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Cayley Transformation and Numerical Stability of Calibration Equation

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Cayley Transformation and Numerical Stability of Calibration Equation Int J Comput Vis (2009) 82: 156–184 DOI 10.1007/s11263-008-0193-x Cayley Transformation and Numerical Stability of Calibration Equation F.C. Wu · Z.H. Wang · Z.Y. Hu Received: 25 May 2008 / Accepted: 22 October 2008 / Published online: 5 December 2008 © US Government 2008 Abstract The application of Cayley transformation to en- it is proved that the standard calibration equation, the Cay- hance the numerical stability of camera calibration is inves- ley calibration family and the S-Cayley calibration family tigated. First, a new calibration equation, called the standard are all some special cases of this generic calibration family. calibration equation, is introduced using the Cayley transfor- mation and its analytical solution is obtained. The standard Keywords Camera calibration · The absolute conic · calibration equation is equivalent to the classical calibration Calibration equation · Cayley transformation · Numerical equation, but it exhibits remarkable better numerical stabil- stability ity. Second, a one-parameter calibration family, called the Cayley calibration family which is equivalent to the stan- dard calibration equation, is obtained using also the Cayley 1 Introduction transformation and it is found that this family is composed Camera calibration is a necessary step to recover 3D met- of those infinite homographies whose rotation has the same ric information from 2D images. Many methods and tech- axis with the rotation between the two given views. The con- niques for camera calibration have been proposed in the last dition number of equations in the Cayley calibration family decades. The existing methods can be roughly classified into varies with the parameter value, and an algorithm to deter- two categories: methods based on a reference calibration mine the best parameter value is provided. Third, the gen- object and self-calibration methods. In the first kind, there eralized Cayley calibration families equivalent to the stan- are mainly the methods based on 3D object (Brown 1971; dard calibration equation are also introduced via general- Faig 1975;Tsai1986; Faugeras 1993), 2D pattern (Zhang ized Cayley transformations. An example of the generalized 1999, 2000; Sturm and Maybank 1999) and 1D object Cayley transformations is illustrated, called the S-Cayley (Zhang 2004;Wuetal.2005; Wang et al. 2007). In these calibration family. As in the Cayley calibration family, the methods, camera calibration is performed by observing a numerical stability of equations in a generalized Cayley cal- 3D calibration object, a 2D calibration pattern or a 1D ibration family also depends on the parameter value. In addi- calibration object. Since the Euclidean geometries of cal- tion, a more generic calibration family is also proposed and ibration objects are used, the methods can provide a cali- bration of high-precision. In the second kind, the methods widely used are the calibration based on the Kruppa equa- · · F.C. Wu ( ) Z.H. Wang Z.Y. Hu tion (Faugeras et al. 1992; Maybank and Faugeras 1992; National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, PO Box 2728, Hartley 1997a; Luong and Faugeras 1997), the absolute Beijing 100190, People’s Republic of China conic (Pollefeys et al. 1996; Hartley 1997b; Pollefeys and e-mail: [email protected] Gool 1999) and the dual absolute quadric (Triggs 1997; Z.H. Wang Ponce et al. 2000). These methods do not need any refer- e-mail: [email protected] ence calibration objects and use only the correspondences of Z.Y. Hu image entities, such as points, lines. It is remarkable and de- e-mail: [email protected] sirable that the self-calibration methods require no reference Int J Comput Vis (2009) 82: 156–184 157 objects. Regrettably, in addition to their low calibration pre- family, the numerical stability of equations in this general cision, the numerical stabilities of self-calibration methods family varies also with the parameter value. An example are usually poor. Hartley and Zisserman (2000) pointed out of the generalized Cayley transformations is illustrated, that the algorithms absolutely depended on self-calibration called the S-Cayley calibration family. should be restrainedly used in practice. Though the people • A more generic calibration family is proposed and it is have paid great attention to improve the precision and the proved that the standard calibration equation, the Cayley numerical stability of self-calibration methods, no satisfac- calibration family and the S-Cayley calibration family are tory results are obtained up to now. all some special cases of this generic calibration family. In this paper, we mainly investigate the numerical sta- This paper is organized as follows. Section 2 reviews bilities of the self-calibration methods based on the ab- briefly the classical calibration algorithm based on the ab- solute conic. The calibration equation induced by the ab- solute conic. In the Sect. 3, the Cayley transformation is in- T = T = −1 solute conic, H∞ωH∞ ω or, equivalently H∞ω ωH∞ , troduced. The standard calibration equation, the Cayley cal- is called the classical calibration equation in this paper. ibration family, and the generalized Cayley calibration fam- Here H∞ is the infinite homography and ω the image of ily are elaborated in Sects. 4, 5 and 6, respectively. Section 7 the absolute conic. The previous studies are mainly fo- gives more generic equivalent equations. Section 8 reports cused on the conditions and computational means to solve the experiment results. Section 9 concludes this paper. the infinite homography (such as, Pollefeys et al. 1996; Pollefeys and Gool 1999; Hartley 1997b, etc.). For a long time the numerical stability of the calibration equation itself 2 Classical Calibration Equation has been neglected in research community. In computation term, generally speaking different forms of the same equa- The infinite homography between the two views is denoted tion can usually have different numerical stabilities. Thus we as H∞ in this paper. If the motion between the two views is have the following problem: For the self-calibration based (R, t), R is the rotation and t the translation, and the intrin- on the absolute conic, do there exist more stable forms than sic matrix of camera is K, then the infinite homography can the classical calibration equation in computation? This prob- be expressed as lem is answered affirmatively in this paper. More specifi- = −1 cally, we use the Cayley transformation to enhance the nu- H∞ KRK . (1) merical stability of camera calibration and here are our main Let ω be the image of the absolute conic (IAC) i.e., ω = contributions: K−T K−1. Then there is the classical calibration equation • Using the Cayley transformation of the infinite homogra- (CCE): phy, a new calibration equation, called the standard cali- T H ωH∞ = ω. (2) bration equation, is introduced and its analytical solution ∞ is obtained. The Cayley transformation of the infinite ho- Evidently, this equation can be changed to the following mography transforms image point to its conjugate point equivalent form: on the image of absolute conic. The standard calibration T −1 equation is equivalent to the classical calibration equa- H∞ω = ωH∞ . (3) tion, but it exhibits remarkable better numerical stability. For the convenience of statement, (2) is called CCE of the • A one-parameter family equivalent to the standard cali- first type (CCE-I), and (3) is called CCE of the second type bration equation, called the Cayley calibration family, is (CCE-II). introduced using also the Cayley transformation and it is Hartley (1997b) used the CCE-II to compute ω and shown that the Cayley calibration family is composed of pointed out that the CCE can only provide 4 independent those infinite homographies whose rotation has the same constraints on ω. Thus at least 3 views are necessary to lin- axis with the rotation between the two given views. The early solve ω.Afterω is determined, the intrinsic matrix K condition number of equations in the Cayley calibration can be determined by the Cholesky factorization of ω, then family varies with the parameter value, and an algorithm the rotation R can in turn be determined. The classical cali- is also provided to determine the best parameter which bration algorithm can be sketched as: gives the best numerical stability among the Cayley fam- ily. Classical Algorithm • Generalized Cayley calibration family equivalent to the standard calibration equation is introduced via general- Goal Given N views of a scene and assume the infinite ho- ized Cayley transformations. Given a generalized Cay- mographies Hi∞ between the 0th and ith views are known. ley transformation, a generalized Cayley calibration fam- Compute the intrinsic matrix K and the rotation Ri between ily can be obtained. Similarly to the Cayley calibration the 0th and ith views. 158 Int J Comput Vis (2009) 82: 156–184 Algorithm The algorithm consists of the following three Here e is the epipole on the 2nd view, steps: F T e = 0, ˜ −1/3 1. Let Hi∞ = Det (Hi∞)Hi∞. Compute the least squares solution of the following set of linear equations a is an unknown 3-vector. Given an infinite correspondence on ω: m(0) ↔ m(1),wehave: − ˜ T ˜ = ˜ T = ˜ 1 (1) T (0) Hi∞ωHi∞ ω(or Hi∞ω ωHi∞), [m ]×([e ]F + e a )m = 0. i = 1, 2,...,N − 1. In the above three equations on a, there are only one in- 2. Determine the intrinsic matrix K by taking the Cholesky dependent constraint. And thus, at least 3 infinite correspon- factorization of ω. dences are necessary to determine linearly the vector a. 3. Determine the rotation Ri by finding the best rota- ˜ K = −1 ˜ tion approximation of Hi∞ K Hi∞K, i.e., com- Method 3 Computation using the modulus constraint. For pute firstly the singular value decomposition of the more than 2 views, using the modulus constraint shown in ˜ K ˜ K = T matrix Hi∞, Hi∞ Udiag(t1,t2,t3)V , then set up Pollefeys et al.
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