G. N. Newsam, D. Q. Huynh, M. J. Brooks, and H.-P. Pan. Recovering Unknown Focal Lengths in Self-Calibration: An Essentially Linear Algorithm and Degenerate Configurations. International Archives of Photogrammetry and Remote Sensing, 31(B3-III):575--580, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

...., of two cameras viewing the same scene. The algorithm first estimates the Fundamental matrix through eight point correspondences, and then uses Singular Value Decomposition and properties of rotation matrices to solve for the focal lengths [8] 6 There is however a simpler algorithm presented in [14] that also uses Singular Value Decomposition to solve for the two unknown focal lengths in a system. As in [8] the first step is to estimate the Fundamental matrix, F , through su#ciently many point correspondences [14] and then compute the Singular Value Decomposition and obtain the ....

....the focal lengths [8] 6 There is however a simpler algorithm presented in [14] that also uses Singular Value Decomposition to solve for the two unknown focal lengths in a system. As in [8] the first step is to estimate the Fundamental matrix, F , through su#ciently many point correspondences [14] and then compute the Singular Value Decomposition and obtain the eigenvectors u i . Using the characteristics of the Essential and Fundamental matrices and the translation of the camera shown in equations 15 through 17 E K #T FK (15) tt (16) t = K # 1 u 3 #K # 1 u 3 (17) the ....

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G. Newsam, D. Huynh, M. Brooks, and H. Pan. Recovering unknown focal lengths in self-calibration: An essentially linear algorithm and degenerate configurations. In ISPRS, volume 18, pages 575--580, 1996.

....adjustment, which usually also includes parameters that model deviations from the perfect pinhole assumption. Hartley has shown that the values of a varying focal length can already be estimated from two views of an unknown (static) scene [5] Simple algorithms for this purpose are proposed in [1, 7, 12]. Basically, given the fundamental matrix of the two views, one can derive a quadratic equation # This work was supported by the project IST 1999 10756, VISIRE. in any of the two values of the focal length. The main drawback is the existence of singularities, especially the case of coplanar ....

....quadratic equation # This work was supported by the project IST 1999 10756, VISIRE. in any of the two values of the focal length. The main drawback is the existence of singularities, especially the case of coplanar optical axes, which does not allow to compute the focal length, with any algorithm [12]. Brooks et al. have considered the case of identical focal length for the two views [2, 3] Their analysis is restricted to the case of a typical stereo system: the optical axes are perfectly coplanar and the relative motion between the two views is planar motion, the normal of the motion plane ....

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G.N. Newsam, D.Q. Huynh, M.J. Brooks and H.P. Pan. Recovering Unknown Focal Lengths in Self-Calibration: An Essentially Linear Algorithm and Degenerate Configurations. ISPRS-Congress, XXXI(B3):575-580, 1996.

....we study the case of a moving camera with variable and unknown focal length, but whose other intrinsic parameters are known. A practical self calibration algorithm was proposed by Azarbayejani and Pentland [1] Algorithms and closed form solutions for the two view case are given, e.g. in Refs. [3 5,8,12]. Newsam et al. derived the critical motions for the two view case [12] In this paper, we derive a complete characterization of critical motion sequences for any number of views and the critical motions for stereo systems. This paper is an extended version of [19] The paper is organized as ....

....but whose other intrinsic parameters are known. A practical self calibration algorithm was proposed by Azarbayejani and Pentland [1] Algorithms and closed form solutions for the two view case are given, e.g. in Refs. 3 5,8,12] Newsam et al. derived the critical motions for the two view case [12]. In this paper, we derive a complete characterization of critical motion sequences for any number of views and the critical motions for stereo systems. This paper is an extended version of [19] The paper is organized as follows. In Section 2 we provide some theoretical background for our ....

[Article contains additional citation context not shown here]

G.N. Newsam, D.Q. Huynh, M.J. Brooks, H.P. Pan, Recovering unknown focal lengths in self-calibration: an essentially linear algorithm and degenerate configurations. Part B3 of the proceedings of the XVIII ISPRS-Congress, Vienna, Austria, 1996, pp. 575-580.

....two perspective cameras, with unknown and possibly different focal lengths, but known other intrinsic parameters. The two focal lengths can in general be recovered from the epipolar geometry [5] but this is nearly always singular in practice, due to optical axes passing close 11 to each other [7]. The knowledge of a line at infinity in the projective reconstruction, however, can be used to overcome the singularity, as described in the following. Let M and M # be the 3 4 projection matrices of the two 2D views. We suppose that the known parts of the calibration matrices (containing ....

Newsam, G.N., Huynh, D.Q., Brooks, M.J., Pan, H.P.: Recovering Unknown Focal Lengths in Self-Calibration: An Essentially Linear Algorithm and Degenerate Configurations. XVIIIth ISPRS Congress Part B3 (1996) 575--580

....for pairs of views. An alternative is the method proposed in [10] which is based on a global projective reconstruction. In general, the focal length can be computed from a single pair of views, given the epipolar geometry. Our camera configuration, however, is degenerate for this problem [8]. For triplets of views, focal length estimation is no longer degenerate in general [14] For each triplet, it is possible to obtain 12 equations of degree 4 in the focal length, with coefficients depending on the 3 fundamental matrices. These equations can be solved individually and their ....

G.N. Newsam, D.Q. Huynh, M.J. Brooks, and H.P. Pan, "Recovering Unknown Focal Lengths in SelfCalibration: An Essentially Linear Algorithm and Degenerate Configurations," Proc. ISPRS-Congress, Vienna, Vol. XXXI, Part B3, pp. 575-580, 1996.

....adjustment, which usually also includes parameters that model deviations from the perfect pinhole assumption. Hartley has shown that the values of a varying focal length can already be estimated from two views of an unknown (static) scene [5] Simple algorithms for this purpose are proposed in [1, 7, 12]. Basically, given the fundamental matrix of the two views, one can derive a quadratic equation in any of the two values of the focal length. The main drawback is the existence of singularities, especially the case of coplanar optical axes, which does not allow to compute the focal length, with ....

....given the fundamental matrix of the two views, one can derive a quadratic equation in any of the two values of the focal length. The main drawback is the existence of singularities, especially the case of coplanar optical axes, which does not allow to compute the focal length, with any algorithm [12]. # This work was supported by the project IST 1999 10756, VISIRE. Brooks et al. have considered the case of identical focal length for the two views [2, 3] Their analysis is restricted to the case of a typical stereo system: the optical axes are perfectly coplanar and the relative motion ....

[Article contains additional citation context not shown here]

G.N. Newsam, D.Q. Huynh, M.J. Brooks and H.P. Pan. Recovering Unknown Focal Lengths in Self-Calibration: An Essentially Linear Algorithm and Degenerate Configurations. ISPRS-Congress, XXXI(B3):575-580, 1996.

....[x 0 e ] 8. 3 Reliability evaluation of 3 D reconstruction If the principal point (i.e. the point corresponding to the camera optical axis in the image) is at the center of the frame and there is no image skew, we can self calibrate the focal lengths of the two images by an analytical method [2, 5, 11, 19] and thereby reconstruct the 3 D Euclidean structure from two views. Using the images shown in Fig. 6, we reconstructed the 3 D Euclidean shape from the 60 manually selected feature points (not shown) by using 5 Figure 6: The reconstructed 3 D structure with uncertainty ellipsoids (stereogram) ....

G. N. Newsam, D. Q. Huynh, M. J. Brooks and H.-P. Pan, Recovering unknown focal lengths in selfcalibration: An essentially linear algorithm and degenerate configurations, Int. Arch. Photogram. Remote Sensing , 31-B3-III, July 1996, Vienna, Austria, pp. 575--580.

....for pairs of views. An alternative is the method proposed in [10] which is based on a global projective reconstruction. In general, the focal length can be computed from a single pair of views, given the epipolar geometry. Our camera configuration, however, is degenerate for this problem [8]. For triplets of views, focal length estimation is no longer degenerate in general [14] For each triplet, it 7 0 20 40 60 80 100 0 1 2 3 4 Relative error [ Number of views off viewing sphere focal length Figure 4: Between 0 and 4 of the 8 views are translated away from the viewing ....

G.N. Newsam, D.Q. Huynh, M.J. Brooks, and H.P. Pan, "Recovering Unknown Focal Lengths in Self-Calibration: An Essentially Linear Algorithm and Degenerate Configurations, " Proc. ISPRS-Congress, Vienna, Vol. XXXI, Part B3, pp. 575-580, 1996.

....constant parameters is practically solved, much less is known for other auto calibration constraints. In [43] additional scene and calibration constraints are used to resolve ambiguous reconstructions, caused by a xed axis rotation. The case of two cameras with unknown focal lengths is studied in [12, 28, 4, 20]. For the general unknown focal length case, Sturm [38] has independently derived results similar to those presented here and in [20, 19] In this paper, we generalise the work of Sturm [37] by relaxing the constraint constancy on the intrinsic parameters. We show that for a large class of ....

....in several images, there may or may not be a 3D quadric having them as image projections. The constraints which guarantee this in two images are called the Kruppa constraints [22] In the two image case, these constraints have been successfully applied in order to derive the critical sets, e.g. [28]. For the more general case of multiple images, the projection equation given by (4) can be used for each image separately. 3. Problem Formulation The problem of auto calibration is to nd the intrinsic camera parameters (K i ) m i=1 , where m denotes the number of camera positions. In ....

[Article contains additional citation context not shown here]

Newsam, G., D. Huynh, M. Brooks, and H.-P. Pan: 1996, `Recovering Unknown Focal Lengths in Self-Calibration: An Essentially Linear Algorithm and Degenerate Congurations'. In: Int. Arch. Photogrammetry & Remote Sensing, Vol. XXXI-B3. Vienna, Austria, pp. 57580.

....In the case of two views of a static scene it is well known that the Kruppa equations [8] provide two constraints on the camera internal parameters. If all the parameters apart from the focal length are known, then the focal lengths of the two views can be computed from the fundamental matrix [4, 12, 16]. Now suppose that in addition to the moving camera there is an object moving independently. Then both the object and the background scene each provide two Kruppa equations from their associated fundamental matrix. This means that there are four, in general, independent equations available on the ....

G. Newsam, D. Huynh, M. Brooks, and H. Pan. Recovering unknown focal lengths in selfcalibration: an essentially linear algorithm and degenerate configurations. In Proc. 18th ISPRS Congress, (International Archives of Photogrammetry and Remote Sensing (IAPRS), vol. XXXI), 1996.

....(proof omitted due to lack of space) In that case, there are infinitely many pairs of values for the two focal lengths that are mathematically valid solutions. Discussion. It is known that the focal lengths can be estimated from the epipolar geometry, without knowing the infinite homography [3, 7]. However, this problem is subject to more singularities: whenever the two optical axes are coplanar (i.e. the two cameras are fixated) and in some other cases, the calibration problem has no unique solution [7] In practice, static stereo systems used for e.g. surveillance, will nearly always be ....

.... can be estimated from the epipolar geometry, without knowing the infinite homography [3, 7] However, this problem is subject to more singularities: whenever the two optical axes are coplanar (i.e. the two cameras are fixated) and in some other cases, the calibration problem has no unique solution [7]. In practice, static stereo systems used for e.g. surveillance, will nearly always be approximately fixated, which will cause numerical instability for the calibration. Another advantage of being able to use the infinite homography is that more calibration constraints are available: from the ....

G.N. Newsam, D.Q. Huynh, M.J. Brooks, H.P. Pan, "Recovering Unknown Focal Lengths in SelfCalibration: An Essentially Linear Algorithm and Degenerate Configurations," XVIII isprs- Congress, Vienna, Austria, Vol. XXXI, Part B3, 575-580, 1996.

....procedure for computing the focal lengths f and f 0 from the fundamental matrix F . The solution is obtained by applying the singular value decomposition (SVD) and solving linear equations in four unknowns. Pan et al. 16, 17] reduced this problem to solving cubic equations. Newsam et al. [15] refined these algorithms into a combination of SVD and linear equations in three unknowns. They also derived the degeneracy condition for the solution to be indeterminate. Bougnoux [1] presented a closedform formula for f in terms of the fundamental matrix F and the epipoles e and e 0 ....

....scalar invariants with respect to image coordinate rotations. We first describe the algorithm and then give a justification for it. Next, we give a complete analysis for degenerate configurations in which the solution is indeterminate. Our result completely agrees with that of Newsam et al. [15]. Finally, we describe an algorithm for computing a single focal length in the degenerate case. We show that the solution is indeterminate only for the singular configurations that Brooks et al. 2] found for a horizontally constrained stereo head. 2. DESCRIPTION OF THE ALGORITHM The inner ....

[Article contains additional citation context not shown here]

G. N. Newsam, D. Q. Huynh, M. J. Brooks and H.- P. Pan, Recovering unknown focal lengths in selfcalibration: An essentially linear algorithm and degenerate configurations, Int. Arch. Photogram. Remote Sensing , 31-B3-III, July 1996, Vienna, Austria, pp. 575--580.

....case of a moving camera with variable and unknown focal length, but whose other intrinsic parameters are known. A practical self calibration algorithm was proposed by Azarbayejani and Pentland [1] Algorithms and closed form solutions for the two view case were given by Hartley and Brooks et al. [7, 3, 4, 10]. Newsam et al. derived the critical motions for the two view case [10] In this paper, we derive a complete characterization of critical motion sequences, for any number of views, and the critical motions for stereo systems. The paper is organized as follows. In x2 we provide some theoretical ....

....intrinsic parameters are known. A practical self calibration algorithm was proposed by Azarbayejani and Pentland [1] Algorithms and closed form solutions for the two view case were given by Hartley and Brooks et al. 7, 3, 4, 10] Newsam et al. derived the critical motions for the two view case [10]. In this paper, we derive a complete characterization of critical motion sequences, for any number of views, and the critical motions for stereo systems. The paper is organized as follows. In x2 we provide some theoretical background for our approach. The problem of deriving critical motion ....

[Article contains additional citation context not shown here]

G.N. Newsam, D.Q. Huynh, M.J. Brooks and H.P. Pan, "Recovering Unknown Focal Lengths in Self-Calibration: An Essentially Linear Algorithm and Degenerate Configurations," Proc. of the XVIII ISPRS-Congress, Vienna, Austria, Vol. XXXI, Part B3, 575-580, 1996.

....only manages to give a rather implicit description of the corresponding critical motions. For practical purposes a more explicit description would be useful. This paper derives explicit critical motions for Euclidean SFM under several simple two image unknown focal length calibration constraints [6, 16, 24, 2, 9]. However, we start by giving a complete description of criticality for known calibrations, for both perspective and orthographic cameras in multiple images. Although this analysis does not result in any new ambiguities, it rules out the possibility of any further unknown ones. A second goal of ....

....Hence, from 2 projective images we might hope to estimate Euclidean structure plus two additional calibration parameters. Hartley [6] gave a method for the case where the only unknown calibration parameters are the focal lengths of the two cameras. This was later elaborated by Newsam et.al. [16], and Zeller Faugeras and Bougnoux [24, 2] HippisleyCox Porrill [9] give a related method for equal but unknown focal lengths and aspect ratios. All of these methods are Kruppa based. We will give a unified presentation and derive the critical motions for the Hartley NewsamBougnoux (unequal f ....

[Article contains additional citation context not shown here]

G. Newsam, D.Q. Huynh, M. Brooks, and H.-P. Pan. Recovering unknown focal lengths in self-calibration: An essentially linear algorithm and degenerate configurations. In Int. Arch. Photogrammetry & Remote Sensing, volume XXXI-B3, pages 575--80, Vienna, 1996.

No context found.

G. N. Newsam, D. Q. Huynh, M. J. Brooks, and H.-P. Pan, "Recovering unknown focal lengths in self-calibration: An essentially linear algorithm and degenerate configurations," Int. Arch. Photogram. Remote Sensing, 31-B3-III, 575--580 #1996#.

No context found.

G. N. Newsam, D. Q. Huynh, M. J. Brooks, and H.-P. Pan. Recovering Unknown Focal Lengths in Self-Calibration: An Essentially Linear Algorithm and Degenerate Configurations. International Archives of Photogrammetry and Remote Sensing, 31(B3-III):575--580, 1996.

No context found.

G. Newsam, D. Q. Huynh, M. Brooks, and H. P. Pan. Recovering unknown focal lengths in selfcalibration: An essentially linear algorithm and degenerate configurations. In Int. Arch. Photogrammetry & Remote Sensing, volume XXXIB3, pages 575--80, Vienna, 1996. 7

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