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Perspective (geometry)
About: Perspective (geometry) is a research topic. Over the lifetime, 277 publications have been published within this topic receiving 5795 citations. The topic is also known as: perspective (geometry).
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TL;DR: In this article, a new perspective on the geometry of a projective line is provided, which helps clarify and illuminate some classical results about projective plane geometry, such as the nine-point circle theorem.
Abstract: This article provides a new perspective on the geometry of a projective line, which helps clarify and illuminate some classical results about projective plane. As part of the same train of ideas, the article also provides a proof of the nine-point circle theorem valid for any affine plane over any field of characteristic different from 2.
TL;DR: In this paper, a methodology, which is based on the concept of epipolar line, for the reconstruction of a 3-D line, from two arbitrary perspective views is given, and simulation study results, in presence of noise, as well as results of error analysis are given.
Abstract: Reconstruction of a line in 3-D space using arbitrary perspective views involves the problem of obtaining the set of parameters representing the line. This is widely used for many applications of 3-D object recognition and machine inspection. A performance analysis of the reconstruction process in the presence of noise in the image planes is necessary in certain applications which require a large degree of accuracy. In this paper, a methodology, which is based on the concept of epipolar line, for the reconstruction of a 3-D line, from two arbitrary perspective views is given. In this problem the points in the second image plane, which correspond to points in the first image plane are found by using epipolar line method, by considering all the points in the first image plane. Then triangulation law is used to find the points in 3-D space. Using least square regression in 3-D, the parameters of a line in 3-D space are found. This least square regression problem is solved by two different methods. Simulation study results of this epipolar line based method, in presence of noise, as well as results of error analysis are given.
01 Jan 2017
TL;DR: Two new means of deriving the reduction equations in common use are presented, one based on analytic geometry, but takes a more straightforward approach than in past works, and the second to attack the problem from a simpler perspective, using trigonometry.
Abstract: The mathematical ABSTRACT The mathematical solution for the true positions of features in space from the position of their images on pairs of radiographs has been discussed in Ballistic Research Labora tory reports by Grabarek and Herr (BRL TN 1634, September 1966, AD 807619), and by Henry C. Dubin (BRL MR 2470, April 1975, AD B003797L, now available for unlimited distribu tion). A truly general solution should not require, or be limited to, orthogonality of the image pairs. Dubin provides such a solution. Because of errors in setup and measure ment, the two lines presumed to connect the respective source and image points through the common object point do not necessarily intersect. Using vector notation and partial deriv atives, he obtains the line of minimum length between these two vectors. The midpoint of this line is the best estimate for the object position, and the length is a measure of the error of the estimate. The redundant image coordinate in Dubin1s method contributes to increasing the accuracy of the estimate of the position of features. The solution of Gra barek and Herr uses analytic geometry and the assumption that the two lines between the respective sources and images intersect, and requires an orthogonal radiographic setup. This approach forgoes generality and some available accuracy. Driven by the need to provide as simple an approach as possible, this paper presents two similar derivations. The author uses analytic geometry to re-derive the equations of Grabarek and Herr from a simpler perspective. The form of the solution provides a concep tual bridge to a more direct derivation by the author, using trigonometry. Both deriva tions are for orthogonal radiographic setups. The trigonometric approach does not require complicated computation of magnification factors, is more easily understood in terms of the geometry of the setup, and is easily implemented in computer or calculator programs to reduce orthogonal radiographs. Flash radiography is a specialized technique widely used in ballistic experimentation to produce stop-motion images of extremely fast events. A small x-ray source of brief duration produces a stereographic projection of intervening objects to form a shadowgraph on a recording medium. The source luminosity and frequency spectrum are tailored to produce a shadowgraph of usable image quality. When geometric information is desired, the source and image positions are carefully controlled. Timing, too, is carefully control led, if kinematic information is sought. The true position of an object must be calcula ted from the measurements of its image on film, and from the known positions of sources and film. The goal of this paper is to make the mathematical basis of the data reduction understandable to as many people involved in the process as possible. This paper presents two new means of deriving the reduction equations in common use. One approach is based on analytic geometry, but takes a more straightforward approach than in past works. The second approach is to attack the problem from a simpler perspective, using trigonometry. The resulting trigonometric formulae are mathematically equivalent to those currently used. However, the form in which they are presented is quite simple, making implementa tion easy. I hope that by making several different approaches available, this paper will be helpful to those having to learn the subject. 2. BACKGROUND To those with a good background in mathematics, there already exists a very simple and elegant approach to the reduction of flash x-rays, but one which is apparently not in wide use. The procedure has the added advantage of being extremely general. It is given in a Ballistic Research Laboratory (BRL) paper by Henry C. Dubin . In Dubin's method, the pair of tube heads which are flashed simultaneously may be located anywhere, as long as the coordinates of their source points are known. For any feature recorded in any pair of non-parallel shadowgraphs, there are two sets of two orthogonal measurements of position that can be made, but only three independent spatial coordinates to be determined. Dubin's procedure utilizes this redundancy to improve on the accuracy of the estimate of the features1 positions. The equations for two lines are found, one from each source to its respective image point. Due to uncertainties in setup and measurement, these lines do not
TL;DR: In this paper, the same practical examples as mentioned in the paper of Wood (1972) in this journal are worked out and their results are compared, and they treat of the spatial intersection method which does not need known poins, but can use any group of well-defined points in the image plane.
TL;DR: In this paper, the use of six points in the model and the common perspective center is recommended since this gives a more rigid connection to the models than the using of only three model points.