Geometric Triangulation Research Articles

Simulating White-Light Images of Coronal Structures for Parker Solar Probe/WISPR: Study of the Total Brightness Profiles

The Wide-field Imager for Parker Solar Probe (WISPR) captures unprecedented white-light images of the solar corona and inner heliosphere. Thanks to the uniqueness of the Parker Solar Probe’s (PSP) orbit, WISPR is able to image “locally” coronal structures at high spatial and time resolutions. The observed plane of sky, however, rapidly changes because of the PSP’s high orbital speed. Therefore, the interpretation of the dynamics of the coronal structures recorded by WISPR is not straightforward. A first study, undertaken by Liewer et al. (Solar Phys.294, 93, 2019), shows how different coronal features (e.g., streamers, flux ropes) appear in the field-of-view of WISPR by means of raytracing simulations. In particular, they analyze the effects of the spatial resolution changes on both the images and the associated height–time maps, and introduce the fundamentals for geometric triangulation. In this follow-up paper, we focus on the study of the total brightness of a simple, spherical, plasma density structure, to understand how the analysis of Thomson-scattered emission by the electrons in a coronal feature can shed light into the determination of its kinematic properties. We investigate two cases: (i) a density sphere at a constant distance from the Sun for different heliographic longitudes; (ii) a density sphere moving outwardly with constant speed. The study allows us to characterize the effects of the varying heliocentric distance of the observer and scattering angle on the total brightness observed, which we exploit to contribute to a better determination of the position and speed of the coronal features observed by WISPR.

Face-to-Face Multi-Point Constraint Method in 3D problems by Numerical Integration Based on Geometrical Triangulation

Face-to-Face Multi-Point Constraint Method in 3D problems by Numerical Integration Based on Geometrical Triangulation

Indoor Multi-Camera-Based Testbed for 3-D Tracking and Control of UAVs

Flight testbeds with multiple unmanned aerial vehicles (UAVs) are especially important to support research on multi-vehicle-related algorithms. The existing platforms usually lack a generic and complete solution allowing for software and hardware design. For such a purpose, this paper presents the design and implementation of a comprehensive multi-camera-based testbed for 3-D tracking and control of UAVs. First, the testbed software consists of a multi-camera system and a ground control system, which performs image processing, camera calibration, 3-D reconstruction, pose estimation, and motion control. In the multi-camera system, the positions and orientations of UAVs are first reconstructed by using epipolar geometric constraints and triangulation methods and then filtered by an extended Kalman filter (EKF). In the ground control system, a classical proportional–derivative (PD) controller is designed to receive the navigation data from the multi-camera system and then generates control commands to the target vehicles. Then, the testbed hardware employs smart cameras with field-programmable gate array (FPGA) modules to allow for real-time computation at a frame rate of 100 Hz. Lightweight quadcopter Parrot Bebop drones are chosen as the target UAVs, which does not require any modification to the hardware. Artificial infrared reflective markers are asymmetrically mounted on target vehicles and observed by multiple infrared cameras located around the flight region. Finally, extensive experiments are performed to demonstrate that the proposed testbed is a comprehensive and complete platform with good scalability applicable for research on a variety of advanced guidance, navigation, and control algorithms.

Geometric triangulations and flips

We prove that for a given flat surface with conical singularities, any pair of geometric triangulations can be connected by a chain of flips.

Robust phase unwrapping by probabilistic consensus

Structured light scanning works by projecting over the scene a supplement of controlled information: the captured signal is processed to provide a unique label (namely a code) for each observed point, and then proceed to geometrical triangulation. In phase shift profilometry sinusoidal patterns are projected and each point is labelled according to the observed phase. Then, due to the periodic nature of the signal, a disambiguation method (known as phase unwrapping) is needed. Several unwrapping techniques have been proposed in the literature, since noisy signals lead to inaccuracies in phase estimation.This paper presents a novel phase unwrapping approach based on a probabilistic framework. The method involves the projection of multiple sinusoidal patterns with distinct period lengths, encoding different phase values at each point location. Phase values are then modelled as samples from a Wrapped Gaussian distribution with an unknown mean, determined by the projector code that generated the values. This formulation allows us to robustly perform phase unwrapping via Maximum Likelihood Estimation, recovering code values from the observed phases. Furthermore, the same likelihood function can be exploited to identify and correct faulty unwrappings by gauging mutual support in a spatial neighbourhood. An extensive experimental assessment validates the Gaussian distribution hypothesis and verifies the improvements in coding accuracy when compared to other classical unwrapping techniques.

Geometry of biperiodic alternating links

A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi-regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a corresponding Euclidean tiling. Consequently, we determine the exact volumes of semi-regular links. We relate their commensurability and arithmeticity to the corresponding tiling, and assuming a conjecture of Milnor, we show there exist infinitely many pairwise incommensurable semi-regular links with the same invariant trace field. We show that only two semi-regular links have totally geodesic checkerboard surfaces; these two links satisfy the Volume Density Conjecture. Finally, we give conditions implying that many additional biperiodic alternating links are hyperbolic and admit a positively oriented, unimodular geometric triangulation. We also provide sharp upper and lower volume bounds for these links.

Monotone Paths in Geometric Triangulations

(I) We prove that the (maximum) number of monotone paths in a triangulation of n points in the plane is \(O(1.8027^n)\). This improves an earlier upper bound of \(O(1.8393^n)\); the current best lower bound is \(\varOmega (1.7034^n)\). (II) Given a planar straight-line graph G with n vertices, we show that the number of monotone paths in G can be computed in \(O(n^2)\) time.

Depth Estimation Based on Pyramid Normalized Cross-Correlation Algorithm for Vergence Control

A depth estimation algorithm based on vergence vision using a mechanical joint attached to two cameras is proposed. A Gaussian pyramid template-matching approach is used to align the view of the slave camera to the fixation point of the master camera. The master camera uses an object detection algorithm to find the target’s centroid and centers it relative to the image coordinates. Then, the vergence movement of the slave camera is performed using a pyramid normalized cross-correlation algorithm. Simple geometric triangulation is employed to compute the depth of that target. This proposed method was implemented using an active binocular vision platform with five degrees of freedom where four degrees of freedom to control the pan and tilt independently, and one degree of freedom to control the baseline, which is the distance between the camera. This system was designed for implementation in agriculture harvesting applications. The Analysis of field trial results indicates a worst-case precision of a target tomatoes’ depth to be ±1.32 cm at a depth of 85 cm.

The cusped hyperbolic census is complete

From its creation in 1989 through subsequent extensions, the widely-used SnapPea now aims to represent all cusped finite-volume hyperbolic 3-manifolds that can be obtained from <= 8 ideal tetrahedra. Its construction, however, has relied on inexact computations and some unproven (though reasonable) assumptions, and so its completeness was never guaranteed. For the first time, we prove here that the census meets its aim: we rigorously certify that every ideal 3-manifold triangulation with <= 8 tetrahedra is either (i) homeomorphic to one of the census manifolds, or (ii) non-hyperbolic. In addition, we extend the census to 9 tetrahedra, and likewise prove this to be complete. We also present the first list of all minimal triangulations of all census manifolds, including non-geometric as well as geometric triangulations.

The Eye and Refractive Geography in Pericles

AbstractThe paper highlights the cultural constructedness of vision in the early modern period by drawing on heteroglossic representations of the eye in early English texts, ranging from anatomy and physiology treatises to philosophy, poetry, emblems, and geometrical perspective in astronomy and land surveying. The argument is based on the association of word and image in early modern representations of space, mirrored in Ortelius’s notion of geography as the eye of history, which shows the importance of the visual element in the system of acquisition and transmission of knowledge in the Renaissance. In the particular case ofPericles, the play unfolds over a vast international geography and creates powerful visual effects. The imaginative spatial conventions of the play can be assimilated to the system of geometrical projection on which maps depended. Locations are used according to a geometric triangulation system to refract the imaginative and spatial vision. As in emblems, the locations unfolding in the play give the action meaning in the process of involved spectatorship. Moreover, in the theatre, the lone monocular beholder of mathematical linear perspective is multiplied into a choric array of spectators.

Convex Polygons in Geometric Triangulations

We show that the maximum number of convex polygons in a triangulation ofnpoints in the plane isO(1.5029n). This improves an earlier bound ofO(1.6181n) established by van Kreveld, Löffler and Pach (2012), and almost matches the current best lower bound of Ω(1.5028n) due to the same authors. Given a planar straight-line graphGwithnvertices, we also show how to compute efficiently the number of convex polygons inG.

The Voronoi functional is maximized by the Delaunay triangulation in the plane

We introduce the Voronoi functional of a triangulation of a finite set of points in the Euclidean plane and prove that among all geometric triangulations of the point set, the Delaunay triangulation maximizes the functional. This result neither extends to topological triangulations in the plane nor to geometric triangulations in three and higher dimensions.

COMPARISON OF CME/SHOCK PROPAGATION MODELS WITH HELIOSPHERIC IMAGING AND IN SITU OBSERVATIONS

ABSTRACT The prediction of the arrival time for fast coronal mass ejections (CMEs) and their associated shocks is highly desirable in space weather studies. In this paper, we use two shock propagation models, i.e., Data Guided Shock Time Of Arrival (DGSTOA) and Data Guided Shock Propagation Model (DGSPM), to predict the kinematical evolution of interplanetary shocks associated with fast CMEs. DGSTOA is based on the similarity theory of shock waves in the solar wind reference frame, and DGSPM is based on the non-similarity theory in the stationary reference frame. The inputs are the kinematics of the CME front at the maximum speed moment obtained from the geometric triangulation method applied to STEREO imaging observations together with the Harmonic Mean approximation. The outputs provide the subsequent propagation of the associated shock. We apply these models to the CMEs on 2012 January 19, January 23, and March 7. We find that the shock models predict reasonably well the shock’s propagation after the impulsive acceleration. The shock’s arrival time and local propagation speed at Earth predicted by these models are consistent with in situ measurements of WIND. We also employ the Drag-Based Model (DBM) as a comparison, and find that it predicts a steeper deceleration than the shock models after the rapid deceleration phase. The predictions of DBM at 1 au agree with the following ICME or sheath structure, not the preceding shock. These results demonstrate the applicability of the shock models used here for future arrival time prediction of interplanetary shocks associated with fast CMEs.

Constructing infinitely many geometric triangulations of the figure eight knot complement

This paper considers “geometric” ideal triangulations of cusped hyperbolic 3-manifolds, i.e. decompositions into positive volume ideal hyperbolic tetrahedra. We exhibit infinitely many geometric ideal triangulations of the figure eight knot complement. As far as we know, this is the first construction of infinitely many geometric triangulations of a cusped hyperbolic 3-manifold. In contrast, our approach does not extend to the figure eight sister manifold, and it is unknown if there are infinitely many geometric triangulations for this manifold.

Are 3‐D coronal mass ejection parameters from single‐view observations consistent with multiview ones?

AbstractTo prepare for when only single‐view observations are available, we have made a test whether the 3‐D parameters (radial velocity, angular width, and source location) of halo coronal mass ejections (HCMEs) from single‐view observations are consistent with those from multiview observations. For this test, we select 44 HCMEs from December 2010 to June 2011 with the following conditions: partial and full HCMEs by SOHO and limb CMEs by twin STEREO spacecraft when they were approximately in quadrature. In this study, we compare the 3‐D parameters of the HCMEs from three different methods: (1) a geometrical triangulation method, the STEREO CAT tool developed by NASA/CCMC, for multiview observations using STEREO/SECCHI and SOHO/LASCO data, (2) the graduated cylindrical shell (GCS) flux rope model for multiview observations using STEREO/SECCHI data, and (3) an ice cream cone model for single‐view observations using SOHO/LASCO data. We find that the radial velocities and the source locations of the HCMEs from three methods are well consistent with one another with high correlation coefficients (≥0.9). However, the angular widths by the ice cream cone model are noticeably underestimated for broad CMEs larger than 100° and several partial HCMEs. A comparison between the 3‐D CME parameters directly measured from twin STEREO spacecraft and the above 3‐D parameters shows that the parameters from multiview are more consistent with the STEREO measurements than those from single view.

Rigidity of infinite hexagonal triangulation of the plane

In the paper, we consider the rigidity problem of the infinite hexagonal triangulation of the plane under the piecewise linear conformal changes introduced by Luo in [5]. Our result shows that if a geometric hexagonal triangulation of the plane is PL conformal to the regular hexagonal triangulation and all inner angles are in $[\delta, \pi/2 -\delta]$ for any constant $\delta > 0$, then it is the regular hexagonal triangulation. This partially solves a conjecture of Luo [4]. The proof uses the concept of \emph{quasi-harmonic} functions to unfold the properties of the mesh.

Generic design methodology for the development of three-dimensional structured-light sensory systems for measuring complex objects

Structured-light (SL) techniques are emerging as popular noncontact approaches for obtaining three-dimensional (3-D) measurements of complex objects for real-time applications in manufacturing, bioengineering, and robotics. The performance of SL systems is determined by the emitting (i.e., projector) and capturing (i.e., camera) hardware components and the triangulation configuration between them and an object of interest. A generic design methodology is presented to determine optimal triangulation configurations for SL systems. These optimal configurations are determined with respect to a set of performance metrics: (1) minimizing the 3-D reconstruction errors, (2) maximizing the pixel-to-pixel correspondence between the projector and camera, and (3) maximizing the dispersion of the measured 3-D points within a measurement volume, while satisfying design constraints based on hardware and user-defined specifications. The proposed methodology utilizes a 3-D geometric triangulation model based on ray-tracing geometry and pin-hole models for the projector and camera. Using the methodology, a set of optimal system configurations can be determined for a given set of hardware components. The design methodology was applied to a real-time SL system for surface profiling of complex objects. Experiments were conducted with an optimal sensor configuration and its performance verified with respect to a nonoptimal hardware configuration.

ESTIMATING THE ARRIVAL TIME OF EARTH-DIRECTED CORONAL MASS EJECTIONS AT IN SITU SPACECRAFT USING COR AND HI OBSERVATIONS FROMSTEREO

The prediction of the arrival time and transit speed of CMEs near the Earth is one of the key problems in understanding the solar terrestrial relationship. Although, STEREO observations now provide a multiple view of CMEs in the heliosphere, the true speeds derived from stereoscopic reconstruction of SECCHI coronagraph data are not quite sufficient in accurate forecasting of arrival time of a majority of CMEs at the Earth. This is due to many factors which change the CME kinematics, like interaction of two or more CMEs or the interaction of CMEs with the pervading solar wind. In order to understand the propagation of CMEs, we have used the 3D triangulation method on SECCHI coronagraph (COR2) images, and geometric triangulation on the J-maps constructed from Heliospheric Imagers HI1 and HI2 images for eight Earth-directed CMEs observed during 2008-2010. Based on the reconstruction, and implementing the Drag Based Model for the distance where the CMEs could not be tracked unambiguously in the interplanetary medium, the arrival time of these CMEs have been estimated. These arrival times have also been compared with the actual arrival time as observed by \textit{in-situ} instruments. The analysis reveals the importance of heliospheric imaging for improved forecasting of the arrival time and direction of propagation of CMEs in the interplanetary medium.

Three-dimensional point-cloud room model in room acoustics simulations

Telepresence applications require communication with the feeling of being together and sharing the same environment. One important task in these applications is to render the acoustics of the distant room for the telepresence system user. This paper presents a fast method for the room geometry acquisition and its representation with a 3D point-cloud model, as well as utilization of such a model for the room acoustics simulations. A room is scanned with a commercially available input device (Kinect for Xbox360) in two different ways; the first one involves the device placed in the middle of the room and rotated around the vertical axis while for the second one the device is moved within the room. Benefits of both approaches were analyzed. The device's depth sensor provides a set of points in a three-dimensional coordinate system which represents scanned surfaces of the room interior. These data are used to build a 3D point-cloud model of the room. Several models are created to meet requirements of different room acoustics simulation algorithms: plane fitting and uniform voxel grid for geometric methods and triangulation mesh for the numerical methods. Advantages of the proposed method over the traditional approaches are discussed.

WHITE-LIGHT OBSERVATIONS OF SOLAR WIND TRANSIENTS AND COMPARISON WITH AUXILIARY DATA SETS

This paper presents results utilizing a new data processing pipeline for STEREO/SECCHI. The pipeline is used to identify and track 24 large- and small-scale solar wind transients from the Sun out to 1 AU. This comparison was performed during a few weeks around the minimum at the end of Solar Cycle 23 and the start of Cycle 24 (2008 December to 2009 January). We use coronagraph data to identify features near the Sun, track them through HI-2A, and identify their signatures with in situ data at the Earth and STEREO-B. We provide measurements and preliminary analysis of the in situ signatures of these features near 1 AU. Along with the demonstration of the utility of heliospheric imagers for tracking even small-scale structures, we identify and discuss an important limitation in using geometric triangulation for determining three-dimensional properties.