Geometry of Single Axis Motions Using Conic Fitting

Transcription

1 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 25, NO., OCTOBER Geometry of Single Axis Motions Using Conic Fitting Guang Jiang, Hung-tat Tsui, Member, IEEE, Long Quan, Senior Member, IEEE, and Andrew Zisserman Abstract Previous algorithms for recovering 3D geometry from an uncalibrated image sequence of a single axis motion of unknown rotation angles are mainly based on the computation of two-view fundamental matrices and three-view trifocal tensors. In this paper, we propose three new methods that are based on fitting a conic locus to corresponding image points over multiple views. The main advantage is that determining only five parameters of a conic from one corresponding point over at least five views is simpler and more robust than determining a fundamental matrix from two views or a trifocal tensor from three views. It is shown that the geometry of single axis motion can be recovered either by computing one conic locus and one fundamental matrix or by computing at least two conic loci. A maximum likelihood solution based on this parametrization of the single axis motion is also described for optimal estimation using three or more loci. The experiments on real image sequences demonstrate the simplicity, accuracy, and robustness of the new methods. Index Terms Turntable, structure from motion, single axis motion, geometry, conic, fundamental matrix. INTRODUCTION æ ESTIMATING 3D models of objects from image sequences of single axis motion, particularly turntable sequences, has been widely studied by computer vision and graphics researchers. Generally, the whole reconstruction procedure includes the determination of camera positions at different viewpoints, the detection of object boundaries, and the extraction of surface models from the volume representation. The estimation of the camera positions or simply the rotation angles relative to a still camera is the most important and difficult part of the modeling process. Traditionally, rotation angles are obtained by careful calibration [3], [8], [9]. Fitzgibbon et al. [5] extended the single axis approach to recover unknown rotation angles from uncalibrated image sequence based on a projective geometry approach. It has been shown that 3D metric reconstruction up to a two-parameter family can be obtained from uncalibrated sequence with unknown rotation angles. However, fundamental matrices and/or trifocal tensors have to be computed for each pair of images or each triplet of images. Mendonça et al. [], Wong et al. [2] recovered the rotation angles from profiles of surfaces with a calibrated camera. The search for corresponding points is transformed into a search for epipolar tangencies. In the new algorithms, we try to improve the accuracy by avoiding the computation of twoview fundamental matrices, three-view trifocal tensors, and epipolar tangencies. Instead, we fit the corresponding points in the whole sequence to its conic locus. It is then shown that all single axis geometry can be directly computed either from one conic and one fundamental matrix or from at least two conics. The rotation angles. G. Jiang and H.-t. Tsui are with the Department of Electronic Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong. {gjiang, httsui}@ee.cuhk.edu.hk.. L. Quan is with the Department of Computer Science, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. quan@cs.ust.hk.. A. Zisserman is with the Department of Engineering Science, University of Oxford, Parks Road, Oxford OX 3PJ, UK. az@robots.ox.ac.uk. Manuscript received 9 Oct. 22; revised 3 May 23; accepted 25 May 23. Recommended for acceptance by Z. Zhang. For information on obtaining reprints of this article, please send to: tpami@computer.org, and reference IEEECS Log Number can then be calculated directly. The advantage of these new methods over the existing ones is significant. First, it is intrinsically a multiple view approach as all geometric information from the whole sequence is nicely summarized in the conics. This contrasts with the computation of fundamental matrices and trifocal tensors that use only a subsequence of two and three views, respectively. Second, as will be shown in Section 7, the essential single axis geometry may be specified by six parameters and this may be estimated from one conic and one fundamental matrix (a total of 2 parameters) or may be estimated from two conics (a total of parameters). Previous methods have involved estimating more than this number of parameters. For example, 8 tensor parameters from each triplet of images are required. Partial results of our techniques have been published in conferences [7], [8]. The paper is organized as follows: Section 2 gives a review of single axis motion and its associated fixed image entities. Section 3 presents an algorithm for estimating the geometry from one conic and one fundamental matrix. Section 4 presents an algorithm for estimating the geometry from two conics and the proof of unique solution from at least three conics is given in Section 5. An important practical issue of recovering rotation angles with missing points are described in Section 6. A Maximum Likelihood Estimation method for calculating these fixed image entities is given in Section 7. Section 8 describes the 3D reconstruction based on single axis motion. Experimental results are presented in Section 9 and, finally, a short conclusion is given in Section. 2 REVIEW OF SINGLE AXIS MOTION AND ITS ASSOCIATED FIXED IMAGE ENTITIES A typical set up for single axis motion consists of a stationary CCD camera pointing at an object to be modeled on a turntable [3]. The internal parameters of the camera are unknown but assumed fixed. The situation is equivalent to that of a fixed object with a camera rotating around it. It is a special case of the more general planar motion [], [4] as it restricts all motions to be generated from rotation around a single axis. Without loss of generality, following [5], the rotation axis of the turntable is chosen to be the z-axis of the world coordinates. Each point on the object is moving in a plane that is perpendicular to the z-axis and all these planes for different points form a pencil of horizontal parallel planes. These planes intersect at a line at infinity in 3D, which is imaged as the vanishing line of the planes and denoted as l. The line l is fixed in all the images. Since the image of the absolute conic! is fixed under rigid motion, the two intersections of the image of the absolute conic! with the line l, remain fixed in all images. Actually, these two fixed points are the images of the two circular points on the horizontal planes. The fixed image entities of a single axis motion are summarized as follows:. The line l s which is the image of the rotation axis. Note that l s is a line of fixed points for all images. 2. The line l which is the image of the vanishing line of all the horizontal planes. Unlike the image of rotation axis, l is a fixed line, but not a line of fixed points. 3. The images of these two circular points i and j, which are the intersection of the image of the absolute conic! with the image of the vanishing line l. 3 METHOD : SINGLE AXIS MOTION FROM ONE CONIC AND ONE FUNDAMENTAL MATRIX One of the key observations in this paper is that the trajectory of the corresponding points in different images of any given space point, visualized in any image plane, gives a conic locus by the very definition of single axis motion (Fig. ). This has been observed by Sawhney et al. [6], whose approach was developed within the traditional calibrated setting. Their primary goal was to recover the ego-motion parameters of the moving point in space with its conic /3/$7. ß 23 IEEE Published by the IEEE Computer Society

2 344 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 25, NO., OCTOBER 23 Fig.. Dinosaur sequence and conics. trajectory, but not to recover the camera motion or camera geometry. The intrinsic invariants for uncalibrated camera geometry have been developed more recently in [5]. Our new development is dependent, to a large extent, on the geometry of these invariants. As long as a point can be tracked over at least five images, a conic C can be fitted [4]. Useful geometric information is then extracted from the fitted conic. Our first method consists of an algorithm using one fitted conic and one fundamental matrix [2], [23]. The algorithm is dependent on the fixed image entities described in Section 2 and can be summarized as follows:. Fit a conic C to all tracked image points corresponding to an object point in an image sequence. Only one point is tracked over at least five images as we need only one conic at this stage. 2. Compute only one fundamental matrix F from any two views. Select two views at a large angle in the sequence while sharing enough corresponding points. 3. Compute the matrix F þ F T. In the case of single axis motion, this matrix will be a degenerate conic consisting two lines. One is the image of rotation axis l s and the other is the image of the vanishing line l [7], [5]. 4. Obtain the pole of the conic C with respect the line l : o ¼ C l : 5. Compute i and j as the intersections of the conic C with the line l. 6. Compute the angular motion between two views from the tracked points a and a on the conic using the Laguerre s formula [7]: ¼ 2i logðfl oa; l oa ; l oi ; l oj gþ. The computation of invariants using the fundamental matrix is the same as in Fitzgibbon et al. [5]. In their approach, the three-view trifocal tensor has to be computed to obtain the rotation angles between the three. In our method, one conic is used to replace the information contained in the trifocal tensors for all triplet of images. Further, the estimation of trifocal tensors is known to be difficult, while conic fitting is an accurate and robust procedure. 4 METHOD 2: SINGLE AXIS MOTION FROM TWO CONICS It is easy to prove the fact that the trajectory of a given object point in space from a single axis motion is always a circle and different points give circles of different radii on different horizontal planes. All circles go through their respective circular points of the plane they are lying on by the very definition of the circular points [7]. All these supporting planes are parallel and they intersect at a common line at infinity on which the circular points lie. We may thus conclude that all circles of different radii on different parallel planes from the single axis motion share the common pair of circular points. Because this is a projective property, it remains true for the projection onto any image plane. In any particular image plane, this means that all conic loci of corresponding points meet in the pair of common points i and j. Thus, by just computing the intersection points of at least two conic loci, we can obtain the images of the circular points and the image of the vanishing line of the parallel planes. Consider the intersection of a pair of conics. There are always four intersection points, including complex and infinite points according to Bezout s theorem. In terms of real intersection points, they may be, 2, or 4 according to the configurations.. For the case of two real intersection points, it is straightforward that the only pair of complex conjugate points is the images of the circular points.. For the case of four real intersection points, it is generally impossible if the conics are a real perspective image of the circles from the single axis motion. There must be at least one pair of complex conjugate circular points if the geometry is correct.. The case of no real intersection point is a common case as illustrated by the two conic loci in Fig. 3. In this case, we obtain two pairs of complex conjugate points. There is a reconstruction ambiguity coming from the ambiguity of the two pairs of complex conjugate points. An investigation of this will be made in Section 5. For the other cases, once we have obtained the images of the pair of circular points as described above, the determination of the image of the vanishing line is given by: l ¼ i j. The trajectory of any given object point in space from the single axis motion is a circle with its center lying on the rotation axis. Under projective geometry and within the uncalibrated framework, the pole of the conic locus with respect to the image of the vanishing line will lie on the image of the rotation axis. The poles of the conics C p and C q with respect to the image of the vanishing line l are simply given by o p ¼ C p l and o q ¼ C q l, respectively. The image of the rotation axis is just the image line going through these poles, i.e., l s ¼ o p o q. Up to this point, we have computed all invariant entities resulting from a single axis motion purely from two conic loci without using any multiple-view geometry constraints such as fundamental matrices and trifocal tensors as in the previous approaches [5]. To recover the rotation angles, we could use the Laguerre s formula as in the previous section. The whole geometry of the single axis motion is then obtained from at least two conics if there is only one pair of complex conjugate points. 5 UNIQUE SOLUTION FROM AT LEAST THREE CONICS As it was shown in the previous section that, using at least two conics may still lead to multiple solutions, if the two conics have no real intersection point. It is natural to introduce a third conic to remove this ambiguity. If we have three fitted conics, there are two possible situations. In one situation, there is one pair of common conjugate points in the intersections of three pairs of conics. In this case, these conjugate points are the images of the circular points. In the other possible situation, there are two common pairs of conjugate points in the intersections of three pairs of conics. If the three conics have two common pairs of complex conjugate intersection points, the solution is not unique for the selection of the image of the circular points. The uniqueness of the solution could not be established. If the three conics have four common points, they

3 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 25, NO., OCTOBER Fig. 2. Invariants deduced from one fundamental matrix and one conic. form a pencil of conics. However, all these conics from a circular motion are related by a special type of projective transformations called homologies, they do not form a pencil. Generally, a unique solution could be expected with at least three conics if these conics are not forming a critical configuration of pencil of conics with common complex conjugate points. What is the exact necessary condition leading to this critical configuration is still unknown to us. Fig. 4. Recovering rotation angles with missing points. In this matrix, the only unknown is the radius r. The image coordinates of the pole of the conic C i with respect to the line l is o ¼ H i ð; ; Þ T ¼ðrh 3 ;rh 23 ;rh 33 Þ T, or in inhomogeneous coordinates o ¼ðh 3 =h 33 ;h 23 =h 33 Þ T. The rotation angle 23 between the view m 2 and m 3 can be obtained as 23 ¼ 2i logðfl o b 2 ; l ob 3 ; l oi; l ojgþ. 6 RECOVERING ROTATION ANGLES WITH MISSING POINTS As mentioned in Section, the key component of the single axis motion is the one-dimensional rotation angles. In practice, some points are often missing in some views along a conic locus. The rotation angles for these views cannot be recovered directly from the fitted conics. In this section, we will discuss how these rotation angles could be recovered when some points cannot be tracked in some views. Fig. 4 illustrates this configuration to help the following development. The conic C corresponding to a circle in the space plane has been obtained from tracking a point over five or more images. The point a is visible in views m and m 2 as a and a 2, but missing in view m 3. So, the point a 3 is not available. However, a point b is available in all three views m, m 2, and m 3 as b, b 2, and b 3. Let us assume its unknown conic locus is the conic C i corresponding to a circle on the space plane i. As we know, the two planes and i are parallel in space, and C and C i will both intersect the image of the vanishing line l at the images of the circular points i and j in the image plane. The rotation angle between the view m and m 2 is known from the conic C. The same rotation angle also fits for the point b on the plane i : 2 ¼ 2i logðfl o b ; l ob 2 ; l oi; l ojgþ. Without loss of generality, let the coordinates of point b in space plane i be ðr; ; Þ T. Then, the coordinates of point b 2 are ðr cosð 2 Þ;rsinð 2 Þ; Þ T. The coordinates of the images of the circular points i and j in this space plane are ð;i;þ T and ð; i; Þ T, respectively. Since there are four pairs of corresponding points between the space plane i and the image plane, the projection matrix H i which maps points in space plane to points in image plane can be obtained. 7 METHOD 3: MAXIMUM LIKELIHOOD ESTIMATION WITH PARAMETRIZATION It has been shown in Section 5 that usually more than two conics have to be tracked along the image sequence to have a unique solution. In this section, a Maximum Likelihood Estimation (MLE) method is given for simultaneous estimation of the fixed geometric entities and n conics, where n 3. As mentioned earlier, each point on the object is moving along a circle on a horizontal plane. For each such plane, there is a plane homography H mapping the conic from the image plane to the circle on the supporting horizontal plane, C cirle ¼ H > C conic H. Without loss of generality, assume the radius of the circle to be unity and its center be at the origin of the plane coordinate frame. The homography can be decomposed into a concatenation of five matrices R, S, T, A, and P u, representing rotation, isotropic scaling, translation, affine, and pure projective transformations, respectively, [9], []: H ¼ RSTAP u. P u A l l 2 l 3 can be determined by the image of the vanishing line l ¼ðl ;l 2 ;l 3 Þ >. = = A A; where ð i; ; Þ T is the pair of the circular points in the affine plane which can be obtained by P u through the points i and j. The Fig. 3. The intersections of two conics.

4 346 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 25, NO., OCTOBER 23 TABLE Fixed Image Entities Computed from Three Different Methods Fig. 5. Recovered rotation angles for the whole sequence of 36 views. The dotted line with the circle mark is calculated from one fundamental matrix and one conic, the dash dot line with diamond mark is calculated from two conics, and the solid line with the square mark is calculated from MLE. images of the circular points count for 4 degrees of freedom: two for determining the pure projective matrix and two for determining the affine matrix. As shown in Fig. 2, the other fixed image entity is the image of the rotation axis l s, which has 2 degrees of freedom. t T t 2 A; where ðt ;t 2 ; Þ > is the pole of the line l with respect to the conic C n on the metric plane. Since the pole is constrained by the rotation axis l s, only degree of freedom exists in the translation matrix. After having applied the transformations by the matrices of P u, A, and T, the conic is back-projected into a circle with center at the original point. The isotropic scaling matrix s S s A scales the circle to a circle with radius unity. So, another degree of freedom is counted for each conic. Since the rotation matrix leaves the circle centered at the origin invariant, no free degree of freedom for the matrix R. In summary, there are in total 6 degrees of freedom for the fixed entities (4 for the images of the two circular point, 2 for l s ) and 2 for each conic (which correspond to where the center is along the rotation axis and its radius of the corresponding circle). The cost function for the MLE involves minimizing the sum of squared geometric distances (one for each of the m measured points) over all 6 þ 2n parameters for the fixed entities and P m d?ðx; CÞ 2. n conics: C¼ P n This is a classical conic fitting problem whose cost function might be highly nonlinear. One way to tackle this is to take a first order approximation as suggested in [2], [5], [6]. Let d? ðx; CÞ 2 ðx T CxÞ 2 ¼ 4ððCxÞ 2 þðcxþ2 2 Þ ; where ðcxþ i denotes the ith component of the 3-vector Cx. This cost function is minimized using the standard Levenberg- Marquart algorithm. The 6 þ 2n parameters are initialized as follows: First, each conic is fitted to corresponding points from at least five views [2], [5]. Then, the pole of each conic with respect to the image of the vanishing line is calculated and the point on the image of the rotation axis that is nearest to the pole is used to estimate the initial value of t ( degree of freedom). Now, each conic can be transformed into a circle. The radius of this circle determines the initial value of s ( degree of freedom). Finally, each conic is mapped to a unit circle centered at the origin and the points on the conic is mapped to the points near to the optimal unit circle. 8 3DRECONSTRUCTION AND METRIC AMBIGUITY Fitzgibbon et al. [5] have shown that the metric reconstruction can be obtained up to a two parameter family. This means that metric structures in all planes perpendicular to the rotation axis can be perfectly determined, the reconstruction along the rotation axis is only up to a D projective transformation. Let P be the projection matrix of the first view with the rotation axis as the z-axis of the world coordinates. Then, the projective Fig. 6. Two views of the 3D reconstructed dinosaur model from the single axis motion computed by the methods discussed in this paper.

5 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 25, NO., OCTOBER Fig. 7. (a) Three images captured from the video. (b) The video is a woman in a rotating chair. (c) A vertical line in the background is marked out for searching the vanishing point along the rotation axis. Fig. 8. Recovered rotation angles of 46 images selected from the video sequence. matrix of any view PðÞ is related to P by a rotation ðþ about the z-axis. In other word, PðÞ ¼PR 4 ðþ, where cos sin sin cos R 4 ðþ ¼B A : These projection matrices can be used for 3D reconstruction by Shape-from-Silhouettes methods [3], [3]. The projection matrix P can be parameterized by two fitted conics [7], where p p 2 h ðr k 3 þ r 2 k 32 Þ r k 4 P p 2 p 22 h ðr k 23 þ r 2 k 232 Þ r k 24 A: p 3 p 32 h ðr k 33 þ r 2 k 332 Þ r k 34 Parameters r, r 2, and h are the only unknown left in the matrix. These two conics are mapped from two circles on two different horizontal planes. r and r 2 are radii of these two circles and h is the distance between centers of the circles. Since we are interested only in a similarity reconstruction, there are two unknown parameters remain. As mentioned by Fitzgibbon et al. [5] and other researchers [2], [22], the two-parameter ambiguity may be further removed by specifying the camera aspect ratio and parallel scene lines. Fig. 7 shows three images of the sequence. Vertical lines in the background with the rotation axis (Fig. 7c) are used to locate the vanishing point v along the rotation axis. We also assume the camera aspect ratio is one [5], [7]. Fig. 8 shows the recovered angles and Fig. 9 the final reconstruction. Another example of the rotating chair sequence is a man s head. Fig. shows three images captured from the video. The reconstructed man s head is shown in Fig.. CONCLUSION We have presented three new methods of determining the geometry of single axis motions. The main idea is to use the fitted conic locus of corresponding points to estimate the motions. This makes the proposed methods more concise, robust, and accurate than the traditional ones which are basing on the estimation of 2-view fundamental matrix and 3-view trifocal tensor. The first method is based on the fitting of one conic and the estimation of one fundamental matrix. The second method is based on the fitting of at least two conics. The third one is a global MLE estimation from at least three conics using a minimal parametrization of the single axis motion. The key advantage of the new methods is that they are intrinsically multiple view approaches as either one, two, or three 9 EXPERIMENTAL RESULTS Dinosaur sequence from a turntable. We first tested our methods on the popular dinosaur sequence from University of Hannover. The sequence contains 36 views from a turntable with a constant angular motion of degrees. The angular accuracy is about.5 degrees [3]. Table lists the recovered invariant quantities computed from one conic and one fundamental matrix (denoted as F þ C), two conics (denoted as 2C), and more than two conics with MLE (denoted as nc), respectively. Fig. 5 displays the recovered rotation angles based on these invariant quantities. With a reasonable choice of these two unknown parameters in projection matrix, the reconstruction results are shown in Fig. 6. Sequences from a rotating chair. We captured a couple of video sequences of a student sitting on a normal rotating chair. The chair is rotated manually. Several marked points are used for tracking the conics. One example is a woman sitting in the chair. Fig. 9. Two views of the 3D reconstructed woman computed by the methods discussed in this paper.

6 348 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 25, NO., OCTOBER 23 Fig.. Three images captured from video. The video is a man in a rotating chair. Fig.. Four views of the 3D reconstructed man s head computed by the methods discussed in this paper. points are tracked over at least five images. In some of our experiments for the MLE method, we had fitted over conics using information from 36 images. This makes the method very robust. However, for each conic, the point was tracked over 5 to 5 views only. Using a small number of parameters, these methods give very robust computations as have been demonstrated in the examples using real images. ACKNOWLEDGMENTS This project is partially supported by the Hong Kong RGC Grant CUHK 4378/2E. REFERENCES [] M. Armstrong, A. Zisserman, and R. Hartley, Self-Calibration from Image Triplets, Proc. European Conf. Computer Vision, pp. 3-6, 996. [2] F. Bookstein, Fitting Conic Sections to Scattered Data, Computer Vision, Graphics and Image Processing, vol. 9, pp. 56-7, 979. [3] C.H. Chien and J.K. Aggarwal, Identification of 3D Objects from Multiple Silhouettes Using Quadtrees/Octrees, Computer Vision, Graphics, and Image Processing, vol. 36, pp , 986. [4] O.D. Faugeras, L. Quan, and P. Sturm, Self-Calibration of a D Projective Camera and Its Application to the Self-Calibration of a 2D Projective Camera, Proc. European Conf. Computer Vision, pp , 998. [5] A.W. Fitzgibbon, G. Cross, and A. Zisserman, Automatic 3D Model Construction for Turn-Table Sequences, Proc. European Workshop SMILE 98, pp. 55-7, 998. [6] R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision. Cambridge Univ. Press, 2. [7] G. Jiang, H.T. Tsui, L. Quan, and S.Q. Liu, Recovering the Geometry of Single Axis Motions by Conic Fitting, Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp , 2. [8] G. Jiang, H.T. Tsui, L. Quan, and A. Zisserman, Single Axis Geometry by Fitting Conics, Proc. European Conf. Computer Vision, pp , 22. [9] J.J. Koenderink and A.J. van Doorn, Affine Structure from Motion, J. Optical Soc. Am., A, vol. 8, no. 2, pp , 99. [] D. Liebowitz and A. Zisserman, Metric Rectification for Perspective Images of Planes, Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp , 998. [] P.R.S. Mendonça, K.-Y.K. Wong, and R. Cipolla, Epipolar Geometry from Profiles under Circular Motion, IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 23, no. 6, pp , June 2. [2] J. Mundy and A. Zisserman, Repeated Structures: Image Correspondence Constraints and Ambiguity of 3D Reconstruction, Applications of Invariance in Computer Vision, Springer Verlag, 994. [3] W. Niem, Robust and Fast Modelling of 3D Natural Objects from Multiple Views, Proc. SPIE. Image and Video Processing II, vol. 282, pp , 994. [4] L. Quan, Conic Reconstruction and Correspondence from Two Views, IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, pp. 5-6, 996. [5] P.D. Sampson, Fitting Conic Sections to Very Scattered Data: An Iterative Refinement of the Bookstein Algorithm, Computer Vision, Graphics, and Image Processing, vol. 8, no. 2, pp. 97-8, 982. [6] H.S. Sawhney, J. Oliensis, and A.R. Hanson, Image Description and 3D Reconstruction from Image Trajectories of Rotational Motion, IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 5, no. 9, pp , Sept [7] J. Semple and G. Kneebone, Algebraic Projective Geometry. Oxford Univ. Press, 952. [8] S. Sullivan and J. Ponce, Automatic Model Construction, Pose Estimation, and Object Recognition from Photographs Using Triangular Splines, IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 2, no., pp. 9-97, Oct [9] R. Szeliski, Shape from Rotation, Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp , 99. [2] K.-Y.K. Wong, P.R.S. Mendonça, and R. Cipolla, Structure and Motion Estimation from Apparent Contours Under Circular Motion, Image and Vision Computing, pp [2] Z. Zhang, R. Deriche, O. Faugeras, and Q.-T. Luong, A Robust Technique for Matching Two Uncalibrated Images through the Recovery of the Unknown Epipolar Geometry, Artificial Intelligence, vol. 78, pp. 87-9, 995. [22] A. Zisserman, D. Liebowitz, and M. Armstrong, Resolving Ambiguities in Auto-Calibration, Philosophical Trans. Royal Soc. London, A, vol. 356, no. 74, pp. 93-2, 998. [23] T. Vieville and D. Lingrand, Using Specific Displacements to Analyze Motion without Calibration, Int l J. Computer Vision, vol. 3, no., pp. 5-29, For more information on this or any other computing topic, please visit our Digital Library at