
Linear calibration of a rotating and zooming camera ∗ Lourdes de Agapito, Richard I. Hartley and Eric Hayman Department of Engineering, Oxford University and G.E. Corporate Research and Development 1 Research Circle, Niskayuna, NY 12309 Abtract apply in case of a minimal assumption of zero-skew. A linear self-calibration method is given for com- These methods applied to moving cameras, and were puting the calibration of a stationary but rotating iterative1. camera. The internal parameters of the camera are al- Recent papers ([1, 9]) have extended calibration ca- lowed to vary from image to image, allowing for zoom- pability by giving algorithms for rotating cameras, al- ing (change of focal length) and possible variation of lowing changing internal parameters, similar in intent the principal point of the camera. In order for calibra- to the algorithms of [5, 6, 8]. The particular scenario tion to be possible some constraints must be placed on of a rotating and zooming camera is common in prac- the calibration of each image. The method works un- tice. For instance a camera at a sports arena under- der the minimal assumption of zero-skew (rectangular goes this sort of motion as it rotates and zooms to pixels), or the more restrictive but reasonable condi- follow a game. Unlike the algorithm of Hartley ([4]), tions of square pixels, known pixel aspect ratio, and these methods use interation, based on the Levenberg- known principal point. Being linear, the algorithm is Marquardt method to minimize a non-linear cost func- extremely rapid, and avoids the convergence problems tion. The linear methods of ([4]) do not appear to be characteristic of iterative algorithms. immediately extendable to the case of varying internal parameters. 1 Introduction In this paper however it is shown that a simple trick, The subject of self-calibration of a camera has re- first used in a different context in [11] transfers the ba- ceived considerable attention, following the ground- sic calibration equation ((3) below) to one for which breaking paper of Maybank and Faugeras [7]. This a simple linear method applies. The resulting linear idea opens the possibility of calibration of a camera method is extremely simple involving a least-squares in the field, without the aid of jigs, or knowledge of solution of a set of homogeneous equations in 6 un- the position of world-points. A subsequent paper of knowns. Each image in the sequence leads to one to Hartley ([4]) gave a method for the self-calibration of four equations, depending on the amount of assumed a rotating, but stationary camera, for which the the- knowledge of the camera. It turns out that the linear ory of [7] does not apply. The algorithm of [4] had the method, apart from being quicker than the iterative advantage of being linear and hence very simple and methods gives results of comparable quality. rapid, unlike the Maybank-Faugeras method, which was somewhat complex. 2 Rotating cameras The methods of these two papers required the cal- We consider a set of images taken with cameras ibration of the camera to be fixed over a sequence all located at the same point in space, which will be of images – no zooming was allowed. Subsequently taken to be the coordinate origin. As has been shown interest in zooming cameras led to a method for self- in (for instance) [4] one may analyse this situation by calibration of cameras with changing internal param- representing each of the cameras as a 3×3matrixPi.A eters ([5]). These results were strengthened in [6, 8] to 1It may be noted that these papers were all predated by ∗This work was sponsored by DARPA contract F33615-94- the paper [3] that gave a non-iterative algorithmfor two-view C-1549 calibration in the case of known aspect ratio and principal point. 1 point in the i-th image, represented by a homogeneous 2. The square-pixel constraint : For each cam- 3-vector xi corresponds to a ray in space consisting era, s =0andαx = αy. −1 of points of the form λPi xi. Points on this ray are −1 3. The known principal point constraint : For mapped into the j-th image to a point xj = PjPi xi. Denoting the transformation each camera, (x0,y0)=(0, 0). − If the pixels have a known aspect ratio other than H = P P 1 (1) ij j i 1, or the principal point is at a different know point one sees that the i-th and j-th images are related by other than the origin, then a simple change of image coordinates converts to one of the cases above. a projective transformation Hij . In practice, one may compute these projective transformations between im- Now, a constraint such as the zero-skew constraint is not reflected in any simple way in the entries of ages by finding matching points in the two images and ∗ computing the projective transformation that relates ω = KK . There seems to be no easy way to use the points. At least four matched points are necessary (3) to develop an algorithm to compute the camera for computing the projective transformation between calibration, enforcing constraints of this type. There- two images. Practical methods for computing these fore, in [1, 9], iterative algorithms are proposed, us- transformations are given in [4, 2]. ing Levenberg-Marquardt iteration to find a non-linear The projective transformation may be related to least squares solution, parametrized directly by the en- the calibration matrices of each of the cameras, as tries of the Ki. This may potentially cause problems with lack of convergence, or convergence to local min- follows. Each camera matrix Pi may be decomposed ima, not to mention the greater complexity of coding. into an upper-triangular calibration matrix Ki and a A simple observation, however, leads to a linear al- rotation matrix Ri representing the orientation of the gorithm. Taking the inverse of (3) gives camera : Pi = KiRi. Substituting in (1) one obtains − − − − −1 1 1 1 Hij ωiH = ωj (5) Hij = KjRjRi Ki = KjRij Ki . (2) ij − −1 Since Rij is a rotation, Rij Rij = I. Straight-forward where ωi = Ki Ki . Now, one may verify that with computation then shows that Hij KiKi Hij = KjKj . K of the form (4), with s =0, This formula is often written as − −1 ω = K K ∗ ∗ 2 − 2 Hij ωi Hij = ωj (3) 1/αx 0 x0/αx 2 − 2 = 01/αy y0/αy ∗ − 2 − 2 2 2 2 2 where ωi = KiKi is the dual image of the absolute x0/αx y0/αy 1+x0/αx + y0/αy conic ([4]) in the i-th image. This formula then repre- (6) sents the transformation of a conic under a projective transformation. In the case where Ki = Kj,thisfor- so ω represents a conic of the form mula may be used to generate a set of linear equations ∗ − 2 2 − 2 2 in the entries of ωi which may be used to solve for (x x0) /αx +(y y0) /αy +1=0 , ∗ ωi , and subsequently for Ki by Choleski factorization. This is briefly the calibration method described in [4]. the image of the absolute conic. The important points The difficulty with extending this linear solution to to note here are the case of a camera with varying intrinsic parame- ters under minimal assumptions such as zero skew or Proposition 2.1. 1. If s = K12 =0,thenω12 =0. known aspect ratio is that such constraints are not eas- ∗ 2. If s =0and αx = αy,thenω11 = ω22. ily related to the entries of ωi . Consider a calibration matrix 3. If s =0and x0 =0,thenω13 =0. Similarly if αx sx0 y =0then ω =0. 0 23 K = αy y0 (4) 1 3 Generation and solution of equations Three constraints are possible, of which the first two Select one image as being a reference image, and were tested in detail in this paper. let H0j be the homography relating the reference im- age to the j-th image. For j = 0, one has H00 = I. 1. The zero-skew constraint : For each camera, Let the image of the absolute conic in the refer- s =0. ence image be ω0. Since it is symmetric, it may be parametrized by its sixdiagonal and above-diagonal camera motion contains some component of rotation entries, which may be denoted in some designated or- about the pricipal axis (Z-axis)ofthecamera.Ifsome −1 der as (a1,a2,...,a6). Since the homographies H0j Z axis motion is included, however, then results are are known, the entries of ωj may be expressed lin- good. Thus, to obtain good results, one must either early in terms of the entries ai of ω0.Noweachof have some Z-rotations, or else include square-pixel the equation types in Proposition 2.1 gives one linear constraints. It should be realized that this failure equation in the entries of ωj, hence a linear equation mode does not represent a weaknesses in the linear in the entries ai of ω0. For each image each condition algorithm of this paper, but rather arises from a situ- gives one equation, and the set of all equations may ation in which self-calibration is intrinsically unstable. be written as Ea =0,wherea =(a ,a ,...,a ), 1 2 6 X-axis rotation. Consider the case where the and each row of E represents one equation. Given at rotation is about the X axis of the camera and the least five equations one can find a solution up to (non- cameras have zero skew. In addition, suppose for the essential) scale.
Details
-
File Typepdf
-
Upload Time-
-
Content LanguagesEnglish
-
Upload UserAnonymous/Not logged-in
-
File Pages8 Page
-
File Size-