cvpr cvpr2013 cvpr2013-400 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Amit Agrawal, Srikumar Ramalingam
Abstract: Imaging systems consisting of a camera looking at multiple spherical mirrors (reflection) or multiple refractive spheres (refraction) have been used for wide-angle imaging applications. We describe such setups as multi-axial imaging systems, since a single sphere results in an axial system. Assuming an internally calibrated camera, calibration of such multi-axial systems involves estimating the sphere radii and locations in the camera coordinate system. However, previous calibration approaches require manual intervention or constrained setups. We present a fully automatic approach using a single photo of a 2D calibration grid. The pose of the calibration grid is assumed to be unknown and is also recovered. Our approach can handle unconstrained setups, where the mirrors/refractive balls can be arranged in any fashion, not necessarily on a grid. The axial nature of rays allows us to compute the axis of each sphere separately. We then show that by choosing rays from two or more spheres, the unknown pose of the calibration grid can be obtained linearly and independently of sphere radii and locations. Knowing the pose, we derive analytical solutions for obtaining the sphere radius and location. This leads to an interesting result that 6-DOF pose estimation of a multi-axial camera can be done without the knowledge of full calibration. Simulations and real experiments demonstrate the applicability of our algorithm.
Reference: text
sentIndex sentText sentNum sentScore
1 Abstract Imaging systems consisting of a camera looking at multiple spherical mirrors (reflection) or multiple refractive spheres (refraction) have been used for wide-angle imaging applications. [sent-2, score-1.121]
2 We describe such setups as multi-axial imaging systems, since a single sphere results in an axial system. [sent-3, score-0.735]
3 Assuming an internally calibrated camera, calibration of such multi-axial systems involves estimating the sphere radii and locations in the camera coordinate system. [sent-4, score-0.99]
4 The pose of the calibration grid is assumed to be unknown and is also recovered. [sent-7, score-0.402]
5 The axial nature of rays allows us to compute the axis of each sphere separately. [sent-9, score-0.802]
6 We then show that by choosing rays from two or more spheres, the unknown pose of the calibration grid can be obtained linearly and independently of sphere radii and locations. [sent-10, score-1.023]
7 Knowing the pose, we derive analytical solutions for obtaining the sphere radius and location. [sent-11, score-0.576]
8 Introduction Catadioptric imaging systems consist of a camera looking at single or multiple mirrors for wide-angle imaging. [sent-15, score-0.439]
9 The special configurations include a perspective camera placed at the focal point of a hyperbolic or an elliptical mirror, or an orthographic camera placed on the axis of a parabolic mirror. [sent-17, score-0.384]
10 Common configurations such as camera viewing a spherical mirror or multiple mirrors always lead to a non-central imaging system [23]. [sent-18, score-0.79]
11 In this paper, we describe such setups (see Figure 1) as multi-axial imaging systems, since a single spherical mirror/refractive ball results in an axial system [2, 24]. [sent-54, score-0.549]
12 Without loss of generality, we use ‘sphere’ to refer to the spherical mirror as well as the refractive sphere. [sent-55, score-0.683]
13 [24] manually marked the outline of each sphere in the captured image to get an initial estimate of sphere center and radius. [sent-59, score-0.886]
14 Techniques such as [8, 19] also used the image contour of sphere for localization. [sent-61, score-0.449]
15 Nayar’s [15] sphereo system require sphere centers to lie on an plane orthogonal to the imaging plane, along with coplanar light sources such that their centroid lies on the optical axis for calibration. [sent-65, score-0.676]
16 This along with mechanical design data is used to estimate the initial sphere locations [13, 7]. [sent-67, score-0.427]
17 We show that calibration of such multi-axial systems can be done using a single photo of a 2D calibration grid (e. [sent-69, score-0.534]
18 The pose of the checkerboard is assumed to be unknown, and is also recovered. [sent-72, score-0.501]
19 It does not require estimation of the sphere contour in the captured photo. [sent-75, score-0.471]
20 Our approach can handle spheres of different radii and is marker-less. [sent-76, score-0.386]
21 While a single sphere results in a multi-perspective image and corresponds to an axial camera, multiple spheres correspond to a multi-axial camera where each 3D point is imaged multiple times via underlying axial cameras. [sent-79, score-1.22]
22 The axial nature of rays allows us to compute the axis of each sphere separately. [sent-80, score-0.802]
23 By choosing rays from two or more spheres, we demonstrate that the unknown pose of the checkerboard can be obtained linearly, and independently of sphere radii and locations. [sent-81, score-1.164]
24 Thus, 6DOF pose estimation can be done in a semi-calibrated setting, without the knowledge of sphere locations and radii. [sent-82, score-0.568]
25 Finally, we derive analytical solutions for estimating the radii and center of spheres with known checkerboard pose and axes. [sent-84, score-0.948]
26 Related Work Axial Systems: A camera looking at a single mirror can be categorized as an (a) axial or (b) off-axis system, depending on its placement on the mirror axis. [sent-87, score-0.902]
27 However, they require three images of a checkerboard in different orientations and their parameterization involves two rotations/translations. [sent-90, score-0.382]
28 Instead, we use a global model parameterized via sphere radius and center. [sent-93, score-0.547]
29 Micusik and Pajdla [14] proposed autocalibration of single mirror based axial systems starting from a central approximation, using multiple images. [sent-94, score-0.517]
30 A brute force approach to apply axial methods to multi-axial scenario is to calibrate each sphere separately. [sent-95, score-0.674]
31 However, this does not utilize inter-sphere constraints and the fact that the same checkerboard is seen via all spheres. [sent-96, score-0.382]
32 We show that using rays from two or more spheres allow computing the checkerboard pose linearly and independently of sphere parameters. [sent-97, score-1.26]
33 This greatly simplifies the calibration, where the checkerboard pose is estimated first, followed by the esti- mation of sphere parameters. [sent-98, score-1.001]
34 Other imaging systems such as flat refractive systems [1] have also been shown to be axial. [sent-99, score-0.416]
35 We build upon the algorithm proposed in [1] to compute the axes and checkerboard pose. [sent-100, score-0.5]
36 Off-Axis Systems: Off-axis mirror calibration is harder than calibrating axial systems. [sent-101, score-0.749]
37 Multi-Axial Cameras Consider the multi-axial imaging systems, where a perspective camera observes the scene reflected via n spherical mirrors, or refracted via n refractive spheres. [sent-111, score-0.679]
38 Let C(i)in=1 be the center of the ith sphere with radius ri. [sent-113, score-0.579]
39 We assume that the internal camera calibration has been done offline and hence we know the camera ray v(i, j) for each 3D point P(i) and each mirror j. [sent-118, score-0.795]
40 Our goal is to compute the center and radius of each sphere, as well as the unknown pose of the calibration grid given nK 3D-2D correspondences. [sent-119, score-0.554]
41 As discussed, we use the term ‘sphere’ to describe both the spherical mirror and the refractive sphere. [sent-120, score-0.683]
42 The estimation of axes and checkerboard pose is identical for both cases, and they differ only in the estimation of sphere radii and locations. [sent-121, score-1.214]
43 Axial and Multi-Axial Geometry: It is well-known that a single sphere corresponds to an axial imaging system [2]. [sent-122, score-0.723]
44 be the distance of the ith sphere center from the CO=P. [sent-125, score-0.459]
45 x bise Athei d=i sCtain/cdei oisf dtheefin ied as the normalized vector joining the sphere center to the COP. [sent-129, score-0.459]
46 [1] derived the coplanarity constraint for axial cameras considering flat refractive geometry. [sent-132, score-0.609]
47 Since each sphere also corresponds to an axial camera, the coplanarity constraint can be used to compute the axis Ai of each sphere. [sent-133, score-0.842]
48 Next, we show that the translation ambiguity along any individual axis can be removed by using rays from two or more spheres, since two or more axes will not be parallel in general. [sent-166, score-0.435]
49 Recovering the Checkerboard Pose A brute-force approach for calibrating multi-axial system would be to calibrate each sphere separately, as an axial system. [sent-169, score-0.764]
50 Note that the checkerboard pose in the camera coordinate system is identical with respect to all the spheres. [sent-170, score-0.67]
51 In addition, calibrating each sphere separately leads to a more difficult and computationally expensive algorithm. [sent-173, score-0.492]
52 Thus, there remains three calibration parameters for each mirror: (a) mirror radius rj, (b) mirror distance dj and (c) tA (j). [sent-179, score-0.892]
53 The pose can be obtained in (a) linear fashion, and (b) independently of sphere radii and distances. [sent-184, score-0.67]
54 Secondly, the full pose can be recovered linearly without estimating sphere parameters, thus enabling pose estimation in a semi-calibrated setting. [sent-187, score-0.687]
55 Finally, since the translation ambiguity is resolved, two parameters remain for each sphere × × (radius and distance) instead of three. [sent-188, score-0.576]
56 Solving for sphere radius rj and distance dj along with tA (j) proved to be difficult and we were not able to find an analytical solution to estimate rj, dj and tA (j) simultaneously. [sent-190, score-0.679]
57 Linear Estimation of Pose We assume that the axes of spheres can be estimated as explained in Section 3. [sent-193, score-0.421]
58 If we combine the coplanarity constraints from two mirrors j and k whose axes are not parallel, Kt has rank 3 and G has rank 5. [sent-215, score-0.387]
59 Using a second sphere k, we get another constraint t = sk + βAk. [sent-224, score-0.427]
60 Let Q(i) = RP(i) + t be the transformed checkerboard × points in the camera coordinate system. [sent-227, score-0.526]
61 Solving for Spherical Mirror Parameters Now we consider the mirror case and describe how to solve for the sphere radius and distance assuming known checkerboard pose and axes. [sent-231, score-1.311]
62 Estimation of sphere radius r and distance d for spherical mirror (left) and refractive ball (right) assuming known checkerboard pose and mirror axis. [sent-240, score-2.048]
63 Referring to Figure 2, the spherical mirror is represented as a circle on π with center at S = [0, d] and radius r. [sent-244, score-0.577]
64 Let M = kw be the common point on the mirror and the camera ray for some scalar k. [sent-245, score-0.464]
65 Known Radius: In practice, a good initial estimate of mirror radius may be known. [sent-262, score-0.383]
66 Non-linear refinement Non-linear refinement can be done by minimizing the image re-projection error for all checkerboard points. [sent-267, score-0.516]
67 The projection of a 3D point can be obtained by solving a 4th degree equation for a spherical mirror as shown in [2]. [sent-268, score-0.502]
68 L,je)t b pe(i t,h je) breo tehcecorresponding detected checkerboard corner in the image. [sent-272, score-0.382]
69 Error in sphere locations and sphere radius after non-linear refinement for all four mirrors. [sent-307, score-1.012]
70 We use a realistic scenario with four spherical × mirrors of radius 0. [sent-310, score-0.441]
71 After estimating the axes and pose, the mirror parameters are estimated, followed by non-linear refinement. [sent-322, score-0.381]
72 Figure 5 shows the error in sphere locations and radii for all four mirrors, as a percentage of the true values. [sent-324, score-0.609]
73 Multiple Refractive Spheres Now we consider a camera looking through multiple refractive spheres (ball lenses) as shown in Figure 8. [sent-328, score-0.689]
74 Similar to mirrors, the axis is defined as the vector joining the camera center to the center of the refractive sphere. [sent-329, score-0.537]
75 The calibration parameters consist of centers, radii and refractive indices of the balls along with the unknown checkerboard pose. [sent-330, score-1.069]
76 1, we can compute the checkerboard pose and axis of each refractive ball in exactly the same manner using the multi-axial property. [sent-333, score-0.911]
77 Let M1 = kw be the common point on the refractive ball and the camera ray w. [sent-339, score-0.513]
78 We therefore assume that the radius of refractive ball (and hence β) is known a-priori2 and solve for the single unknown d by solving the above derived 12th degree equation. [sent-360, score-0.514]
79 The projection of a 3D point can be obtained by solving a 10th degree equation for refractive sphere as shown in [2]. [sent-362, score-0.762]
80 Simulation: We use similar settings as in Section 6 to simulate a camera looking through four refractive spheres of radius 12. [sent-363, score-0.809]
81 Neither the spherical array nor the checkerboard is fronto-parallel to the imaging plane. [sent-373, score-0.608]
82 Figure 6 and Figure 7 shows the error in estimated axes, sphere distances and checkerboard pose using our algorithm after non-linear refinement, with Gaussian noise of increasing variance added to checkerboard corners. [sent-374, score-1.409]
83 For example, the maximum error in sphere distance is less than 0. [sent-376, score-0.485]
84 6ersa8um1Mean Error in Estmiated Axsi ing known sphere radii. [sent-388, score-0.427]
85 Simulation results for multiple refractive spheres assuming known sphere radii. [sent-401, score-0.947]
86 Figure 8 also shows the detected corners (red) and re-projected corners as well as the sphere boundary (green) using the estimated calibration parameters. [sent-414, score-0.746]
87 Spherical Mirrors: In Figure 9, the camera looks at the scene reflected from four spherical mirrors4 of radius 25. [sent-417, score-0.443]
88 For the case of reflection, the camera does not view the checkerboard directly. [sent-420, score-0.499]
89 To obtain the ground truth checkerboard pose, we need to estimate its extrinsic without a direct view [21, 10]. [sent-421, score-0.41]
90 [25], which requires capturing three images of the checkerboard using a planar mirror in different orientations. [sent-423, score-0.645]
91 The algorithm in [25] estimates the planar mirror poses as well as the checkerboard pose. [sent-424, score-0.645]
92 1 1 14 4 40 0 024 2 and re-projected checkerboard points/sphere boundary (green) using estimated calibration parameters are shown. [sent-428, score-0.637]
93 Comparison of checkerboard pose obtained using our method with ground truth (GT) pose for Figure 8. [sent-448, score-0.62]
94 Figure 8 also shows the detected corners (red) and the re-projected corners as well as the sphere boundary (green) using the estimated calibration parameters. [sent-463, score-0.746]
95 Notice that the estimated contour matches well with the actual sphere boundary. [sent-464, score-0.49]
96 Table 3 shows the estimated checkerboard pose along with the ground truth pose estimated using [25]. [sent-475, score-0.702]
97 Captured photo with detected (red) and re-projected checkerboard points (green) after calibration. [sent-484, score-0.414]
98 Discussions and Conclusions We have presented a single image based calibration approach for multi-axial systems, consisting of a camera looking at multiple spherical mirrors or multiple refractive spheres. [sent-517, score-0.962]
99 We believe that ours is the first algorithm that allows calibrating multiple refractive spheres based imaging system using a single photo. [sent-518, score-0.674]
100 We also showed an interesting theoretical result that full 6-DOF pose estimation of such multi-axial systems can be done in a semi-calibrated setting (without the knowledge of complete calibration), by choosing rays from two or more of the underlying axial systems. [sent-519, score-0.465]
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