Geometric Interpretations of the Relation between the Image of the Absolute Conic and Sphere Images

Transcription

1 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 8, NO. 1, DECEMBER Geometric Interpretations of the Relation between the Image of the Absolute Conic and Sphere Images Xianghua Ying and Hongbin Zha Abstract A spherical object has been introduced into camera calibration for several years through utilizing the properties of an image conic, which is the projection of the occluding contour of a sphere in the perspective image. However, in literature, only an algebraic interpretation was presented for the relation between the image of the absolute conic and sphere images. In this paper, we propose two geometric interpretations of this relation and two novel camera calibration methods using sphere images are derived from these geometric interpretations. Index Terms Camera calibration, geometric interpretation, sphere image, image of the absolute conic, double-contact theorem. 1 INTRODUCTION Ç CAMERA calibration is a process of modeling the mapping between D objects and their D images, and this process is often required when recovering D information from D images. The parameters of a camera to be calibrated are divided into two classes: intrinsic and extrinsic. The intrinsic parameters describe the camera s imaging geometric characteristics, and the extrinsic parameters represent the camera s orientation and position with respect to the world coordinate system. Many approaches to camera calibration have been proposed and they can be classified into two categories: using calibration objects [8], [18], [0], [15], [1], [4], [1], [16], [19], and self-calibration [11], [7], [17], [1]. As we know, the occluding contour of a sphere is projected to a conic in the perspective image [1], [4], [8, p. 190], [16], [19]. The reason is as follows: The occluding contour of a sphere seen from any direction is always a circle. The circle and the camera center form a right circular cone. The line passing through the camera and sphere s centers is the revolution axis of the right circular cone. The image of the occluding contour of a sphere is obtained by intersecting the right circular cone with the image plane. It is clear that this is a classical conic section, as illustrated in Fig. 1. The image conic of a sphere is called a sphere image in this paper. Existing methods for camera calibration using sphere images may be broadly divided into the following two categories from a mathematical viewpoint: 1. Using the properties of the principal axes of sphere images. Daucher et al. [4] found that the major axis of a sphere image passes through the principal point. Based on this observation, they further proposed to first determine the aspect ratio using three sphere images, then determine the principal point, and, finally, determine the focal length. Note that this method can only recover four intrinsic parameters while assuming the skew factor equal to zero. A similar strategy is adopted in [], []. Recently, a geometric invariant-based method using sphere images proposed in [19] (this method was originally proposed for. The authors are with the National Laboratory on Machine Perception, Peking University, Beijing , China. {xhying, zha}@cis.pku.edu.cn. Manuscript received Sept. 005; revised 7 Apr. 006; accepted 4 May 006; published online 1 Oct Recommended for acceptance by M. Pollefeys. For information on obtaining reprints of this article, please send to: tpami@computer.org, and reference IEEECS Log Number TPAMI catadioptric camera calibration) directly gave two constraint equations in the intrinsic parameters arising from one sphere image. Therefore, three sphere images may be used to recover all the five intrinsic parameters with nonlinear optimization techniques provided good initial guesses.. Using the relation with the image of the absolute conic. The image of the absolute conic (IAC) plays a central role in camera calibration. Teramoto and Xu [16] first discovered the algebraic relation between the sphere image and the IAC and then provided an efficient algorithm to solve for the camera parameters. However, in their approach, the minimization is accomplished by means of a generalpurpose nonlinear minimization and requires a good initial estimation to start the minimization. Agrawal and Davis [1] utilized the dual representation instead, i.e., the algebraic relation between the dual form of a sphere image and the dual image of the absolute conic (DIAC), and then employed semidefinite programming (SDP) to solve for the intrinsic parameters without requiring initial estimations. However, the geometric interpretations of the relation between sphere images and the IAC are as yet undiscovered in the literature. The overlooking of these geometric interpretations may be due to the fact that this relation is not evident from the geometric viewpoint. The main contribution of this paper is that we discovered that these geometric interpretations give us new insights into the fundamental properties of sphere images and two linear calibration methods using sphere images are derived from these geometric interpretations. The first geometric interpretation is that each sphere image is double-contact with the IAC and, after finding six double-contact points from three sphere images, we may determine the IAC using some conic fitting method. The second geometric interpretation is that three sphere images and the IAC satisfy the double-contact theorem, and the IAC is directly determined from the double-contact theorem which does not require finding the double-contact points. Although the calibration problem may be solved by SDP without initial estimations [1], the novel methods proposed in this paper seem more straightforward and are an order of magnitude faster than that using SDP. This paper is organized as follows: Section briefly introduces some notations and basic principles. An algebraic interpretation of the relation between the sphere image and the IAC is introduced in Section. In Section 4, two geometric interpretations are proposed, and two novel camera calibration algorithms are derived from these interpretations. Experimental results are shown in Section 5. Finally, Section 6 presents some concluding remarks. NOTATIONS AND BASIC PRINCIPLES.1 Pinhole Camera Model Let M ¼ðX; Y ; Z; 1Þ T be a world point and m ¼ðu; v; 1Þ T be its image point, both in the homogeneous coordinates. They satisfy m ¼ PM; where P is a 4 projection matrix describing the perspective projection process. is an unknown scale factor. The projection matrix can be decomposed as where ð1þ P ¼ KRjt ½ Š; ðþ f x s u 0 K ¼ 4 0 f y v 0 5: ðþ /06/$0.00 ß 006 IEEE Published by the IEEE Computer Society

2 0 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 8, NO. 1, DECEMBER 006 where is an unknown scale factor and the image conic C is a sphere image obtained from Q.. The IAC and the DIAC The absolute conic 1 is a conic with purely imaginary points on the plane at infinity 1 ¼ð0; 0; 0; 1Þ T, and its matrix form is ¼ : ð1þ Even though 1 does not have any real points, it shares the properties of any conic such as that a line intersects a conic in two points, the pole-polar relationship, etc. Here are a few particular properties of 1 [8, p. 6]: Fig. 1. A sphere image C obtained from a right circular cone Q and the circle C is an occluding contour of a sphere. Here, the upper triangular matrix K is the intrinsic parameter matrix, and ðr; tþ denotes a rigid transformation (i.e., R is a rotation matrix and t is a translation vector) which indicates the orientation and position of the camera with respect to the world coordinate system.. The Equation of a Sphere Image Let the origin of the world coordinate system be located in the vertex of a right circular cone Q, and the z-axis of the world coordinate system coincide with the revolution axis of the right cone. Then, the right cone Q represented in the world coordinate system is Q ¼ ; ð4þ where ¼ tanð=þ, and is the apex angle of the cone. A D point M ¼ðX; Y ; Z; 1Þ T on the cone Q satisfies or M T QM ¼ 0 M T QM ¼ 0; where M ¼ðX; Y ; ZÞ T are the inhomogeneous coordinates of the world point, and Q ¼ : ð7þ 0 0 From Fig. 1, we know the vertex of the cone Q is located in the camera s optical center. Therefore, only rotation exists between the world coordinate system and the camera coordinate system, i.e., t ¼ ð0; 0; 0Þ T. Then, from (1) and (), the image of a world point M on the cone Q satisfies m ¼ PM ¼ KRj0 ½ ŠM ¼ KRM: ð8þ Since KR is invertible, we have M ¼ R 1 K 1 m: Substituting (9) into (6), we obtain or m T K T R T QR 1 K 1 m ¼ 0 C ¼ K T R T QR 1 K 1 ; ð5þ ð6þ ð9þ ð10þ ð11þ 1. All circles intersect 1 in two points. Suppose that the support plane of the circle is, then intersects 1 in a line, and this line intersects 1 in two points. These two points are the circular points of.. All spheres intersect 1 in 1. The mapping between 1 and its perspective image is given by the planar homography H ¼ KR. Since the absolute conic 1 is on 1, one may compute the image of the absolute conic under H as:! ¼ H T 1 H 1 ¼ ðkrþ T IKR ð Þ 1 ¼ K T K 1 : ð1þ Here are a few remarks for the image of the absolute conic [8, pp ]: 1. The image of the absolute conic,!, depends only on the intrinsic parameter of the camera; it does not depend on the camera orientation or position.. The images of the circular points of a plane lie on!, and the two image points are the intersection of the vanishing line of the plane and!. The dual of the absolute conic 1 is a degenerate dual imaginary quadric in -space called the absolute dual quadric and denoted as [8, p. 65]: Q 1 ¼ : ð14þ We may define the dual image of the absolute conic as:! ¼! 1 ¼ KK T : ð15þ It is a dual (line) conic, whereas! is a point conic (though it contains no real points). The conic! is the image of Q 1 [8, p. 01]. ALGEBRAIC INTERPRETATIONS.1 The Algebraic Relation between a Sphere Image and the IAC Expand the right side of (11) using Q ¼ þ ð16þ ð1 þ Þ and, after some manipulations, we obtain where! is the IAC, and r is the third column of R. C ¼! vv T ; p v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ K T r ; ð17þ ð18þ

3 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 8, NO. 1, DECEMBER Fig.. The geometric interpretation of the relation between a sphere image C and the IAC!. Note that the figure is illustrated on the complex projective plane. For visualization purposes of the pole-polar relationship between m C and l, m C is shown located outside C rather than inside on the real projective plane.. The Algebraic Relation between a Sphere Image and the DIAC Inverse both sides of (11) and, after some manipulations as in (16), we obtain 0 C ¼! v 0 v 0T ; ð19þ where 0 is an unknown scale factor and C is the inversion of the conic C, i.e., the dual conic.! is the DIAC and p v 0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1= Kr : ð0þ By means of (8), we obtain that the projection of the sphere s center (i.e., the projection of the resolution axis of the right cone), m C, satisfies From (0) and (1), we obtain m C ¼ KRð0; 0; 1Þ T ¼ Kr : m C / v 0 ; ð1þ ðþ where / indicates equality up to a nonzero scale factor. From (17) and (19), it is not difficult to find that the two equations have the same mathematical form, no matter whether the dual representation is adopted or not. In the rest of paper, we only discuss the geometric interpretation for! and how to determine! from this geometric interpretation because! can be interpreted and determined in the same way. 4 GEOMETRIC INTERPRETATIONS 4.1 Geometric Interpretation of the Relation between the IAC and a Sphere Image Equation (17) can be rewritten as C! ¼ vv T : ðþ Since the rank of the matrix vv T is one, the rank of the matrix C! is one too. Consider the pencil of two conics S 1 and S ; S 1 þ S represents a conic which passes through all the common points of S 1 and S [14, p. 156]. Since two coincident lines (i.e., a repeated line) can be seen as a degenerate conic with rank 1, from the properties of a pencil of two conics described in [14, pp ], we know that C is tangent to! at two image points, i.e., two double-contact points, m I and m J, and the line l / v (derived from ()) passes through the two tangent points, m I and m J (see Fig. ). From (17), (1), (18), and (0), we can obtain: Cv 0 /ð! vv T Þv 0 / v: ð4þ It means that the line l / v and the point m C / v 0 (as defined in ()) satisfy the pole-polar relationship with respect to C, i.e., l is the polar line of m C and m C is the pole of l [14, p. 108]. From the definition described in (1) and (), m C can be seen as the image of the center of a circle C, which is an occluding contour of a sphere (as shown in Fig. 1). Since the center of a circle and the line at infinity on the support plane of the circle satisfy the pole-polar relationship with respect to the circle, l is the vanishing line of the support plane of C and the two tangent points, m I and m J, are the images of the circular points on the support plane. Similar results can be obtained for C and!. Based on the previous analysis, we obtain: Proposition 1. A sphere image and the IAC are mutually tangential at two double-contact image points, and the two tangent points are the images of the two circular points on the support plane for the occluding contour of the sphere. Proposition. The dual of a sphere image and the DIAC are mutually tangential at two double-contact image points. As we know, for any circle, it passes through the two circular points lying on the absolute conic. There are four intersection points (real and imaginary) of two coplanar conics [14]. Therefore, two among the four intersection points of the image of the circle and the IAC are the images of the two circular points, and the other two intersection points are undetermined. In Proposition 1, we claim that if and only if the optical center of the camera and the circle form a right circular cone, e.g., the occluding contour of a sphere, the image of the circle is double-contact with the IAC at the images of the two circular points. There exists a simple proof for Proposition 1: On the one hand, the intersection of the right cone Q represented in the world coordinate system and the plane at infinity is a conic with equation equal to Q as defined in (7). This conic has two double-contact points with the absolute conic, and this property is a projective invariant. On the other hand, the support plane intersects the right cone at a circle. As we know, any circle passes through the circular points, and the circular points lie on the absolute conic. Obviously, the image of the conic with equation Q on the plane at infinity and the image of the circle on the support plane are both identical with that of the sphere. Proposition 1 may also be interpreted by the properties of the images of two concentric circles [10]. Here, we consider the conic with equation Q on the plane at infinity and the absolute conic as two concentric circles. Note that, generally speaking, the IAC is not a circle. Kanatani and Liu [9] gave a geometric interpretation of the image of a circle, but did not relate it to the IAC. Due to lack of space, detailed discussions are not shown here. 4. Determining the IAC with the Double-Contact Points For each sphere image C, its corresponding line l / v can be determined (how to find v up to an unknown scale factor has been given in [1], i.e., the norm of v, kvk, is as yet unknown). Therefore, the two intersection points of the obtained line l and the conic C, i.e., m I and m J, may be determined. From Proposition 1, we know the two intersection points m I and m J should also lie on the IAC!. Therefore, six points on! may be obtained from three sphere images. Since five points are required to determine a conic,! may be computed from these six points. After obtaining!, it is not difficult to determine v, i.e., kvk from (). 4. The Calibration Algorithm by Finding the Double-Contact Points The complete calibration algorithm by finding the double-contact points consists of the following steps: 1. Fit conic curves, then obtain C i.. Find l i / v i.. Find the intersection points of C i and l i, then determine! and obtain K using the Cholesky factorization. 4. Obtain v i, then solve for the extrinsic parameters of the camera (see [16] and [1] for details).

4 04 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 8, NO. 1, DECEMBER 006 purely imaginary points, but it shares the properties of any conic, such as the double-contact theorem. Therefore, three sphere images and the IAC may be interpreted by the double-contact theorem. An example of three sphere images is shown in Fig. 4a. Three pairs of common chords of each two of the three sphere images are drawn in Fig. 4b. Though the intersection points of each two sphere images are all imaginary points in Fig. 4a, the common chords may be real lines shown as in Fig. 4b. Note that the IAC is not shown in Fig. 4 since it cannot be drawn in the real plane. 4.6 Determining the IAC Using the Double-Contact Theorem Given three sphere images C i ði ¼ 1; ; Þ, from (5), we can determine the IAC! as:! / 1 ð4kc 1 þðm þ M M 1 Þ þ 4kC þðm þ M 1 M Þ þ 4kC þðm 1 þ M M Þ Þ: ð6þ Fig.. Geometry for the double-contact theorem. 4.4 The Double-Contact Theorem From the double-contact theorem [5, pp. 18-], we know that if three conics C 1, C, and C all have double-contact with another conic!, then each two of C 1, C, and C have a distinguished pair of opposite common chords (shown as solid lines in Fig. ), and the three such pairs of common chords are the pairs of opposite sides of a complete quadrangle. Let L 1 and M 1 be a pair of opposite common chords of C and C, L and M be a pair of opposite common chords of C 1 and C, and L and M be a pair of opposite common chords of C 1 and C. We assume that L 1, L, and L are concurrent. Then, from the double-contact theorem, we have,! / 4kC 1 þðm þ M M 1 Þ 4kC þðm þ M 1 M Þ 4kC þðm 1 þ M M Þ ; ð5þ where the notation, the square of a -vector x, means x ¼ xx T, and k is a scale factor which can be determined from L 1, L, L, M 1, M, and M (see [5, p. 19, (6), (7), and (8)] for details). 4.5 Geometric Interpretation of the Relation between the IAC and Three Sphere Images Using the Double-Contact Theorem From discussions in Section 4.1, we know that each sphere image is tangent to the IAC at two double-contact points. The IAC has only Note that we do not required finding the double-contact points here. 4.7 The Calibration Algorithm Using the Double-Contact Theorem The complete calibration algorithm using the double-contact theorem consists of the following steps: 1. Fit conic curves, then obtain C i.. Determine! using the double-contact theorem.. Obtain K using the Cholesky factorization of!. 4. Obtain v i, then solve for the extrinsic parameters of the camera. (See [16] and [1] for details.) 4.8 Singularities Like almost any algorithm, these two calibration algorithms have singularities. In practice, it is important to be aware of the singularities in order to obtain reliable results by avoiding them. As stated in Section 4.1, each sphere image provides two double-contact points on the IAC. However, if two sphere images are double-contact with the IAC at the same two points, there are only two constraints on the IAC obtained from the two sphere images, not four in general cases. It is not difficult to prove that if the axes of the two right circular cones corresponding to the two sphere images are coincident, it is degenerate. In this degenerate case, the two common chords of the two sphere images are coincident, not distinct, and the repeated common chord is the vanishing line of the two support planes corresponding to the two spheres (since the two support planes are parallel, they have the same vanishing line). Moreover, the double-contact theorem Fig. 4. (a) A simulated image containing three sphere images. (b) Three pairs of opposite common chords of each two of these three sphere images are shown as solid lines. Obviously, these six chords form a complete quadrangle. (c) Obtained by zooming in on (b). Note that only three common chords are visible in(c).

5 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 8, NO. 1, DECEMBER Fig. 5. The estimated results of simulated experiments. See text for details. requires that the common chords must be distinct. Therefore, this case is also degenerate if the double-contact theorem is used. 5 EXPERIMENTS We perform a number of experiments, both simulated and real, to test our algorithms with respect to noise sensitivity and make comparisons with the following algorithms:. DCP and DDCP: By finding the double-contact points with the IAC and the DIAC, respectively.. DCT and DDCT: Using the double-contact theorem related to the IAC and the DIAC, respectively.. SDP and DSDP: Employing semidefinite programming with the representation of the IAC and the DIAC, respectively [1]. 5.1 Calibration with Simulated Data The simulated camera has the following parameters: f x ¼ 1;00, f y ¼ 1;000, s ¼ 0, u 0 ¼ 400, and v 0 ¼ 00. The resolution of the simulated image is We generate an image containing three sphere images uniformly distributed within the image as shown in Fig. 4a. On each sphere image, we choose 100 points. Gaussian noise with zero-mean and standard deviation is added to these image points. We vary the noise level from 0 to pixels. The conic fitting algorithm presented in [6] is used here. For each noise level, we perform 1,000 independent trials, and the mean values and standard deviations of these recovered parameters are computed over each run. The estimated results of these experiments are shown in Fig. 5. Since the performances of f x and f y and u 0 and v 0 are both very similar, the estimated results for f y and v 0 are not shown here. From Fig. 5, it is not difficult to find that the estimated results from SDP and DSDP are almost identical to each other. In fact, there are only very small differences among the estimated results from these six different methods. We compare the runtimes of these methods using MATLAB implementations of all algorithms on a 1.7 GHz Pentium IV processor. Note that realtime performance is not expected for any of the algorithms under MATLAB, and our only goal is to provide comparison. All results are averaged over 1,000 trials and recorded in Table 1. Since SDP is a convex optimization problem and has polynomial worst-case complexity, the runtimes of SDP and DSDP are about 10 times slower than that using DCP, DDCP, DCT, and DDCT. However, the runtimes of DCP and DDCP are a little slower than that using DCT and DDCT. The reason for this seems to be that the doublecontact points are not required to be found in DCT and DDCT. 5. Calibration with Real Data The test sphere for the real experiments is a billiard ball. The ball was placed in front of a white screen. We took images of the ball using a Sony DSC-F717 digital camera. Three sphere images are taken for calibration purposes. One of the three images is shown in Fig. 6a. The resolution of these images is Edges were extracted using Canny s edge detector and the ellipses were obtained using a least squares ellipse fitting algorithm [6]. In order to obtain unbiased results, these sphere images should be uniformly distributed within the image. The extracted edges of the contours from the three real sphere images are shown in Fig. 6b. The ground truths for the camera parameters are not known, but the approach in [0] is applied before the experiments using a calibration pattern which serves as a reference. The calibration results with real data are listed in Table. From Table, one may find that the calibration results using these six methods are similar to one another. TABLE 1 Runtimes (in Milliseconds) for the Six Algorithms

6 06 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 8, NO. 1, DECEMBER 006 Fig. 6. (a) A real image used in the real experiments. (b) Conic fitting results from three sphere images. 6 CONCLUSIONS In this paper, two geometric interpretations are proposed for the relation between sphere images and the IAC and also the relation between the duals of sphere images and the DIAC. For one sphere image, the geometric interpretation is that the sphere image is tangent to the IAC at two double-contact image points, and the two tangent points are the images of the two circular points on the support plane for the occluding contour of the sphere. The images of the two circular points may be determined by intersections of the vanishing line of the support plane and the sphere image, while the vanishing line may be obtained from the image of the sphere s center by using the pole-polar relationship with respect to the sphere image. For three sphere images, the geometric interpretation is that the three sphere images and the IAC satisfy the double-contact theorem, and the IAC can be directly determined from the double-contact theorem. These two geometric interpretations proposed in this paper provide new insights into the fundamental properties of sphere images, especially from the aspect of their providing constraints on the camera parameters, i.e., the reasons why the sphere images can be used for camera calibration from the geometric viewpoint. Two linear calibration approaches using sphere images are derived from these geometric interpretations, while the previous ones are all nonlinear. Only three sphere images are required, and all five intrinsic parameters are recovered linearly without making assumptions, such as, zeroskew or unitary aspect ratio. The minimum number for the previous nonlinear optimization methods is also three. Extensive experiments on simulated and real data were performed and shown that the two calibration methods are an order of magnitude faster than the previous nonlinear optimization methods while maintaining comparable accuracy. Although we have studied singularities of the proposed algorithms, a more thorough investigation may need to be pursued. TABLE Calibration Results with Real Data ACKNOWLEDGMENTS The authors would like to thank the anonymous reviewers for their constructive comments, which have contributed to a vast improvement of the paper. This work was supported in part by an NSFC Grant (No ) and the NKBRPC (No. 004CB18000, and No. 006CB0100). REFERENCES [1] M. Agrawal and L.S. Davis, Camera Calibration Using Spheres: A Semi- Definite Programming Approach, Proc. Ninth Int l Conf. Computer Vision, pp , 00. [] P. Beardsley, D. Murray, and A. Zisserman, Camera Calibration Using Multiple Images, Proc. Second European Conf. Computer Vision, pp. 1-0, 199. [] K. Daniilidis and J. 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