K. Astrom, R. Cipolla, and P. J. Giblin. Generalised epipolar constraints. In B. F. Buxton and R. Cipolla, editors, Proc. 4th European Conf. on Computer Vision, volume II of Lecture Notes in Computer Science 1065, pages 97--108, Cambridge, UK, April 1996. Springer--Verlag.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the epipolar geometry if the solution was unsatisfactory. A partial solution was found by pre conditioning the data. In its unconditioned form, the problem is generally poorly conditioned. The depth of all the points is only 16 taken to be 1 pixel, resulting in vectors of, for example, [200 150 1] T . All vectors in the image plane thus have a very shallow depth gradient in the x and y directions. A slight uncertainty in this gradient can give rise to an error of tens or even hundreds of pixels in the image point to which the vector refers and so it is not surprising that the solutions ....

....in a different direction (but still, obviously, from the same place) This provides a stereoscopic pair and is a natural stepping stone towards random dot stereograms and 3D structure recovery. 3.3. 1 Initial investigations An method of performing this warping has been proposed by strm et al. [1] for the epipolar geometry of curved surfaces in motion. The approach is a fairly intuitive one: calculate the vectors of the actual image rays from the objects to the camera centre, rotate the camera image plane and re project the rays onto the new plane. However, for an uncalibrated camera the ....

[Article contains additional citation context not shown here]

K strm, R Cipolla and P J Giblin. Generalised Epipolar Constraints. In B Buxton, R Cipolla, editors, Proc. ECCV '96, Volume II, pages 97-108. Springer-Verlag, 1996.

....three dimensional curves obey several constraints at each point. These constraints involve high order spatio temporal derivatives of the image curve motion and camera motion and are thus difficult to handle in practice due to numerical problems. Yet another type of reconstruction can be found in [1, 3, 7], where image motion constraints that hold for certain points on the curve are developed. These constraints are more robust and can be applied to the silhouette of curved surfaces, but they do not exploit the full structure of the curve reconstruction problem. 120 Berthilsson and Astr om Another ....

....2 ( 0, 2#] weget s(# X ) linhull # cos(kt) # 2 # sin(kt) # 2 # and d(# X ) linhull(1, cos(t) sin(t) 122 Berthilsson and Astr om Example 3. Let X be a bounded piece of a straight line in R 2 . The set X can then be transformed to the line segment X # = I = [ 1, 1], on a coordinate axis, say the x axis, by a nonsingular affine transformation, leaving its shape unchanged. If #X (x) x, 0, 0) for x # I , then s(# X ) # # # # # f # # # # # # # # 1 1 fxdx=0, # 1 1 fdx=0, f#L 2 (I) # # # # # . If we parametrise L 2 (I ) by the ....

[Article contains additional citation context not shown here]

K. Astrom, R. Cipolla, and P.J. Giblin, "Generalised Epipolar Constraints," in Proc. 4th European Conf. on Computer Vision, Cambridge, England, 1996, pp. 97--108.

....is shown there that the image of threedimensional curves obey several constraints at each point. These constraints involve high order spatiotemporal derivatives of the image curve motion and camera motion, which therefore might lead to numerical problems. Yet another contribution can be found in [PP91, Car94, #CG96], where image motion constraints that hold for certain points on the curve are developed. These constraints seem more robust than the ones mentioned above and can be applied to the silhouette of curved surfaces, but they do not exploit the full structure of the curve reconstruction problem. 2 ....

K. #str#m, R. Cipolla, and P. J. Giblin. Generalised epipolar constraints. In Proc. 4th European Conf. on Computer Vision, Cambridge, UK, 1996.

.... especially concentrated on using point correspondences to estimate either the epipolar or the trifocal geometry, see [6, 9, 5, 21, 8] Recently, attention has also turned to the use of other features, such as lines, conics, general curves or even silhouettes of threedimensional bodies, see [18, 17, 2, 1]. For points the geometric situation is fairly well understood. For example, in the uncalibrated case there exist three solutions to the problem of estimating the fundamental matrix from 7 point correspondences in 2 views as well as to the problem of estimating the trifocal tensor from 6 point ....

....is that they are computationally expensive and in the rst case, higher order derivatives have to be calculated. An even more general situation is to use images of silhouettes of three dimensional bodies. This approach involves complicated geometrical considerations, but solutions exist, see [11, 1]. In [18] conic correspondences were used to reconstruct a conic in space when the epipolar geometry is known. In this case the object in space is assumed to be a degenerate quadric, i.e. a planar conic. When the object in space is a non degenerate quadric, three images are needed to make ....

[Article contains additional citation context not shown here]

K. #str#m, R. Cipolla, and P. J. Giblin. Generalised epipolar constraints. In Proc. 4th European Conf. on Computer Vision, Cambridge, UK, 1996.

....of three dimensional curves obey several constraints at each point. These constraints involve high order spatio temporal derivatives of the image curve motion and camera motion and are thus diOEcult to handle in practice due to numerical problems. Yet another type of reconstruction can be found in [PP91, Car94, #CG96], where image motion constraints that hold for certain points on the curve are developed. These constraints are more robust and can be applied to the silhouette of curved surfaces, but they do not exploit the full structure of the curve reconstruction problem. Another application of shape is ....

K. #str#m, R. Cipolla, and P. J. Giblin. Generalised epipolar constraints. In Proc. 4th European Conf. on Computer Vision, Cambridge,

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K. Astrom, R. Cipolla, and P. J. Giblin. Generalised epipolar constraints. In B. F. Buxton and R. Cipolla, editors, Proc. 4th European Conf. on Computer Vision, volume II, pages 97--108. Springer--Verlag, 1996.

No context found.

K. Astrom, R. Cipolla, and P. J. Giblin. Generalised epipolar constraints. In B. F. Buxton and R. Cipolla, editors, Proc. 4th European Conf. on Computer Vision, volume II of Lecture Notes in Computer Science 1065, pages 97--108, Cambridge, UK, April 1996. Springer--Verlag.

No context found.

K. Astrom, R. Cipolla, and P. Giblin. Generalised epipolar constraints. Int. Journal of Computer Vision, 33(1):51--72, September 1999. 31

No context found.

K. Astrom, R. Cipolla, and P. J. Giblin. Generalised epipolar constraints. In B. F. Buxton and R. Cipolla, editors, Proc. 4th European Conf. on Computer Vision, volume II of Lecture Notes in Computer Science 1065, pages 97--108, Cambridge, UK, April 1996. Springer--Verlag.

No context found.

K. Astrom, R. Cipolla, and P. Giblin. Generalised epipolar constraints. Int. Journal of Computer Vision, 33(1):51--72, September 1999. 31

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K. Astrom, R. Cipolla, and P. Giblin, "Generalised epipolar constraints," Int. Journal of Computer Vision, vol. 33, no. 1, pp. 51--72, September 1999.

.... to the intersection of two consecutive contour generators [7] The connection between the epipolar ge ometry and the frontier points was established in [17] and an algorithm for motion estimation from profiles under perspective projection was introduced in [6] Relat ed works also include [2], where a technique based on registering the images using a planar curve was first developed. This method was implemented in [11] which also showed results of reconstruction from the estimated motion. The work in [ 19] presents a method where the affine approximation is used to bootstrap the full ....

.... by the rotating object to overcome the main difficulties and drawbacks present in other methods which have attempted to estimate motion from profiles, namely: 1) the need for a very good initialization for the epipolar geometry and an unrealistic demand for a large number of epipolar tangencies [6, 2, 1] (here as few as two epipolar tangencies are needed) 2) restriction to linear motion [31] whereas circular motion is a more practical situation) or (3) the use of an affine approximation [39] which may be used only for shallow scenes) An interesting comparison can be made between the work ....

[Article contains additional citation context not shown here]

K. AstrOm, R. Cipolla, and P. J. Giblin. Generalised epipolar constraints. In B. E Buxton and R. Cipolla, editors, Proc. 4th European Conf on Computer Vision, volume II of Lecture Notes in Computer Science 1065, pages 97-108, Cambridge, UK, April 1996. Springer-Verlag.

.... by the rotating object to overcome the main difficulties and drawbacks present in other methods which have attempted to estimate motion from profiles, namely: 1) the need for a very good initialization for the epipolar geometry and an unrealistic demand for a large number of epipolar tangencies [6, 2, 1] (here as few as two epipolar tangencies are needed) 2) restriction to linear motion [31] whereas circular motion is a more practical situation) or (3) the use of an affine approximation [39] which may be used only for shallow scenes) An interesting comparison can be made between the work ....

.... a particular case of Theorem 1 in which the pinhole camera P [13] is giv en by = lr I t ] where t = 0 00Z] T, for any a 0, symmetry considerations show that the profile of will be bilaterally symmetric with respect to the image of the y axis [28, 26] which corresponds to the line qs = [1 0 0] T in (homogeneous) image coordinates. Proof of Theorem 1 (particular case) Since is bilaterally symmetric about qs, there is a transformation T that maps each point of onto its symmetric counterpart, 3) However, as any bilateral symmetry transformation, T is also a harmonic homology, ....

[Article contains additional citation context not shown here]

K. AstrOm, R. Cipolla, and P. Giblin. Generalised epipolar constraints. Int. Journal of Computer Vision, 33(1):51-72, September 1999. 31

.... to the intersection of two consecutive contour generators [7] The connection between the epipolar geometry and the frontier points was established in [17] and an algorithm for motion estimation from profiles under perspective projection was introduced in [6] Related works also include [2], where a technique based on registering the images using a planar curve was first developed. This method was implemented in [11] which also showed results of reconstruction from the estimated motion. The work in [19] presents a method where the affine approximation is used to bootstrap the full ....

.... by the rotating object to overcome the main difficulties and drawbacks present in other methods which have attempted to estimate motion from profiles, namely: 1) the need for a very good initialization for the epipolar geometry and an unrealistic demand for a large number of epipolar tangencies [6, 2, 1] (here as few as two epipolar tangencies are needed) 2) restriction to linear motion [31] whereas circular motion is a more practical situation) or (3) the use of an affine approximation [39] which may be used only for shallow scenes) An interesting comparison can be made between the work ....

[Article contains additional citation context not shown here]

K. Astrom, R. Cipolla, and P. J. Giblin. Generalised epipolar constraints. In B. F. Buxton and R. Cipolla, editors, Proc. 4th European Conf. on Computer Vision, volume II of Lecture Notes in Computer Science 1065, pages 97--108, Cambridge, UK, April 1996. Springer--Verlag.

.... by the rotating object to overcome the main difficulties and drawbacks present in other methods which have attempted to estimate motion from profiles, namely: 1) the need for a very good initialization for the epipolar geometry and an unrealistic demand for a large number of epipolar tangencies [6, 2, 1] (here as few as two epipolar tangencies are needed) 2) restriction to linear motion [31] whereas circular motion is a more practical situation) or (3) the use of an affine approximation [39] which may be used only for shallow scenes) An interesting comparison can be made between the work ....

.... case of Theorem 1 in which the pinhole camera P [13] is given by P = I jt ] where t = 0 0 ] T , for any 0, symmetry considerations show that the profile s of S will be bilaterally symmetric with respect to the image of the y axis [28, 26] which corresponds to the line q s = [1 0 0] T in (homogeneous) image coordinates. Proof of Theorem 1 (particular case) Since s is bilaterally symmetric about q s , there is a transformation T that maps each point of s onto its symmetric counterpart, given by T = 2 6 6 6 6 4 1 0 0 0 1 0 0 0 1 3 7 7 7 7 5 : 3) ....

[Article contains additional citation context not shown here]

K. Astrom, R. Cipolla, and P. Giblin. Generalised epipolar constraints. Int. Journal of Computer Vision, 33(1):51--72, September 1999. 31

....instances. After the motions have been recovered, the local models can then be integrated into a global surface description. Their method requires a calibrated trinocular stereo system. The algorithms developed in this thesis use a single uncalibrated camera, making them much more flexible. In [1], Astrom, Cipolla and Giblin presented an iterative method to recover camera motion from an image sequence of a curved surface using the generalized epipolar constraints. A maximum likelihood method is employed to minimize the objective function based on the constraints on the motion parameters ....

K. Astrom, R. Cipolla, and P.J. Giblin. Generalised epipolar constraint. In Proc. 4th European Conf. on Computer Vision, volume 2, pages 97--108, 1996.

....by the intersection of two contour generators. A frontier point projects on its associated images as an epipolar tangency. The use of frontier points and epipolar tangencies for motion recovery was first shown in [2] A parallax based technique, using a reference planar contour was shown in [1], where the images are registered using the reference contour and common tangents are used 2 Paulo R. S. Mendonca, Kwan Yee K. Wong, Roberto Cipolla to determine the projections of the frontier point. The techniques described above face two main difficulties: the likely non uniqueness of the ....

.... becomes C C 0 with C # (17) Thus, we have proved that the transformation corresponds to a plane induced homography (see [9] This means that the registration of the images can be done by using instead of a planar contour as proposed in [1, 6]. It is known that different choices of the plane that induces the homography in a plane plus parallax parameterization of the fundamental matrix will result in different homographies, although they will all generate the same fundamental matrix, since C C D C ....

K. Astrom, R. Cipolla, and P. J. Giblin. Generalised epipolar constraints. In B. F. Buxton and R. Cipolla, editors, Proc. 4th European Conf. on Computer Vision, volume II, pages 97--108. Springer--Verlag, 1996.

....criteria [11] as shown in fig. 3, given by C = X i # #m 0T i Fm i # 2 #Fm i # 2 1 #Fm i # 2 2 #m 0T i Fm i # 2 #F T m 0 i # 2 1 #F T m 0 i # 2 2 ; 3) to be minimized by searching for the epipoles. Details of the procedure and a similar idea can be found in [2, 5]. The point to be stressed is that the algorithm is quite prone to converge to local minima, and needs at least 8 tangencies in each image, a requirement not satisfied by most images of real scenes. 3. The Affine Case When the field of view is narrow or the depth variation is small compared with ....

K. Astrom, R. Cipolla, and P.J. Giblin. Generalised epipolar constraint. In Proc. 4th European Conf. on Computer Vision, volume 2, pages 97--108, 1996.

.... (aOEne transformation) A, such that q = A(t)w: For simplicity the relationship between w and p will be expressed by a single matrix S representing both intrinsic calibration and orientation of the camera, i.e. p = R(t)A(t)w = S(t)w: A thorough treatment of the other cases can be found in [1]. The results are summarised in the following two tables. The direction of motion is represented with a three vector, Deltac, in the central projection cases and Deltak = cos( sin( 0) T in the parallel projection case. In the tables DeltaR is a three by three rotation matrix, DeltaS ....

....approximately the same as minimising g = X ff 2 i oe 2 i : The estimate is the motion parameters that minimise this weighted sum of squared residuals. In our implementations minimisation is performed using either a modi ed Newton Raphson method or Gauss Newton method. For more details see [1]. A B C D Fig. 2. Recti cation of uncalibrated images. Two images (A B) are projected onto the viewing sphere and recti ed using the motion parameters (C D) After recti cation The epipolar tangency planes all intersect at the x axis. The two sets of epipolar tangency planes should be equal. The ....

[Article contains additional citation context not shown here]

K. #str#m, R. Cipolla, and P. J. Giblin. Generalised epipolar constraints. Technical report, Dept. of Mathematics, Lund University, 1996.

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Astrom, K., Cipolla, R. & Giblin, P.J. 1996 Generalised epipolar constraint. In Proc. 4th . Conf.

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K. Astrom, R. Cipolla, and P. J. Giblin. Generalised epipolar constraints. In B. F. Buxton and R. Cipolla, editors, Proc. 4th European Conf. on Computer Vision, volume II of Lecture Notes in Computer Science 1065, pages 97--108, Cambridge, UK, April 1996. Springer--Verlag.

No context found.

K. Astrom, R. Cipolla, and P. Giblin. Generalised epipolar constraints. Int. Journal of Computer Vision, 33(1):51--72, September 1999. 129

No context found.

K. Astrom, R. Cipolla, and P. Giblin. Generalised epipolar constraints. Int. Journal of Computer Vision, 33(1):51--72, September 1999.

No context found.

K. Astrom, R. Cipolla and P. J. Giblin, Generalised Epipolar Constraints, In Proc. of ECCV, 1996, pages 97 - 108.

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K strm, R Cipolla and P Giblin. Generalised Epipolar Constraints. In B Buxton, R Cipolla, editors, Proc. ECCV'96, Volume II, pages 97-108. Springer-Verlag, 1996.

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