Conjugate Points Research Articles

Technique to model the movement of the scene using image sequence

The paper proposes a technique to model the movement of a scene based on backward stochastic gradient pixel-by-pixel estimation of the deformation field. Calculation of the moving object’s area is considered as the task of testing the hypothesis that the image grid nodes belong to the area of motion. The paper presents the estimated two kinds of errors: false positive and false negative. The obtained results are compared with the results of the MVFAST algorithm. The increase in the signal-to-noise ratio of the image of a moving object is achieved by combining the frames of the video sequence under study. To combine conjugate points, high-speed recurrent algorithms are used that do not require a priori information. The paper presents an example of estimation of object’s trajectory parameters using the technique. The interframe geometric deformations of the video sequence are used as intermediate parameters of the trajectory.

Convex and Starshaped Sets in Manifolds Without Conjugate Points

Let $\mathcal{W}^{n}$ be the class of $C^{\infty }$ complete simply connected $n-$dimensional manifolds without conjugate points. The hyperbolic space as well as Euclidean space are good examples of such manifolds. Let $% W\in \mathcal{W}^{n}$ and let $A$ be a subset of $W$. This article aims at characterization and building convex and starshaped sets in this class from inside. For example, it is proven that, for a compact starshaped set, the convex kernel is the intersection of stars of extreme points only. Also, if a closed unbounded convex set $A$ does not contain a totally geodesic hypersurface and its boundary has no geodesic ray, then $A$ is the convex hull of its extreme points. This result is a refinement of the well-known Karein-Millman theorem.

Improved Piecewise Linear Transformation for Precise Warping of Very-High-Resolution Remote Sensing Images

A large number of evenly distributed conjugate points (CPs) in entirely overlapping regions of the images are required to achieve successful co-registration between very-high-resolution (VHR) remote sensing images. The CPs are then used to construct a non-linear transformation model that locally warps a sensed image to a reference image’s coordinates. Piecewise linear (PL) transformation is largely exploited for warping VHR images because of its superior performance as compared to the other methods. The PL transformation constructs triangular regions on a sensed image from the CPs by applying the Delaunay algorithm, after which the corresponding triangular regions in a reference image are constructed using the same CPs on the image. Each corresponding region in the sensed image is then locally warped to the regions of the reference image through an affine transformation estimated from the CPs on the triangle vertices. The warping performance of the PL transformation shows reliable results, particularly in regions inside the triangles, i.e., within the convex hulls. However, the regions outside the triangles, which are warped when the extrapolated boundary planes are extended using CPs located close to the regions, incur severe geometric distortion. In this study, we propose an effective approach that focuses on the improvement of the warping performance of the PL transformation over the external area of the triangles. Accordingly, the proposed improved piecewise linear (IPL) transformation uses additional pseudo-CPs intentionally extracted from positions on the boundary of the sensed image. The corresponding pseudo-CPs on the reference image are determined by estimating the affine transformation from CPs located close to the pseudo-CPs. The latter are simultaneously used with the former to construct the triangular regions, which are enlarged accordingly. Experiments on both simulated and real datasets, constructed from Worldview-3 and Kompsat-3A satellite images, were conducted to validate the effectiveness of the proposed IPL transformation. That transformation was shown to outperform the existing linear/non-linear transformation models such as an affine, third and fourth polynomials, local weighted mean, and PL. Moreover, we demonstrated that the IPL transformation improved the warping performance over the PL transformation outside the triangular regions by increasing the correlation coefficient values from 0.259 to 0.304, 0.603 to 0.657, and 0.180 to 0.338 in the first, second, and third real datasets, respectively.

Horosphere slab separation theorems in manifolds without conjugate points

Let mathcal {W}^{n} be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let Win mathcal {W}^{n} and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in Win mathcal {W} are considered.

Cosmological singularities from high matter density without global topological assumptions

Cosmological singularity theorems such as that of Hawking and Penrose assume local curvature conditions as well as global ones like the existence of a compact (achronal) slice. Here, we prove a new singularity theorem for chronological spacetimes that satisfy what we call a ‘past null focusing’ condition. Such a condition forces all null geodesics gamma :[0,a)rightarrow M with future endpoint gamma (0) to develop a pair of conjugate points if past complete. By the Einstein field equations, such a condition will be satisfied if the density of matter fields remains sufficiently high towards the past of the spacetime, as may be expected in certain cosmological scenarios. The theorem obtained doesn’t make starting assumptions about the spacetime’s topology, such as the existence of a compact achronal slice, and if in addition to a ‘past null focusing’ condition we assume the timelike convergence condition, then further consequences pertaining to the existence of CMC foliations and the character of the singularity are obtained. With the addition of the timelike convergence condition, we obtain the conclusion that all timelike geodesics are past incomplete, rather than the existence of a single incomplete non-spacelike geodesic.

Sobolev H1 geometry of the symplectomorphism group

For a closed symplectic manifold (M,ω) with compatible Riemannian metric g we study the Sobolev H1 geometry of the group of all Hs diffeomorphisms on M which preserve the symplectic structure. We show that, for sufficiently large s, the H1 metric admits globally defined geodesics and the corresponding exponential map is a non-linear Fredholm map of index zero. Finally, we show that the H1 metric carries conjugate points via some simple examples.

On the Group of Almost Periodic Diffeomorphisms and its Exponential Map

Abstract We define the group of almost periodic diffeomorphisms on $\mathbb{R}^n$ and on an arbitrary Lie group. We then study the properties of its Riemannian and Lie group exponential maps and provide applications to fluid equations. In particular, we show that there exists a geodesic of a weak Riemannian metric on the group of almost periodic diffeomorphisms of the line that consists entirely of conjugate points.

The integrability of symplectic twist maps without conjugate points

It is proved that a symplectic twist map of the cotangent bundle $T^{\ast }\mathbb{T}^{d}$ of the $d$-dimensional torus that is without conjugate points is $C^{0}$-integrable, that is $T^{\ast }\mathbb{T}^{d}$ is foliated by a family of invariant $C^{0}$ Lagrangian graphs.

Riemannian rigidity of the parallel postulate in total curvature

Without any symmetry assumptions and under natural integrability conditions on the Gaussian curvature, we show that the Euclidean plane is unique among all the complete 2-dimensional Riemannian manifolds that satisfy the Euclid’s fifth postulate. Namely, we prove that the Euclidean plane is the only Riemannian surface free of conjugate points and admitting total curvature that satisfies Playfair’s version of the parallel postulate.

Excessive backlog probabilities of two parallel queues

Let X be the constrained random walk on $${\mathbb Z}_+^2$$ with increments (1, 0), $$(-1,0)$$ , (0, 1) and $$(0,-1)$$ ; X represents, at arrivals and service completions, the lengths of two queues (or two stacks in computer science applications) working in parallel whose service and interarrival times are exponentially distributed with arrival rates $$\lambda _i$$ and service rates $$\mu _i$$ , $$i=1,2$$ ; we assume $$\lambda _i < \mu _i$$ , $$i=1,2$$ , i.e., X is assumed stable. Without loss of generality we assume $$\rho _1 =\lambda _1/\mu _1 \geqslant \rho _2 = \lambda _2/\mu _2$$ . Let $$\tau _n$$ be the first time X hits the line $$\partial A_n = \{x \in {\mathbb Z}^2:x(1)+x(2) = n \}$$ , i.e., when the sum of the components of X equals n for the first time. Let Y be the same random walk as X but only constrained on $$\{y \in {{\mathbb {Z}}}^2: y(2)=0\}$$ and its jump probabilities for the first component reversed. Let $$\partial B =\{y \in {{\mathbb {Z}}}^2: y(1) = y(2) \}$$ and let $$\tau $$ be the first time Y hits $$\partial B$$ . The probability $$p_n = P_x(\tau _n < \tau _0)$$ is a key performance measure of the queueing system (or the two stacks) represented by X (if the queues/stacks share a common buffer, then $$p_n$$ is the probability that this buffer overflows during the system’s first busy cycle). Stability of the process implies that $$p_n$$ decays exponentially in n when the process starts off the exit boundary $$\partial A_n.$$ We show that, for $$x_n= \lfloor nx \rfloor $$ , $$x \in {{\mathbb {R}}}_+^2$$ , $$x(1)+x(2) \leqslant 1$$ , $$x(1) > 0$$ , $$P_{(n-x_n(1),x_n(2))}( \tau < \infty )$$ approximates $$P_{x_n}(\tau _n < \tau _0)$$ with exponentially vanishing relative error. Let $$r = (\lambda _1 + \lambda _2)/(\mu _1 + \mu _2)$$ ; for $$r^2 < \rho _2$$ and $$\rho _1 \ne \rho _2$$ , we construct a class of harmonic functions from single and conjugate points on a related characteristic surface for Y with which the probability $$P_y(\tau < \infty )$$ can be approximated with bounded relative error. For $$r^2 = \rho _1 \rho _2$$ , we obtain the exact formula $$P_y(\tau < \infty ) = r^{y(1)-y(2)} +\frac{r(1-r)}{r-\rho _2}\left( \rho _1^{y(1)} - r^{y(1)-y(2)} \rho _1^{y(2)}\right) .$$

The Source Regions of Whistlers

AbstractWe present a new method for identifying the source regions of lightning‐generated whistlers observed at a fixed location. In addition to the spatial distribution of causative lightning discharges, we calculate the ratio of lightning discharges transmitted into ground detectable whistlers as a function of location. Our method relies on the time of the whistlers and the time and source location of spherics from a global lightning database. We apply this method to whistlers recorded at 15 ground‐based stations in the Automatic Whistler Detector and Analyzer Network operating between 2007 and 2018 and to located lightning strokes from the World Wide Lightning Location Network database. We present the obtained maps of causative lightning and transmission rates. Our results show that the source region of whistlers corresponding to each ground station is around the magnetic conjugate point of the respective station. The size of the source region is typically less than 2,000 km in radius with a small fraction of sources extending to up to 3,500 km. The transmission ratio is maximal at the conjugate point and decreases with increasing distance from it. This conforms to the theory that whistlers detected on the ground propagated in a ducted mode through the plasmasphere, and thus, the lightning strokes of their causative spherics must cluster around the footprint of the ducts in the other hemisphere. Our method applied resolves the whistler excitation region mystery that resulted from correlation‐based analysis methods, concerning the source region of whistlers detected in Dunedin, New Zealand.

Indications of Ground-based Electromagnetic Observations to A Possible Lithosphere–Atmosphere–Ionosphere Electromagnetic Coupling before the 12 May 2008 Wenchuan MS 8.0 Earthquake

A large number of various precursors have been reported since the Wenchuan MS 8.0 earthquake (EQ) took place on 12 May 2008 in China. In this work, previous investigations of both ground-based electromagnetic (EM) parameters and spatial ionospheric parameters were first examined. The statistical results showed that various anomalies presented different time-scale variations but tended to be characterized by a common feature – reaching their climax on 9 May, three days before the Wenchuan event, which indicates a lithosphere–atmosphere–ionosphere (LAI) electromagnetic coupling. Second, the fluctuations on 9 May based on the observational ground-based ultra low frequency (ULF) electrical field at the Gaobeidian (GBD) station and the direct current/ultra low frequency (DC–ULF) geomagnetic vertical Z field at the Chengdu (CD) station were comparably analyzed with those of ionospheric disturbances reported previously. The results showed that distinct electromagnetic changes, geomagnetic “double low-point” phenomena, and ionospheric disturbances above both sides of the Earth started in turn, respectively, but reached their climax simultaneously within dozens of hours on 9 May. This evolutionary process increases the probability that electromagnetic energy propagates from the epicentral area, via the atmosphere and ionosphere, to the equatorial plane, and through this plane finally to its magnetically conjugated area in the opposite hemisphere, causing electromagnetic disturbances on the Earth’s surface, in the atmosphere, and in the ionosphere and its conjugate point, in that order.

Some Classes of Harmonic Mapping with a Symmetric Conjecture Point Defined by Subordination

In the paper, we introduce some subclasses of harmonic mapping, the analytic part of which is related to general starlike (or convex) functions with a symmetric conjecture point defined by subordination. Using the conditions satisfied by the analytic part, we obtain the integral expressions, the coefficient estimates, distortion estimates and the growth estimates of the co-analytic part g, and Jacobian estimates, the growth estimates and covering theorem of the harmonic function f. Through the above research, the geometric properties of the classes are obtained. In particular, we draw figures of extremum functions to better reflect the geometric properties of the classes. For the first time, we introduce and obtain the properties of harmonic univalent functions with respect to symmetric conjugate points. The conclusion of this paper extends the original research.

Stability and local minimality of spherical harmonic twists u={mathbf {Q}}(|x|) x|x|^{-1}, positivity of second variations and conjugate points on mathbf{SO}(n)

In this paper we discuss the stability and local minimising properties of spherical twists that arise as solutions to the harmonic map equation HME[u;Xn,Sn-1]:=Δu+|∇u|2u=0inXn,|u|=1inXn,u=φon∂Xn,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\mathbf{HME}[u;\\, \\mathbb {X}^n, \\mathbb {S}^{n-1}] :=\\left\\{ \\begin{array}{ll} \\Delta u + |\\nabla u|^2 u =0 &{} \\qquad \\text { in } \\mathbb {X}^n, \\\\ |u|=1 &{}\\qquad \\text { in } \\mathbb {X}^n , \\\\ u = \\varphi &{}\\qquad \\text { on } \\partial \\mathbb {X}^n, \\end{array}\\right. \\end{aligned}$$\\end{document}by way of examining the positivityof the second variation of the associated Dirichlet energy. Here, following [31], by a spherical twist we mean a map u in mathscr {W}^{1,2}(mathbb {X}^n, mathbb {S}^{n-1}) of the form x mapsto {mathbf {Q}}(|x|)x|x|^{-1} where {mathbf {Q}}={mathbf {Q}}(r) lies in mathscr {C}([a, b], mathbf{SO}(n)) and {mathbb {X}}^n={x in mathbb {R}^n : a<|x|<b} (nge 2). It is shown that subject to a structural condition on the twist path the energy at the associated spherical twist solution to the system has a positive definite second variation and subsequently proven to furnish a strong local energy minimiser.A detailed study of Jacobi fields and conjugate points along the twist path {mathbf {Q}}(r)=mathrm{exp}(mathscr {G}(r) {mathbf {H}}) and geodesics on mathbf{SO}(n) is undertaken and its remarkable implication and interplay on the minimalityof spherical harmonic twists exploited.

On total flexibility of local structures of Finsler tori without conjugate points

We show that given a point on a Finsler surface, one can always find a neighborhood of the point and isometrically embed this neighborhood into a Finsler torus without conjugate points.

Geomagnetic conjugate observations of ionospheric disturbances in response to a North Korean underground nuclear explosion on 3 September 2017

Abstract. We report observations of ionospheric disturbances in response to a North Korean underground nuclear explosion (UNE) on 3 September 2017. By using data from IGS (International GNSS Service) stations and Swarm satellites, geomagnetic conjugate ionospheric disturbances were observed. The observational evidence showed that UNE-generated ionospheric disturbances propagated radially from the UNE epicenter with a velocity of ∼280 m s−1. We propose that the ionospheric disturbances are results of electrodynamic process caused by LAIC (lithosphere–atmosphere–ionosphere coupling) electric field penetration. The LAIC electric field can also be mapped to the conjugate hemispheres along the magnetic field line and consequently cause ionospheric disturbances in conjugate regions. The UNE-generated LAIC electric field penetration plays an important role in the ionospheric disturbances in the region of the nuclear test site nearby and the corresponding geomagnetic conjugate points.

Memory-Efficient Implementation of Elliptic Curve Cryptography for the Internet-of-Things

In this paper, we present memory-efficient and scalable implementations of NIST standardized elliptic curves P-256, P-384 and P-521 on three ARMv6-M processors (i.e. Cortex-M0, M0+, and M1). Specifically, we propose a refined approach to perform the Multiply-ACcumulate (MAC) operation using hardware multiplier provided by ARMv6-M processor, and a compact doubling routine for multi-precision squaring that executes both doubling and partial product operations in an efficient way. We demonstrate that the proposed squaring implementation achieves a speed up of 28 percent compared to the same operation employed in Micro-ECC. Then, we reduce one modular reduction in co-Z conjugate point addition by using lazy reduction and special form representation (CD-AB, EF-AB), which further reduces the execution time of both P-256 and P-384 implementations. Finally, we propose scalable implementations of ECC scalar multiplication on ARMv6-M processors that are widely used for Internet of Things applications.

Accuracy Improvement of Water Level Measuring Method Using Stereo CCTV System

최근 농업용 중소규모 저수지는 노후화 및 안전성 저하 등으로 국지성 호우 발생 시 붕괴사고와 같은 문제가 발생되고 있다. 본 연구는 입체형 CCTV를 이용하여 입체영상을 획득하고, 자동으로 수위 변화를 측정할 수 있는 시스템을 개발하여 재해에 취약한 중소저수지를 효율적으로 관리하고자 한다. 이를 위하여 시스템 개발은 UMI(unified modeling language)를 이용하여 시스템을 개발하였고, 시스템 구현은 CCTV 검정, 취득한 영상의 특징점 및 공액점을 독취 후 에피폴라 기하관계 구현, 표준 상호 상관성(nomalized cross-correlation) 알고리즘을 이용한 영상정합 등을 통하여 시차맵를 제작하여 수위 변화를 분석하였다. 그 결과 본 연구에서 개발한 입체형 CCTV 자동수위측정시스템의 정확도가 기존 시스템에 비하여 좋은 결과를 얻을 수 있었다.Recently, problems have arisen in small and medium-sized reservoirs used for agriculture, such as cave-in incidents during local heavy rainfalls, owing to its deterioration and safety degradation. In this study, we acquired stereo images using stereo closed-circuit television (CCTV) and developed an automatic system that can measure water level changes in order to efficiently manage small and medium-sized reservoirs that are vulnerable to disasters. The system was developed using the unified modeling language (UML) and implemented by testing the CCTV. Epipolar geometry was utilized after reading the key points and conjugate points of the acquired images. The water level changes were then analyzed by producing a disparity map through the image matching using a normalized cross-correlation algorithm. The stereo CCTV-based automatic water level measurement system developed in this study exhibited a high accuracy compared to the conventional system.

A support theorem for the X-ray transform on manifolds with plane covers

AbstractThis paper is concerned with support theorems of the X-ray transform on non-compact manifolds with conjugate points. In particular, we prove that all simply connected 2-step nilpotent Lie groups have a support theorem. Important ingredients of the proof are the concept of plane covers and a support theorem for simple manifolds by Krishnan. We also provide examples of non-homogeneous 3-dimensional simply connected manifolds with conjugate points which have support theorems.

Automated Coregistration of Multisensor Orthophotos Generated from Unmanned Aerial Vehicle Platforms

Image coregistration is a key preprocessing step to ensure the effective application of very-high-resolution (VHR) orthophotos generated from multisensor images acquired from unmanned aerial vehicle (UAV) platforms. The most accurate method to align an orthophoto is the installation of air-photo targets at a test site prior to flight image acquisition, and these targets were used as ground control points (GCPs) for georeferencing and georectification. However, there are time and cost limitations related to installing the targets and conducting field surveys on the targets during every flight. To address this problem, this paper presents an automated coregistration approach for orthophotos generated from VHR images acquired from multisensors mounted on UAV platforms. Spatial information from the orthophotos, provided by the global navigation satellite system (GNSS) at each image’s acquisition time, is used as ancillary information for phase correlation-based coregistration. A transformation function between the multisensor orthophotos is then estimated based on conjugate points (CPs), which are locally extracted over orthophotos using the phase correlation approach. Two multisensor datasets are constructed to evaluate the proposed approach. These visual and quantitative evaluations confirm the superiority of the proposed method.