Relating Vanishing Points to Catadioptric Camera Calibration

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2 Parabolic mirror equation: Hyperbolic mirrorr equation: = = 1 (1) (2) The properties of VPs we derive in this paper are for the more complicated hyperbolic h case. In fact, as described in Section 2, our derivation also covers thee parabolic case since the parabolic case is one of the degenerate forms in central catadioptric cameras. In this paper, we analyse a the image formation of VPs corresponding to the XYZ axes of the 3D real world coordinate system, and then demonstrate the calibration of central catadioptric systemss through the detected VPs. In Section 2, the unifying model for catadioptric image formation and the projection of two parallel 3D lines are briefly explained. In Section 3, we analyse the geometric properties of vanishing points and howw they relate to the cameraa parameters. In Section 4, we show how to use vanishing points for camera calibration. Section 5 outlines the experimental results for both synthetic and a real data. Finally, we conclude in Section UNIFYING MODEL The unifying model of central catadioptric projection developed by Barreto 10 is shown s in Figure 1. A 3D world point is projected on the unit sphere at. The unit sphere iss centre at the focal point of o mirror and denoted as. The unifying model represents this transformation by a 3 4 matrix =[ ] (3) Figure 1. The unifying model for r image formation of central catadioptric cameras. After the transformation =, the point is then mapped to the point p in the sensor plane. This transformation is modelledd using function, in whichh the non-linearity of the mapping m is embedded. The point with coordinates (0, 0, ξ) T is the other projection centre which re-projects the pointt on the unit sphere to in the sensor plane. The function is written as = + (4) Finally, a collineation is applied to transform to obtain the point in the catadioptric image plane, i.e. SPIE-IS&T/ Vol C-2

3 is written as = (5) 0 0 = where =0 0 (6) changes according to the mirror type and shape, is the camera calibration matrix and is the rotation matrix of the mirror relative to the camera. The majority of the catadioptric sensors commercially available have their mirror accurately aligned with the camera, i.e. the conventional camera is not rotated with relation to the mirror surface. Therefore, the rotation matrix =. Generally, the calibration of a central catadioptric system obtains the mirror parameters ξ and the intrinsic parameter matrix. The parameter = 1. For the parabolic mirror case, ξ =1 and the calibration is much easier. The lines under the projection of parabolic sensor are mapped to circles in the image plane if the aspect ratio of the camera is unity and the skew factor is zero 9. Our target is the more complicated hyperbolic case where ξ is unknown.. = 0 (7) is the aspect ratio, is the effective focal length, is the skew factor and, is the principal point. 3. CATADIOPTRIC PROJECTION OF VANISHING POINTS Section 2 described how a general point in the 3D world is mapped to the image point in the catadioptric image plane. In this section, the catadioptric projection of the three dominant vanishing points is introduced. The three ideal points, and associated with the XYZ axes of the 3D world coordinate system are the so-called dominant vanishing points. They are firstly projected on the unit sphere by =[ ] (8) The centre of unit sphere can be assumed to be located at [0,0,0], then equation (3) becomes =. The ideal points, and associated with the XYZ axes of the 3D world coordinate system are located at infinity along the direction of their associated axis. As shown in Figure 2, when, and are projected to the unit sphere, their projections are simply = [1 0 0], = [0 1 0] and = [0 0 1], respectively. 4 v,, Figure 2. The projection of the three dominant VPs from 3D world to the unit sphere. SPIE-IS&T/ Vol C-3

4 We now have =0 1 0=[ ] (9) where, and correspond to the columns of the rotation matrix. We then map the points on the sphere to the 2D canonical plane using equation (4). Since _ = _ (10) _ and = = =1, the vanishing points on the canonical plane equate to _ = _ (11) _ + _ + _ + The last step in the image formation is to apply the collineation matrix to. It is written as. 0 0 = (12) If the dominant vanishing points in the image plane are denoted as, then in matrix form we have. 0 0 _ = _ (13) _ + _ + _ + Most commercially available cameras have square pixels with zero skew, hence we can assume =1 and =0. The vanishing points projected to the image plane can be expressed as = _ + _ (14) = _ + _ (15) where the index =1, 2 or 3 denote the VPs associated X, Y and Z axes respectively. The superscript + indicate the vanishing points associated with the positive direction of X, Y and Z axes. As mentioned before, a set of parallel lines in the 3D world are projected to a set of conics intersecting at two vanishing points in the catadioptric image plane. We can use the same procedure to work out the other vanishing points which correspond to ideal points [ ], [ ] and [ ] (i.e. ideal points lie at infinity towards the negative directions of X, Y and Z axes). 4. CAMERA CALIBRATION USING VANISHING POINTS Based on the relation between the VPs and camera parameters derived above, individual parameters can be obtained by following some simple equation manipulation. Figure 3 shows an example of catadioptric projection of lines. There are three groups of parallel lines with their associated X, Y and Z axes directions. They are then mapped to the image plane SPIE-IS&T/ Vol C-4

5 as conics, each group of conics intersect at two vanishing points. For each pair of vanishing points detected in the image plane, a line can be drawn to cross both points. If the vanishing points for each group of conics are expressed in homogenous coordinates = 1 and = 1, then is obtained by the cross product of and. = (16) The intersection of these three lines computed from each group of conics gives us the principal point = [ 1]. The principal point can be calculated using the cross product of any two lines obtained from equation (16). = (17) Figure 3. The projection of parallel lines in the image plane and their associated vanishing points. Once the principal point is obtained, the columns of rotation matrix can be derived. From equations (14) and (15), we have = = _ (18) _ Divide the two equations in (18), = = _ (19) _ If = = and we have three pairs of vanishing points, then there are =, =, = (20) Here the properties involved with rotation matrix should be accounted for; that is the norm of each column or row of R is 1; R is an orthogonal matrix. Now we have sufficient constraints to estimate the rotation matrix R. Once R is obtained, and can be easily calculated. SPIE-IS&T/ Vol C-5

6 5. EXPERIMENTAL RESULTS The calibration procedure illustrated in Section 4 is demonstrated and tested on both synthetic and real data with respect to noise. 5.1 Synthetic data For this simulation, the calibration parameters of the catadioptric camera are set to: aspect ratio r = 1, skew factor s = 0, effective focal length = 1000, image centre = 600 and = 400, image size pixels and mirror parameter = The rotation matrix is randomly generated using the formula cos cos cos sin + sin sin cos sin sin + cos sin cos =cos sin cos cos + sin sin sin sin cos + cos sin sin (21) sin sin cos cos cos Gaussian noise with zero mean and standard deviation varying from 0.0 to 5.0 pixels is added to each of the simulated vanishing points in the image plane. For each noise level, we perform 100 independent trials. The means and the standard deviations of the intrinsic parameters,,, are computed and shown in Figure 4. The estimated principal points and mirror parameter are close to the ground truth as the noise level increase, therefore, it is reliable to use vanishing points to obtain the principal points and mirror parameters. Comparably, the focal length error gets larger when noise levels increase, this is due to its high magnitude (i.e.10 ) and the multiplication of uncertainty while calculating the focal length. However, the error in the focal length still stays reasonably small when the noise level is less than 4 pixels. o c 1020 c o 100 có 980 c o«960 d 0.93 ïü E E `mn ó E io -0 0_92 v v c N a, d o a, Gaussian noise level M Gaussian noise level (a) (b) S 610, 410 á009- E To 2' ö ç a u 594- u To > Gaussian noise level Gaussian noise level (c) (d) Figure 4. Experimental results of applying proposed calibration approach on synthetic data. SPIE-IS&T/ Vol C-6

7 5.2 Real data The proposed method was also tested on real images. The conics are fitted to the extracted arc segments (i.e. the projection of straight lines) using the algorithm proposed by Sturm and Gargollo 11 ; then vanishing points are detected as the intersection of parallel conics. Figure 5 shows an example of conic fitting and vanishing points detection from a catadioptric image. We used a perspective camera with a hyperbolic mirror. The hyperbolic mirror employed is a commercially available unit called the Panoramic Optic (0-360.com, Carson City, USA), which has a vertical field of view (FOV) of 115 and its vertical maximum visible angle above horizon is Images of five different manmade environments were taken. For each environment, images were repeated at six different locations. Each image was calibrated independently. Compared to the ground truth, the estimation results are as follows: the average error for the focal point is ±5.24 pixels, for the mirror parameter is ±0.021, for the principal point is ±2.21 pixels, and is ±1.79 pixels. We know that only accurate calibration enables the rectification of the catadioptric image. In Figure 6, some sample rectification results achieved using the calibrated parameters are shown. Figure 5, The detected vanishing points and principal points on real catadioptric image. w 4' iiìi.k w.n,üj ÿ,,ir.... SPIE-IS&T/ Vol C-7

8 Cri Figure 6. Visual assessment of calibration results: corrected perspectives of man-made environments and human faces. SPIE-IS&T/ Vol C-8

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