1] We extend the Bayesloc seismic multiple-event location algorithm for application to global arrival time data sets. Bayesloc is a formulation of the joint probability distribution spanning multiple-event location parameters, including hypocenters, travel time corrections, pick precision, and phase labels. Stochastic priors may be used to constrain any of the Bayesloc parameters. Markov Chain Monte Carlo sampling is used to draw samples from the joint probability distribution, and the posteriori samples are summarized to infer conventional location parameters such as the hypocenter. The first application of the broad area Bayesloc algorithm is to a data set consisting of all well-recorded events in the Middle East and the most well-recorded events with 5° spatial sampling globally. This sampling strategy is designed to provide the ray coverage needed to determine lithospheric-scale P wave velocity structure in the Middle East using the complementary ray geometry provided by regional (subhorizontal) and teleseismic (subvertical) raypaths and to determine a consistent, albeit lower-resolution, image of global mantle structure. The data set consists of 5401 events and 878,535 P, Pn, pP, sP, and PcP arrivals recorded at 4606 stations. Relocated epicenters are an average of 16 km from bulletin locations. The data set included events that are known to an accuracy of 1 km (a.k.a. GT1) based on nonseismic information. The average distance between GT1 epicenters and our relocated epicenters is 5.6 km. For arrivals labeled P, Pn, and PcP, ∼92%, ∼90%, and 96% are properly labeled with probability >0.9, respectively. Incorrect phase labels are found to be erroneous at rates of 0.6%, 0.2%, 1.6%, and 2.5% for P, Pn, PcP, and depth phases (pP and sP), respectively. Labels found to be incorrect, but not erroneous, were reassigned to another phase label. P and Pn residual standard deviation with respect to ak135 travel times are dramatically reduced from 3.45 s to 1.01 s. The differences between travel time residuals for nearly reciprocal raypaths are significantly reduced from the input event locations, suggesting that Bayesloc relocation improves data set consistency. The reciprocity tests suggest that the dominant contribution to travel time residuals calculated from information provided in global bulletins is location and picks errors, not travel time prediction errors due to 3-D structure. Modeling the whole multiple-event system results in accurate locations and an internally consistent data set that is ideal for tomography and other travel time calibration studies. Simmons et al. (2011) (companion paper) use the Bayesloc-processed data set to develop a 3-D tomographic image, which further reduces residual standard deviation to 0.50 s.

Transparent and Specular Object Reconstruction

Abstract This state of the art report covers reconstruction methods for transparent and specular objects or phenomena. While the 3D acquisition of opaque surfaces with Lambertian reflectance is a well‐studied problem, transparent, refractive, specular and potentially dynamic scenes pose challenging problems for acquisition systems. This report reviews and categorizes the literature in this field. Despite tremendous interest in object digitization, the acquisition of digital models of transparent or specular objects is far from being a solved problem. On the other hand, real‐world data is in high demand for applications such as object modelling, preservation of historic artefacts and as input to data‐driven modelling techniques. With this report we aim at providing a reference for and an introduction to the field of transparent and specular object reconstruction. We describe acquisition approaches for different classes of objects. Transparent objects/phenomena that do not change the straight ray geometry can be found foremost in natural phenomena. Refraction effects are usually small and can be considered negligible for these objects. Phenomena as diverse as fire, smoke, and interstellar nebulae can be modelled using a straight ray model of image formation. Refractive and specular surfaces on the other hand change the straight rays into usually piecewise linear ray paths, adding additional complexity to the reconstruction problem. Translucent objects exhibit significant sub‐surface scattering effects rendering traditional acquisition approaches unstable. Different classes of techniques have been developed to deal with these problems and good reconstruction results can be achieved with current state‐of‐the‐art techniques. However, the approaches are still specialized and targeted at very specific object classes. We classify the existing literature and hope to provide an entry point to this exiting field.

From reflection elements to structure — A look at the history of data interpretation

Traditionally, input acquired in the field consisted of the original paper records; output submitted to the client consisted of structural sections and depth-contour maps of selected interfaces. Before the introduction of magnetic recording, it was common practice to do the conversion in the field office. Tools for this conversion ranged from slide rules and desk calculators to wavefront charts. These tools were based on the geometry of rays in media where velocity is a function of depth only. The detailed algorithms underlying the conversion were often developed in the exploration companies and — originally — were carefully guarded. But at least the underlying principles were exchanged throughout the industry through books, journal articles, and presentations at meetings, such as noted in nearly 300 references in C. H. Dix’s Seismic Prospecting for Oil (1952) . The techniques of data acquisition and data interpretation have changed considerably, but the underlying principles of ray geometry are the same. Therefore, many new methods are based on ideas formulated in the early times of the industry.

Seismic tomography with the reversible jump algorithm

SUMMARY The reversible jump algorithm is a statistical method for Bayesian inference with a variable number of unknowns. Here, we apply this method to the seismic tomography problem. The approach lets us consider the issue of model parametrization (i.e. the way of discretizing the velocity field) as part of the inversion process. The model is parametrized using Voronoi cells with mobile geometry and number. The size, position and shape of the cells defining the velocity model are directly determined by the data. The inverse problem is tackled within a Bayesian framework and explicit regularization of model parameters is not required. The mobile position and number of cells means that global damping procedures, controlled by an optimal regularization parameter, are avoided. Many velocity models with variable numbers of cells are generated via a transdimensional Markov chain and information is extracted from the ensemble as a whole. As an aid to interpretation we visualize the expected earth model that is obtained via Monte Carlo integration in a straightforward manner. The procedure is particularly adept at imaging rapid changes or discontinuities in wave speed. While each velocity model in the final ensemble consists of many discontinuities at cell boundaries, these are smoothed out in the averaged ensemble solution while those required by the data are reinforced. The ensemble of models can also be used to produce uncertainty estimates and experiments with synthetic data suggest that they represent actual uncertainty surprisingly well. We use the fast marching method in order to iteratively update the ray geometry and account for the non-linearity of the problem. The method is tested here with synthetic data in a 2-D application and compared with a subspace method that is a more standard matrix-based inversion scheme. Preliminary results illustrate the advantages of the reversible jump algorithm. A real data example is also shown where a tomographic image of Rayleigh wave group velocity for the Australian continent is constructed together with uncertainty estimates.