Evaluation, transformation, and parameterization of epipolar conics

Transcription

1 Evaluation, transformation, and parameterization of epipolar conics Tomáš Svoboda N - CTU CMP July 31, 2000 Available at ftp://cmp.felk.cvut.cz/pub/cmp/articles/svoboda/svoboda-tr pdf This research was supported by the Czech Ministry of Education under the grant VS96049 and 4/11/AIP CR, by the Grant Agency of the Czech Republic under the grant GACR 102/00/1679, by the Research Programme J04/98: Decision and control for industry, and by EU Fifth Framework Programme project Omniviews No Center for Machine Perception, Department of Cybernetics Faculty of Electrical Engineering, Czech Technical University Karlovo náměstí 13, Prague 2, Czech Republic fax , phone , www:

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3 Transformation of planar conics is the topic of this report. The main goal is to find an Euclidean transformation of a conic into its essential position. Appropriate parameterizations of conics for drawing are proposed. Both regular and singular conics are discussed. 1 Intro and motivation Epipolar curves of a pair of central catadioptric panoramic cameras are general conics 1 [4, 9]. A visualization of conics in a panoramic image is sometimes useful. An example is a drawing of curved polygons during a texture extraction needed for a scene reconstruction [11]. Algorithms for drawing of conics, used in computer graphics, usually assume that a conic, is at least axially oriented, better, in its essential position [1, 7, 12]. Epipolar conics transformed to its essential position can be easier parameterized. The choice of proper parameterization is important because of sampling problem in digital images. 2 Transformation A conic is understood as the locus of 2-D points [u, v] T which satisfy the following quadratic equation a u 2 + b uv + c u + d v 2 + e v + f = 0. This quadratic form can be rewritten in the matrix form where A is a symmetric 3 3 matrix A = u T Au = 0, (1) a b/2 c/2 b/2 d e/2 c/2 e/2 f and u = [u, v, 1] T are image pixel coordinates (axis u is consistently oriented with Cartesian 2 x axis, and axis v is oriented in the opposite direction as y axis). Conics have a number of interesting properties that are not discussed here, a reader is referred to [2] or [10]. A type of the conic can be recognized from matrix A. Let = det(a) and [ ] a11 a δ = det 12. a 21 a 22 Type of the conic depends on values of and δ, [6]. Conic types are summarized in Table 1. The goal is to find a transformation that transforms a conic to its essential position. The center, 1 Expression conic is used in this text as an abbreviation of more precise term conical section. 2 Right-handed Cartesian coordinate system is assumed. 1

4 2 Transformation Regular conics ( 0) Singular conics ( = 0) δ > 0 ellipse (could be real or imaginary) two imaginary lines with a real intersection (eg. x 2 + y 2 = 0) δ < 0 hyperbola two transverses δ = 0 parabola two parallel lines (could be real or imaginary) Table 1: The kind of a conic depends on matrix determinants. of the conic in the essential position coincides with the center of the coordinate system and axes of the conics are oriented accordingly to axes of the coordinate system, see Fig 1. u (0, 0) ϕ u v A u v u 0 Figure 1: Transformation of an ellipse from a general to its essential position. v 2.1 Rotation We want find the matrix A = f(a, ϕ), which satisfy u T A u = 0, (2) where u are point coordinates in rotated coordinate system such that u and v are oriented the same as the axis of the conic, see Figure 1. Being the angle of rotation denoted by ϕ, then new coordinates u are u = cos ϕ sin ϕ 0 sin ϕ cos ϕ u. 2

5 2.2 Translation Contrary u = cos ϕ sin ϕ 0 sin ϕ cos ϕ u. (3) Equations for elements of A matrix are derived by substituting of (3) into (1). a 11 = a 11 cos 2 ϕ + a 22 sin 2 ϕ 2a 12 cos ϕ sin ϕ, a 22 = a 11 sin 2 ϕ + a 22 cos 2 ϕ + 2a 12 cos ϕ sin ϕ, a 13 = a 13 cos ϕ a 23 sin ϕ, a 13 = a 13 sin ϕ + a 23 cos ϕ, a 12 = a 11 cos ϕ sin ϕ a 22 sin ϕ cos ϕ + a 12 (cos 2 ϕ sin 2 ϕ), a 33 = a 33. If u and v are oriented the same as the axis of the conic then a 12 equation tan 2ϕ = 2 tan ϕ 1 tan 2 ϕ we can derive an equation for ϕ from = 0 should hold. Using the 0 = a 11 cos ϕ sin ϕ a 22 sin ϕ cos ϕ + a 12 (cos 2 ϕ sin 2 ϕ). After some manipulation we end up with tan 2ϕ = 2a 12 a 22 a 11. (4) Note that the function tan is defined even for a 22 = a 11. Moreover, if this equality holds, the epipolar conic is a circle, so it has no sense to consider rotation. 2.2 Translation The shift of a conic is determined by the position of the conic center u 0, see Figures 1 and 2. The task is to find elements of the matrix A in rotated and shifted coordinate system u T A u = 0. (5) By substituting u = u + u 0 (6) 3

6 3 Parametric expressions of regular conics into (2) we end up with 11 = a 11, 22 = a 22, 13 = 2a 11u 0 + 2a 13, 23 = 2a 22v 0 + 2a 23, 33 = a 33 + u 02 a 11 + v 02 a a 13u 0 + 2a 23v 0. For an ellipse and a hyperbola in the essential position holds 13 = a 23 this nullity condition into equation for elements of A we end up with = 0. By substituting u 0 = a 13 a, (7) 11 v 0 = a 23 a. (8) 22 This derivation has to be slightly modified for a parabola since at least one from the diagonal elements a 11, a 22 equals to zero. Assuming a 22 = 0, u 0 can be estimated from (7). The second coordinate of the shift v 0 is computed by substituting u 0 into (2) v 0 = a 11 u 2 0 2a 13u 0 a 33 2a 23. (9) Similarly, assuming a 11 = 0, v 0 is computed from (8) and u 0 is computed by substituting v 0 into (2). 3 Parametric expressions of regular conics Once the matrix of a conics in the essential position is known, the conic be easily parametrized by a parameter t. In this section we propose parameterization suitable for drawing of conics. 3.1 Ellipse One of the suitable parameterization of an ellipse is [u, v] = [a cos t, b sin t], where 0 t 2π, (10) where a is the semi-major axis and b is the semi-minor axis. Parameters a, b can be estimated from the matrix A. A central conic 3 has, in its essential position, equation 3 Central conics are ellipses and hyperbolas. 11u v 2 = 33. 4

7 3.2 Hyperbola By comparing this equation with known form of an ellipse equation we end up with a = u 2 a 2 a v 2 b 2 = 1, and b = a 33. (11) Hyperbola One of the possible parameterization of a hyperbola could be [u, v] = [ a cos t, b tan t ], where 0 t 2π, (12) see [3]. However, this parameterization is not suitable for drawing due to the significant discontinuity when t approaches π/2. We propose more appropriate parameterization. The essential u ϕ (0, 0) v A u u v u 0 v Figure 2: Transformation of a hyperbola from a general to its essential position. equation of a hyperbola in pixel coordinates is v 2 a 2 u2 b 2 = 1. (13) 5

8 4 Singular conics By parameterizing t = u, we obtain [u, v] = t, ±a 1 + t2, (14) where parameters a, b can be computed alike for an ellipse. The parameter t is bounded by the image resolution. 3.3 Parabola The parameterization by an angle is not convenient for the same reason as for a hyperbola. Depending on the nullity of diagonal elements of A, 11 and a 22, a proper parameterization is chosen. Assuming 22 = 0, one of the possible parameterization is ] [u, v] = [t, a 11 2 t 2, (15) 23 where the parameter t is bounded by the image resolution. 4 Singular conics A conic is said to be singular if b 2 det(a) = 0, (16) where A is from (1). We modified this condition for numerical computation to det(a) < ɛ, where ɛ is floating point relative accuracy 4. We do not need to consider all possibilities of singular conics, see Table 1. An epipolar conic becomes a singular one if the epipolar plane contains the axis of the mirror [4, 9]. Then the singular conic is a radial line going through the image center 5. An epipolar conics also becomes singular if the image plane is parallel to the axis of the mirror [4]. However, this configuration hardly ever appears in real panoramic cameras and it is neglected here. If singularity of a conic is detected, the decision about the type of a conic is made according to the value of the sub-determinant δ, see Table Central conics The matrix A is for a singular central conics A = a 11 a11 a 22 0 a11 a 22 a (17) 4 On machines with IEEE floating-point arithmetic, ɛ = 2 52, which is roughly Detailed analysis of numerical machine precision can be found in [5]. 5 More precisely, it goes through the point where the axis of the mirror pierces the image plane, which point usually coincides with the image center. 6

9 4.2 Parabola The proof can be found in [8]. By substituting (17) into (1) we end up with the equation of the radial line a11 u + a 22 v = 0. (18) This line do not need to be rotated or shifted, since it goes through the origin of the image coordinate system. For drawing, we express the line as a functional dependency of one coordinate on the second one. The range of u and v is limited by the image resolution. 4.2 Parabola The matrix A looks a22 u = v, if a 11 a 22, a 11 v = A = a11 a 22 u, if a 22 > a a a 23 a 13 a (19) By substituting this matrix into (1) the following equation of the radial line is obtained. a 13 u + a 23 v = 0. (20) For drawing, we use the same approach as for singular central conics. References [1] James D. Foley, Andries van Dam, Steven K. Feiner, and John F. Hughes. Computer Graphics- Principles and Practice. Addison-Wesley, 2-nd edition, [2] Michiel Hazewinkel, editor. Encyclopaedia of Mathematics. Kluwer Academic Publishers, [3] I. Herman. The Use of Projective Geometry in Computer Graphics. Springer-Verlag, [4] Tomáš Pajdla, Tomáš Svoboda, and Václav Hlaváč. Epipolar geometry of central panoramic cameras. In Ryad Benosman, editor, Panoramic Imaging. Springer, To appear. [5] William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Numerical Recipes in C. The Art of Scientific Computing. Cambridge University Press, 2-nd edition, [6] Karel Rektorys. Přehled užité matematiky. Prometheus Praha, 6-th edition,

10 Contents [7] Jiří Sochor, Jiří Žára, and Bedřich Beneš. Algoritmy počítačové grafiky. ČVUT Praha, 2-nd edition, [8] Tomáš Svoboda. Central Panoramic Cameras Design, Geometry, Egomotion. PhD Thesis, Center for Machine Perception, Czech Technical University, Prague, Czech Republic, April [9] Tomáš Svoboda, Tomáš Pajdla, and Václav Hlaváč. Epipolar geometry for panoramic cameras. In Hans Burkhardt and Neumann Bernd, editors, the fifth European Conference on Computer Vision, Freiburg, Germany, volume 1406 of Lecture Notes in Computer Science, pages , Berlin, Germany, June Springer. [10] Eric W. Weisstein. Eric Weisstein s world of mathematics. [11] Tomáš Werner, Tomáš Pajdla, and Martin Urban. REC3D: Toolbox for 3D Reconstruction from Uncalibrated 2D Views. Technical Report CTU-CMP , Czech Technical University, FEL ČVUT, Karlovo náměstí 13, Praha, Czech Republic, December [12] Jiří Žára, Bedřich Beneš, and Petr Felkel. Moderní počítačová grafika. Computer Press, Praha, Contents 1 Intro and motivation 1 2 Transformation Rotation Translation Parametric expressions of regular conics Ellipse Hyperbola Parabola Singular conics Central conics Parabola References 7 8