## Relative orientation revisited (1991)

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Venue: | Journal of the Optical Society of America A |

Citations: | 36 - 1 self |

### BibTeX

@ARTICLE{Horn91relativeorientation,

author = {Berthold K. P. Horn},

title = {Relative orientation revisited},

journal = {Journal of the Optical Society of America A},

year = {1991},

volume = {8},

pages = {1630--1638}

}

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### Abstract

Relative Orientation is the recovery of the position and orientation of one imaging system relative to another from correspondences between five or more ray pairs. It is one of four core problems in photogrammetry and is of central importance in binocular stereo, as well as in long range motion vision. While five ray correspondences are sufficient to yield a finite number of solutions, more than five correspondences are used in practice to ensure an accurate solution using least squares methods. Most iterative schemes for minimizing the sum of squares of weighted errors require a good guess as a starting value. The author has previously published a method that finds the best solution without requiring an initial guess. In this paper an even simpler method is presented that utilizes the representation of rotations by unit quaternions. 1. See also: ``Relative Orientation,'' {\it International Journal of Computer Vision}, Vol.~4, No.~1, pp.~59--78, January 1990.

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Citation Context ...tion and unit-quaternion notation, see Ref. 6. See also Appendix A and Ref. 6.) We can write the triple product in the form or, if we let b = (0,b) and r = (0,r), ( = (r x b) • 1', (4) t = i-b • 44*, =-=(5)-=- where we have used the fact that /' = qlq* nas zero scalar part and rb = (-r-b,r x b), (6) since both r and & have zero scalar parts. The triple product can now be further transformed to yield or fin... |

138 | Relative orientation
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- 1990
(Show Context)
Citation Context ...ion q to represent this rotation, we can write r = qly, (2) where I and I' are unit quaternions with zero scalar part and vector part equal to 1 and 1', respectively; that is, I = (0,1), !' = (0,1'). =-=(3)-=- (For use of unit-quatemion notation in a related photogrammetric problem, including a discussion of numerically stable methods for converting between orthonormalmatrix notation and unit-quaternion no... |

136 |
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Citation Context ...wns, being separately linear in the components of 4 and in the components of d. B. Symmetry in the Coplanarity Condition We can rewrite the triple product by using ( = rd • ql = f- • yd* = q*r • Id*, =-=(11)-=- t = q*r • Id* = (4*r)* • (Id*)* = r*q • dl*. (12) Finally, noting that I* = —I and r* = —r, since r and I are quaternions with zero scalar parts, we obtain t^rq-dl. (13) The symmetry can be seen in m... |

130 |
Solving polynomial systems using continuation for engineering and scientific problems
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Citation Context ... • ql = f- • yd* = q*r • Id*, (11) t = q*r • Id* = (4*r)* • (Id*)* = r*q • dl*. (12) Finally, noting that I* = —I and r* = —r, since r and I are quaternions with zero scalar parts, we obtain t^rq-dl. =-=(13)-=- The symmetry can be seen in more detail if the dot product for t is expanded in terms of the scalar and vector components of4= (<7,<l) and d = (d,d): (d • r)(q • 1) + (q • r)(d • 1) + (dq - d • q)(l ... |

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An Introduction to Theoretical Kinematics
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Citation Context ...e the triple product in the form or, if we let b = (0,b) and r = (0,r), ( = (r x b) • 1', (4) t = i-b • 44*, (5) where we have used the fact that /' = qlq* nas zero scalar part and rb = (-r-b,r x b), =-=(6)-=- since both r and & have zero scalar parts. The triple product can now be further transformed to yield or finally t=i-bq-ql (7) t=rd-ql, (8) where d = bq. [In Eqs. (4)-(7) a number of quaternion ident... |

49 |
A homotopy for solving general polynomial systems that respects m-homogeneous structures
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Citation Context ...sponding sets of rays {1,} and {r,} (for i = 1,2,..., n) from the left and the right imaging systems, respectively, the task is to find q and d that minimize where e, = (hd • ql,), subject to s w.e,, =-=(15)-=- 4-4=1, d-d^l, q-d=0. (16) The weight factor is chosen according to the reliability of a particular measurement but also depends on the ray direction. That is, the error contributions that one wishes ... |

36 |
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Citation Context ...orm solution is at hand, let us see how small changes in q and d affect the total error. First, by ignoring second-order terms in (q + Sq) • (q + Sq) = i, (d+ sd)-(d+ 8d)= i, (q + o$) • (d + 8d) = 0, =-=(18)-=- we obtain the following contraints on the increments: q-8q^0, d-sd^O, q-8d^-d-8q=0. (19) We have to find increments 8$ and Set that minimize ^Wi[h(d+8d)-{q+Sq)ly, (20) subject to the three constraint... |

33 |
Computing all solutions to polynomial systems using homotopy continuation
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Citation Context ... and {r,} (for i = 1,2,..., n) from the left and the right imaging systems, respectively, the task is to find q and d that minimize where e, = (hd • ql,), subject to s w.e,, (15) 4-4=1, d-d^l, q-d=0. =-=(16)-=- The weight factor is chosen according to the reliability of a particular measurement but also depends on the ray direction. That is, the error contributions that one wishes to minimize are distances ... |

30 |
Epiconvergence and Continuous
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- 1994
(Show Context)
Citation Context ...en in more detail if the dot product for t is expanded in terms of the scalar and vector components of4= (<7,<l) and d = (d,d): (d • r)(q • 1) + (q • r)(d • 1) + (dq - d • q)(l • r) + d[rql]+ q[rdl]. =-=(14)-=- Certain other symmetries now become apparent. If the parameters {4, d} satisfy the coplanarity condition for corresponding sets of rays {1,} and {r,}, then what follows is true: (1) The set of parame... |

11 |
Algebraic methods in 3D motion estimation from two-view point correspondences
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Citation Context ...4 and in the components of d. B. Symmetry in the Coplanarity Condition We can rewrite the triple product by using ( = rd • ql = f- • yd* = q*r • Id*, (11) t = q*r • Id* = (4*r)* • (Id*)* = r*q • dl*. =-=(12)-=- Finally, noting that I* = —I and r* = —r, since r and I are quaternions with zero scalar parts, we obtain t^rq-dl. (13) The symmetry can be seen in more detail if the dot product for t is expanded in... |

8 |
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Citation Context ...tes. The coplanarity conditions can, of course, be equally well expressed in the coordinates of the left imaging system.) Using the unit quaternion q to represent this rotation, we can write r = qly, =-=(2)-=- where I and I' are unit quaternions with zero scalar part and vector part equal to 1 and 1', respectively; that is, I = (0,1), !' = (0,1'). (3) (For use of unit-quatemion notation in a related photog... |

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Citation Context ...used the fact that /' = qlq* nas zero scalar part and rb = (-r-b,r x b), (6) since both r and & have zero scalar parts. The triple product can now be further transformed to yield or finally t=i-bq-ql =-=(7)-=- t=rd-ql, (8) where d = bq. [In Eqs. (4)-(7) a number of quaternion identities, such as 6q • 6 = & • by, have been used that can be easily checked by the rule for quaternion multiplication in terms of... |

4 |
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(Show Context)
Citation Context ... product is zero. For the ray 1 from the left camera's center of projection and the ray r from the right camera's center of projection to be coplanar with the baseline b, we must have 3 ' 4 [bl'r]=0, =-=(1)-=-sBertholdK.P.Horn Vol. 8, No. 10/October 1991/J. Opt. Soc. Am. A 1631 where 1' is the left camera's ray rotated into the right imaging system's coordinates. (Here the baseline b is also expressed in t... |

1 |
Photogrammetric in Praxis und Theorie
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(Show Context)
Citation Context ...ily checked by the rule for quaternion multiplication in terms of the scalar and vector parts of the quaternions given in Appendix A.] Note that d is orthogonal to 4, since <?• 4= 64 •4=6-44* =6-^=0, =-=(9)-=- where 6 is the identity with respect to quaternion multiplication. (The identity 6 has a unit scalar part and zero vector part.) The baseline can be recovered from S with dq=bqq*=b6=b, (10) so one ma... |

1 |
Sur deux problemes de reconstruction," INRIA Rep. 882 (Institut National de Recherche en Informatique et an Automatique, Les Chesnay
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- 1988
(Show Context)
Citation Context ...4* =6-^=0, (9) where 6 is the identity with respect to quaternion multiplication. (The identity 6 has a unit scalar part and zero vector part.) The baseline can be recovered from S with dq=bqq*=b6=b, =-=(10)-=- so one may as well work with the parameters 4 and d, rather than 4 and b, if this is convenient. Note that the resulting expression is bilinear in the unknowns, being separately linear in the compone... |

1 | Relative orientation," Memo 994 - Horn - 1987 |