Coplanar Circles Research Articles

Camera calibration using the dual double-contact property of circles

To mitigate the influence of the relative positional relationship between circles on camera calibration, this study explores the double contact of dual circles with dual circular points. We demonstrate that one of the generalized eigenvectors of two dual circles corresponds to the line passing through their respective centers. Moreover, we establish a methodology to identify this line from the three generalized eigenvectors and show that, regardless of the positional arrangement of the three co-planar circles, the image of the circular points can be constructed by calculating the image of this line. Consequently, the applicability of circles as calibration templates is expanded, enabling the development of a novel optimization technique for fitting circular images with enhanced calibration precision.

Common pole-polar and common tangent properties of dual coplanar circles and their application in camera calibration

Common pole-polar and common tangent properties of dual coplanar circles and their application in camera calibration

Camera calibration based on the common pole-polar properties between two coplanar circles with various positions.

Computer vision camera calibration is widely performed using parallel circles. Various cases of two coplanar circles are algebraically explained, proving that the common pole is located at the line at infinity for all relative positions, and the corresponding polar passes through the centers of the two circles. The two common poles of the two coplanar circles are the points at infinity when concentric; one common pole of the two coplanar circles is a point at infinity when nonconcentric. Accordingly, the vanishing line can be obtained by using the common pole-polar properties of two groups of two coplanar circles, and the camera's intrinsic parameters are solved according to the constraints between the image of the circular points and the imaged absolute conic. The camera calibration can be solved using only three images of two coplanar circles. Simulation and experiments verify that the proposed algorithms are effective.

Relative pose estimation of a lander using crater detection and matching

Future space exploration missions require precise information about the lander pose during the descent and landing steps. An effective algorithm that utilizes crater detection and matching is presented to determine the lander pose with respect to the planetary surface. First, the projections of the crater circular rims in the descent image are detected and fitted into ellipses based on the geometric distance and coplanar circles constraint. Second, the detected craters are metrically rectified through a two-dimensional homography and matched with the crater database by similarity transformation. Finally, the lander pose is calculated by a norm-based optimization method. The algorithm is tested by synthetic and real trials. The experimental results show that our presented algorithm can determine the lander pose accurately and robustly.

Linear Determination of a Camera's Intrinsic Parameters Using Two Intersecting Circles

An approach using two intersecting circles is proposed as a linear approach for determining a camera's intrinsic parameters. The two intersecting coplanar circles have four intersection points in the projective plane: two real points and two circular points. In the image plane, the diagonal triangle - on which the image of the four intersection points composes a complete quadrangle - is a self-polar triangle for the projection curves of the circles. The vertex of the self-polar triangle is the null space of the degenerate conic formed by the image of the four intersection points. By solving the three vertices of the self-polar triangle using the image coordinates of the two real intersection points, the degenerate conic can be obtained. The image of the two circular points is then computed from the intersection points of the degenerate conic. Using the image of the circular points from the three images of the same planar pattern with different directions, the intrinsic parameters can be linearly determined.

J164023 視体積交差による人物の3次元復元のためのカメラ校正に関する一考察([J164-02]医療・健康・福祉のためのセンシングおよびロボティクス(2))

In order to achieve a marker-free human motion capture system, which is composed of reconstruction of human body in 3D volume using silhouette volume intersection and estimation of 3D skelton information by fitting an articulated human model to the reconstructed 3D volume data, the projection matrices of multiple cameras should be determined in the calibration process in advance. To conduct the multi-camera calibration with ease, this paper investigates a calibration method which estimates the projection matrix of a camera using an image of two coplanar circles. Computational experiment using a virtual camera simulator confirms the effectiveness of the conic-based calibration method for estimating the camera parameters and the projection matrix.

Photographic composite detection using circles

In this work, we propose several new methods for detecting photographic composites using circles. In particular, we focus on three kinds of scenes: (1) two coplanar circles with the same radius; (2) a single circle with its discriminable center; (3) a single circle with geometric constraints for camera calibration. For two circles’ situation, we first estimate the focal length based on the equality of the sizes of two coplanar circles, and then estimate the normal vector of the world circle plane. Inconsistencies in the angles among the normal vectors (Each circle determines a normal vector) are used as evidence of tampering. On the other hand, for the single circle case, we warp the circle to make metric measurement. To demonstrate the effectiveness of the approach, we present results for synthetic and visually plausible composite images.

Camera calibration using one perspective view of two arbitrary coplanar circles

We present a camera calibration technique to determine the focal length and the extrinsic parameters of a camera merely by using one perspective view of two coplanar circles with arbitrary radii and topological configuration. We believe our effort is valuable, in that the position of a camera and its focal length are frequently adjusted in many vision applications. In contrast to the iterative optimization technique in previous work, whose convergence relies heavily on proper initialization, we propose a closed-form solution on the basis of simple matrix operations, which turns out to be computationally efficient. It can also be used to initialize existing algorithms for higher accuracy and speed. Our method is based on the projective equation of a circle, first presented here, which naturally encodes the intrinsic and extrinsic camera parameters and the degenerate conic envelope spanned by the image of circular points. Extensive experiments with both synthetic data and real images verify the efficiency and robustness of our technique.

Coplanar circles, quasi-affine invariance and calibration

We define the lines associated with two coplanar circles, and give the distributions of any two coplanar circles and their associated lines. Further we prove that the distribution of two coplanar circles with no real intersection and their associated lines is a quasi-affine invariance. Then the results are applied to calibrating a camera. The calibration method has the advantages: (1) it is based on conic fitting; (2) it does not need any matching. Experiments with two separate circles validate our quasi-affine invariance and show that the estimated camera intrinsic parameters are as good as those obtained by Zhang's (2000) method.

The mathematical theory of the magnetic field round a circular current, and allied problems

An attempt is made to simplify the theory of the magnetic field round a plane circular current filament. The lines of force round such a filament are identical with the lines of flow round a circular vortex in hydrodynamics. A diagram showing these curves was drawn by Donald Macfarlane under Kelvin's direction in 1869 and has often been reproduced since. The author shows how the curves can be drawn by means of simple bipolar formulae. He shows that the mutual inductance M between two concentric and coplanar circles the radii of which are a and b is given very approximately by the formulaM = 8Π2b2/a + 3√(a2-b2)If b/a is less than 0.5 the maximum inaccuracy of this formula is less than 1 in 40 000. If b/a is less than 0.7 its inaccuracy is less than 4 in 1 000. It is less than 1 per cent when b/a = 0.9. Even when b/a is 0.95 the inaccuracy is less than 2 per cent. A formula is given for the attraction between two coaxial circular filaments. An equation is also given from which the distance between them when their attraction is a maximum can be computed. If 6 is not greater than about the tenth part of a, the distance y between them, when their attraction is a maximum, is given very approximately by2y = a + b.In a mathematical Appendix a fairly complete list is given of formulae for the two fundamental elliptic integrals and of formulae for differentiating them. Some of these formulae are new and some can only be found in treatises which are out of print and are obtainable in very few libraries. They can be usefully applied in electrical theory

XV. On systems of circles and spheres

This Memoir is divided into three Parts: Part I. treats of systems of circles in one plane; Part II. treats of systems of circles on the surface of a sphere; and Part III. of systems of spheres; the method of treatment being that indicated in two papers among Clifford’s ‘Mathematical Papers,ʼ viz., “On Power-Coordinates” (pp. 546—555) and “On the Powers of Spheres” (pp. 332-336). These two papers probably contain the notes of a paper which was read by Clifford to the London Mathematical Society, Feb. 27, 1868, “On Circles and Spheres,” which was not published (‘Lond. Math. Soc. Proc.,ʼ vol. 2, p. 61). The method of treatment indicated in these papers of Clifford’s was successfully applied by the author to prove some theorems given by him in a paper “On the Properties of a Triangle formed by Coplanar Circles” (1885) (‘Quarterly Journal of Mathematics,ʼ vol. 21), and then to the extension of those theorems to the case of spheres. But as Clifford’s papers contained some suggestions as to the application of the same method to the treatment of Bi-circular Quartics, he was induced to develop these ideas and extend the results to the case of the analogous curves on spheres—called by Professor Cayley Spheri-quadrics—and also of cyclides. It is impossible to say whether, if at all, Clifford was indebted to Darboux for any of the ideas contained in the two papers cited above; but it is noticeable that they coincide in a great measure with those expressed by Darboux in several papers published during the years 1869‒1872. In Part I. (§§ 1—124) of this Memoir a general relation is first shown to subsist between the powers of any two groups of five circles; the definition of the power of two circles, as the extension of Steiner’s “power of a point and a circle,” being due to Darboux, but the definition is here slightly modified so as to include the case when the radius of either (or each) circle is infinite. In Chapter II. an extension of the definition so as to apply to a certain system of conics is given; this is practically adapted from Chapter II. in Professor Casey’s Memoir “On Bicircular Quartics” (1867) (‘Irish Acad. Trans.,’ vol. 24). In Chapter III. the general theorem is applied to several interesting cases of circles; some of the results of this chapter are believed to be new. In Chapter IV. the problem of drawing a circle to cut three given circles at given angles is considered, and the circles connected with a triangle formed by three circles, which are analogous to the circumcircle, the inscribed and escribed, and the nine-points circle of an ordinary triangle are discussed. The results are the same, with one or two exceptions which may be new, as arrived at, but in a different manner, in the paper by the author in the ‘Quarterly Journal’ (vol. 21). In Chapter V. the power-coordinates of a point (or circle) are defined, and the equations of circles, &c., discussed; and it is shown that there are two simple coordinate systems of reference; one consisting of four orthogonal circles, mentioned by Clifford (Casey and Darboux consider five orthogonal spheres), the other consisting of two orthogonal circles and their two points of intersection, which seems to have been indicated for the first time by Mr. Homersham Cox in a paper “On Systems of Circles and Bicircular Quartics” (‘Quarterly Journal,’ vol. 19, 1883). In Chapter VI. the general equation of the second degree in power-coordinates is discussed, and in Chapter VII. Bi-circular Quartics are classified according to the number of principal circles which they possess. In Chapter VIII. the connexion between Bi-circular Quartics and their focal conics is briefly indicated, the circle of curvature is found, and an expression for the radius of curvature at any point of a bi-circular quartic is investigated. In these last three chapters the results are probably all old, but as the method employed is different from any previously used to discuss these curves in detail, it may not be without interest.

Mathematical Notes by Professor Tait - 2. On the Ovals of Descartes

The following results were obtained lately while I was considering how most simply to describe by working sections surfaces analogous to that treated in the preceding note. They are so elementary that it is not likely that they can be new, but as they are novel to myself, and to several mathematicians whom I have consulted, I bring them before the Society:—Let two coplanar circles be described, with centres A and B. Take any point, C, in the line of centres, and draw a line CPQ, cutting the circles in P and Q. Find the locus of R, the intersection of AP and BQ.