## Uncalibrated Euclidean Reconstruction (2000)

Venue: | Review. IVC |

Citations: | 1 - 0 self |

### BibTeX

@ARTICLE{Fusiello00uncalibratedeuclidean,

author = {Andrea Fusiello},

title = {Uncalibrated Euclidean Reconstruction },

journal = {Review. IVC},

year = {2000},

pages = {18--6}

}

### OpenURL

### Abstract

This paper presented a review of recent techniques for Euclidean reconstruction from a single moving camera, with unconstrained motion and unknown constant parameters.

### Citations

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Practical Optimization
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Citation Context ...he number of systems to be solved rapidly increases with the number of displacements. The over-constrained system of equation is solved with a non-linear least-squares technique (LevenbergMarquardt [=-=12]-=-, or Iterated Extended Kalman Filter [32]). The problem with non-linear least-squares is that a starting point close to the solution is needed. This can be obtained by applying globally convergent met... |

1302 |
Three-Dimensional Computer Vision: A Geometric Viewpoint
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Citation Context ...icles. This document have been adapted from [11]. 2 Notation and basics This section introduces the mathematical background on perspective projections necessary for our purposes. Our notation follows =-=[6]-=-. X m Z C W R Y Figure 1: The pinhole camera model, with the camera reference frame (X,Y,Z) depicted. Z is also called the optical axis . A pinhole camera is modeled by its optical center C and its re... |

651 |
A computer algorithm for reconstructing a scene from two projections
- Longuet-Higgins
- 1981
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Citation Context ...9, 51]. It can be seen that (9) is equivalent to (A 01 ~ m 0 ) > [t] ^R(A 1 ~ m) = 0: (10) Changing to normalized coordinates, ~ n = A 1 ~ m; one obtain the original formulation of the LonguetHiggins =-=[27]-=- equation, ~ n 0> E~ n = 0 (11) involving the essential matrix E = [t] ^R; (12) which can be obtained when intrinsic parameters are known. E depends uponsve independent parameters (rotation and transl... |

509 |
What can be seen in three dimensions with an uncalibrated stereo rig
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(Show Context)
Citation Context ...,snd the set of camera matrices f ~ P i g and the scene structure f ~ w j g such that ~ m i j ' ~ P i ~ w j : (19) Without further restrictions we will, in general, obtain a projective reconstruction =-=[34, 43, -=-5] dened up to an arbitrary projective transformation. Indeed, if f ~ P i g and f ~ w j g satisfy (19), also f ~ P i ~ Tg and f ~ T 1 ~ w j g satisfy (19) for any 4 4 nonsingular matrix ~ T. A projec... |

367 | Camera self-calibration: theory and experiments
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(Show Context)
Citation Context ...speed ambiguity: it is impossible to determine whether a given image motion is caused by a nearby object with slow relative motion or a distant object with fast relative motion). Maybank and Faugeras =-=[31, 10]-=- proved that, if intrinsic parameters are constant, Euclidean reconstruction is achievable. The procedure is known as autocalibration. In this approach, the internal unchanging parameters of the camer... |

329 | Determining the epipolar geometry and its uncertainty: A review
- Zhang
(Show Context)
Citation Context ...se, the only geometrical information that can be computed from pairs of images is the fundamental matrix. Its computation requires a minimum of eight point correspondences to obtain a unique solution =-=[29, 51]-=-. It can be seen that (9) is equivalent to (A 01 ~ m 0 ) > [t] ^R(A 1 ~ m) = 0: (10) Changing to normalized coordinates, ~ n = A 1 ~ m; one obtain the original formulation of the LonguetHiggins [27] e... |

295 | Faugeras. A theory of selfcalibration of a moving camera - Maybank, D - 1992 |

286 | Estimation of relative camera positions for uncalibrated cameras
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Citation Context ...real matrix E 3 3 can be factorized as product of a nonzero skew-symmetric matrix and a rotation matrix if and only if E has two identical singular values and a zero singular value. For a proof see [=-=13, 8-=-]. 3 2.2 Homography of a plane Given two views of a scene, there is a linear projective transformation (an homography) relating the projection m of the point of a plane in thesrst view to its project... |

286 |
Gool. Selfcalibration and metric reconstruction inspite of varying and unknown internal camera parameters
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Citation Context ...w introduces 5 n k unknowns, therefore the unknown intrinsic parameters can be computed provided that 5n 8 (n 1)(5 n k n c ) + 5 n k ; (55) which is equivalent to the following equation reported in [=-=36-=-]: nn k + (n 1)n c 8: (56) Despite the problem for constant intrinsic parameters is far from being completely solved, the more general one in which intrinsic parameters are varying is gaining the att... |

249 |
Stochastic Models, Estimation and Control, volume 1
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Citation Context ... increases with the number of displacements. The over-constrained system of equation is solved with a non-linear least-squares technique (LevenbergMarquardt [12], or Iterated Extended Kalman Filter [=-=32]-=-). The problem with non-linear least-squares is that a starting point close to the solution is needed. This can be obtained by applying globally convergent methods to subsets of equations (like in the... |

241 | Euclidean reconstruction from uncalibrated views,” in Applications of Invariance in Computer Vision
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(Show Context)
Citation Context ...~ P i ~ Tg and f ~ T 1 ~ w j g satisfy (19) for any 4 4 nonsingular matrix ~ T. A projective reconstruction can be computed starting from points correspondences only, without any a-priori knowledge [=-=17, 18, 45, 44, 19, 3, 2, 48-=-]. Despite it conveys some useful in formations [40, 39], we would like to obtain a Euclidean reconstruction, a very special one that diers from the true reconstruction by an unknown similarity transf... |

239 | The fundamental matrix: theory, algorithms, and stability analysis
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(Show Context)
Citation Context ...se, the only geometrical information that can be computed from pairs of images is the fundamental matrix. Its computation requires a minimum of eight point correspondences to obtain a unique solution =-=[29, 51]-=-. It can be seen that (9) is equivalent to (A 01 ~ m 0 ) > [t] ^R(A 1 ~ m) = 0: (10) Changing to normalized coordinates, ~ n = A 1 ~ m; one obtain the original formulation of the LonguetHiggins [27] e... |

218 | Autocalibration and the absolute quadric
- Triggs
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(Show Context)
Citation Context ... to recover ~ T without additional information, using only the hypothesis of constant intrinsic parameters. The works by Hartley [17], Pollefeys and Van Gool [37], Heyden and Astrom [20], 7 Triggs [4=-=6]-=- and Bougnoux [4] will be reviewed, butsrst we will make some remarks that are common to most of the methods. We can choose thesrst Euclidean-calibrated camera to be ~ P 0 eucl = A[I j 0], therebysxin... |

216 | A factorization based algorithm for multi-image projective structure and motion
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(Show Context)
Citation Context ...~ P i ~ Tg and f ~ T 1 ~ w j g satisfy (19) for any 4 4 nonsingular matrix ~ T. A projective reconstruction can be computed starting from points correspondences only, without any a-priori knowledge [=-=17, 18, 45, 44, 19, 3, 2, 48-=-]. Despite it conveys some useful in formations [40, 39], we would like to obtain a Euclidean reconstruction, a very special one that diers from the true reconstruction by an unknown similarity transf... |

206 |
Algebraic projective geometry
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Citation Context ...on diers from Euclidean by an unknown projective transformation in the 3-D projective space, which can be seen as a suitable change of basis. Thanks to the fundamental theorem of projective geometry [=-=41-=-], a collineation in space is determined bysve points, hence the knowledge of the true (Euclidean) position ofsve points allows to compute the unknown 4 4 matrix ~ T that transform the Euclidean fram... |

178 | Canonic representations for the geometries of multiple projective views
- Luong, Viéville
(Show Context)
Citation Context ...be seen that, given the two projection matrices, ~ P = A[I j 0]; ~ P 0 = A 0 [R j t] (15) (the world reference frame issxed on thesrst camera) and a plane of equation n > x = d, the following holds [30]: H = A 0 (R + t n > d )A 1 : (16) H is the homography matrix for the plane . If d !1, H1 = A 0 RA 1 : (17) This is the homography matrix for the innity plane, which maps vanishing points to vanis... |

164 |
defence of the 8-point algorithm
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(Show Context)
Citation Context ...e non-linear least-squares problem: sequential quadratic programming [12] on N 3 cameras, and a quasi-linear method with SVD factorization on N 4 cameras. He recommend to use data standardization [1=-=5-=-] and to enforce det(s= 3. The sought transformation ~ T is computed by taking the eigen-decomposition of 5.2.5 Bougnoux This methods [4] is dierent from the previous ones, because it does not require... |

159 |
A.N.: Motion and Structure from Feature Correspondences: A Review
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Citation Context ...ers are known, the problem of computing the extrinsic parameters (motion) from point correspondences is the well-known relative orientation problem, for which a variety of methods have been developed =-=[24, 13, 23]-=-. In principle, from the set of correspondences f ~ m i g one can compute the fundamental matrix, from which the essential matrix is immediately obtained with (13). Motion parameters 4 R and the direc... |

133 |
Motion from point matches: multiplicity of solutions
- Faugeras, Maybank
- 1990
(Show Context)
Citation Context ...real matrix E 3 3 can be factorized as product of a nonzero skew-symmetric matrix and a rotation matrix if and only if E has two identical singular values and a zero singular value. For a proof see [=-=13, 8-=-]. 3 2.2 Homography of a plane Given two views of a scene, there is a linear projective transformation (an homography) relating the projection m of the point of a plane in thesrst view to its project... |

109 | Euclidean reconstruction from image sequences with varying and unknown focal length and principal point, in
- Heyden, Astrom
- 1997
(Show Context)
Citation Context ... solved, the more general one in which intrinsic parameters are varying is gaining the attention of researchers. In fact, Bougnoux's method already copes with varying parameters. Heyden and Astrom [2=-=1]-=- proposed a method that works with varying and unknown focal length and principal point. Later, they proved [22] that it is sucient to know any of thesve intrinsic parameters to make Euclidean reconst... |

105 | Faugeras. ”Self-Calibration of a Moving Camera from Point Correspondences and Fundamental Matrices
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- 1997
(Show Context)
Citation Context ...damental matrix, from which the essential matrix is immediately obtained with (13). Motion parameters 4 R and the direction of translation t are obtained directly from the factorization (12) of E. In =-=[28-=-] direct and iterative methods are compared. Recently, new approaches based on the idea of stratication [30, 7] have been introduced. Starting from a projective reconstruction, which can be computed f... |

96 |
From projective to Euclidean space under any practical situation, a criticism ofself-calibration
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(Show Context)
Citation Context ...thout additional information, using only the hypothesis of constant intrinsic parameters. The works by Hartley [17], Pollefeys and Van Gool [37], Heyden and Astrom [20], 7 Triggs [46] and Bougnoux [4=-=]-=- will be reviewed, butsrst we will make some remarks that are common to most of the methods. We can choose thesrst Euclidean-calibrated camera to be ~ P 0 eucl = A[I j 0], therebysxing arbitrarily the... |

72 | Euclidean reconstruction from constant intrinsic parameters
- Heyden, Astrom
- 1996
(Show Context)
Citation Context ...ging problem is to recover ~ T without additional information, using only the hypothesis of constant intrinsic parameters. The works by Hartley [17], Pollefeys and Van Gool [37], Heyden and Astrom [2=-=0]-=-, 7 Triggs [46] and Bougnoux [4] will be reviewed, butsrst we will make some remarks that are common to most of the methods. We can choose thesrst Euclidean-calibrated camera to be ~ P 0 eucl = A[I j ... |

60 |
Some properties of the E matrix in two-view motion estimation
- Huang, Faugeras
- 1989
(Show Context)
Citation Context ...easy to see that F = A 0> EA 1 : (13) Unlike the fundamental matrix, whose only property is being of rank two, the essential matrix is characterized by the two constraints found by Huang and Faugeras =-=[25-=-] which are the nullity of the determinant and the equality of the two non-zero singular values. Indeed, the following Theorem holds: Theorem 2.1 A real matrix E 3 3 can be factorized as product of a... |

57 | Reconstruction from image sequences by means of relative depths
- Heyden
- 1995
(Show Context)
Citation Context ...~ P i ~ Tg and f ~ T 1 ~ w j g satisfy (19) for any 4 4 nonsingular matrix ~ T. A projective reconstruction can be computed starting from points correspondences only, without any a-priori knowledge [=-=17, 18, 45, 44, 19, 3, 2, 48-=-]. Despite it conveys some useful in formations [40, 39], we would like to obtain a Euclidean reconstruction, a very special one that diers from the true reconstruction by an unknown similarity transf... |

55 |
Relative 3D positioning and 3D convex hull computation from a weakly calibrated stereo pair
- Robert, Faugeras
- 1993
(Show Context)
Citation Context ...~ T. A projective reconstruction can be computed starting from points correspondences only, without any a-priori knowledge [17, 18, 45, 44, 19, 3, 2, 48]. Despite it conveys some useful in formations =-=[40, 39-=-], we would like to obtain a Euclidean reconstruction, a very special one that diers from the true reconstruction by an unknown similarity transformation. This is composed by a rigid displacement (due... |

50 | Kruppa’s equations derived from the fundamental matrix
- Hartley
- 1997
(Show Context)
Citation Context ...A = 2 6 6 4 q k 1 k 3 2 (k2 k3k5 ] 2 k4 k5 2 k2 k3k5 p k4 k5 2 k 3 0 p k 4 k 5 2 k 5 0 0 1 3 7 7 5 : (21) Kruppa equations were rediscovered and derived by Maybank and Faugeras [31]. Recently Hartley =-=[16]-=- provided a simpler form, based on the Singular Value Decomposition of the fundamental matrix. Let F be written as F = UDV > (with SVD), and U = 2 4 u > 1 u > 2 u > 3 3 5 V = 2 4 v > 1 v > 2 v > 3 3 5... |

49 |
Zur Ermittlung eines Objektes aus zwei Perspektiven mit innerer Orientierung,” Sitzungsberichte Österreichische Akademie der Wissenschaften
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(Show Context)
Citation Context ...nce redundancy improves stability. 4.3 Kruppa equations With a minimum of three displacements, we can obtain the internal parameters of the camera using a system of polynomial equations due to Kruppa =-=[26]-=-, which are derived from a geometric interpretation of the rigidity constraint [31, 8]. The unknown in the Kruppa equations is the matrix K = AA > , called the Kruppa coecients matrix, that represents... |

48 | Camera self-calibration from video sequences: the Kruppa equations revisited. Research Report 2793
- Zeller, Faugeras
- 1996
(Show Context)
Citation Context ...ras, one has the additional constraint k 3 k 5 = k 2 [28]. There are basically two classes of methods for solving the resulting system of equations (assuming that more than three views are available) =-=[28, 50-=-]: Partition the equations set in groups ofsve and solve each group with a global convergent technique for systems of polynomial equations, like homotopy continuation methods [35, 42]. Each system wi... |

46 |
The modulus constraint: A new constraint for selfcalibration
- Pollefeys, Oosterlinck
- 1996
(Show Context)
Citation Context ...pproximate ane (or quasi-ane) reconstruction. 5.2.2 Pollefeys and Van Gool In this approach [37], a projective reconstruction issrst updated to ane reconstruction by the use of the modulus constraint =-=[30, 38]-=-: since the left-hand part of (31) is conjugated to a (scaled) rotation matrix, all eigenvalues must have equal moduli. Note that this holds if and only if intrinsic parameters are constant. To make t... |

40 | Projective structure and motion from image sequences using subspace methods
- Heyden
- 1997
(Show Context)
Citation Context |

36 | Threading fundamental matrices
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Citation Context |

36 | Uncalibrated Euclidean reconstruction: a review
- Fusiello
- 2000
(Show Context)
Citation Context ...es. Such a comparative account sheds light on the relations between dierent methods, presented in dierent ways and formalisms in the original research articles. This document have been adapted from [1=-=1]-=-. 2 Notation and basics This section introduces the mathematical background on perspective projections necessary for our purposes. Our notation follows [6]. X m Z C W R Y Figure 1: The pinhole camera ... |

35 | Cheirality invariants
- Hartley
- 1993
(Show Context)
Citation Context ...ed in order for the algorithm to converge to the solution. Given that from H i 1 the computation of A is straightforward, a guess for a (that determines H i 1 ) is sucient. The cheirality constraints =-=[14-=-] are exploited by Hartley to estimate the innity plane homography, thereby obtaining an approximate ane (or quasi-ane) reconstruction. 5.2.2 Pollefeys and Van Gool In this approach [37], a projective... |

35 | Minimal conditions on intrinsic parameters for euclidean reconstruction
- Heyden, Äström
- 1997
(Show Context)
Citation Context ... In fact, Bougnoux's method already copes with varying parameters. Heyden and Astrom [21] proposed a method that works with varying and unknown focal length and principal point. Later, they proved [2=-=2]-=- that it is sucient to know any of thesve intrinsic parameters to make Euclidean reconstruction, even if all other parameters are unknown and varying (this can be obtained as a special case of Eq.56).... |

35 | Relative orientation revisited
- Horn
- 1991
(Show Context)
Citation Context ...ers are known, the problem of computing the extrinsic parameters (motion) from point correspondences is the well-known relative orientation problem, for which a variety of methods have been developed =-=[24, 13, 23]-=-. In principle, from the set of correspondences f ~ m i g one can compute the fundamental matrix, from which the essential matrix is immediately obtained with (13). Motion parameters 4 R and the direc... |

34 | Numerical algebraic geometry
- Sommese, Wampler
- 1996
(Show Context)
Citation Context ... are available) [28, 50]: Partition the equations set in groups ofsve and solve each group with a global convergent technique for systems of polynomial equations, like homotopy continuation methods [=-=35, 42]-=-. Each system will give a set of solutions and the solution common to all of them is chosen. This method { presented in [28] { has the great advantage of global convergence, but is computationally exp... |

32 | Euclidean structure from uncalibrated images
- Armstrong, Zisserman, et al.
- 1994
(Show Context)
Citation Context ...that is an ane reconstruction. From ane to Euclidean. Another useful observation is, if H1 is known and the intrinsic parameters are constant, the intrinsic parameters matrix A can easily be computed =-=[1, 17, 30, 49]-=-. Let us consider the case of two cameras. If A 0 = A, then H1 is exactly known (with the right scale), since det(H1)= det(ARA 1 ) = 1: (34) 8 From (17) we obtain R = A 0 1 H1A; and, since RR > = I, i... |

26 | Simultaneous reconstruction of scene structure and camera locations from uncalibrated image sequences
- Sparr
- 1996
(Show Context)
Citation Context |

21 |
Using collineations to compute motion and structure in an uncalibrated image sequence
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- 1994
(Show Context)
Citation Context ...that is an ane reconstruction. From ane to Euclidean. Another useful observation is, if H1 is known and the intrinsic parameters are constant, the intrinsic parameters matrix A can easily be computed =-=[1, 17, 30, 49]-=-. Let us consider the case of two cameras. If A 0 = A, then H1 is exactly known (with the right scale), since det(H1)= det(ARA 1 ) = 1: (34) 8 From (17) we obtain R = A 0 1 H1A; and, since RR > = I, i... |

18 |
It can be done without camera calibration
- Mohr, Arbogast
- 1990
(Show Context)
Citation Context ...,snd the set of camera matrices f ~ P i g and the scene structure f ~ w j g such that ~ m i j ' ~ P i ~ w j : (19) Without further restrictions we will, in general, obtain a projective reconstruction =-=[34, 43, -=-5] dened up to an arbitrary projective transformation. Indeed, if f ~ P i g and f ~ w j g satisfy (19), also f ~ P i ~ Tg and f ~ T 1 ~ w j g satisfy (19) for any 4 4 nonsingular matrix ~ T. A projec... |

16 |
A factorization method for projective and Euclidean reconstruction
- Ueshiba, Tomita
- 1998
(Show Context)
Citation Context |

14 | An algebralc-analytic method for reconstruction from image correspon- dences
- Sparr
(Show Context)
Citation Context ...,snd the set of camera matrices f ~ P i g and the scene structure f ~ w j g such that ~ m i j ' ~ P i ~ w j : (19) Without further restrictions we will, in general, obtain a projective reconstruction =-=[34, 43, -=-5] dened up to an arbitrary projective transformation. Indeed, if f ~ P i g and f ~ w j g satisfy (19), also f ~ P i ~ Tg and f ~ T 1 ~ w j g satisfy (19) for any 4 4 nonsingular matrix ~ T. A projec... |

13 | 3-D reconstruction of urban scenes from image sequences
- Faugeras, Laveau, et al.
- 1998
(Show Context)
Citation Context ...are in the scene, H1 can be computed from point correspondences, like any ordinary plane homography [49]. Moreover, with additional knowledge, it can be estimated from vanishing points or parallelism =-=[7, 9-=-], or constrained motion [1]. In the rest of the section, some of the most promising stratication techniques will be reviewed. 5.2.1 Hartley Hartley [17] pioneered this kind of approach. Starting from... |

11 |
A simple techinique for selfcalibration
- Mendonça, Cipolla
- 1999
(Show Context)
Citation Context ...ments are available. The main limitation of all these methods is the sensitivity to the noise in the localization of points. 4.4 Mendonca and Cipolla Mendonca and Cipolla method for autocalibration [3=-=3]-=- is based on the exploitation of the rigidity constraint (via Theorem 2.1). A cost function is designed, which takes the intrinsic parameters as arguments, and the fundamental matrices as parameters, ... |

10 |
Finding all isolated solutions to polynomial systems using hompack
- Morgan, Sommese, et al.
- 1989
(Show Context)
Citation Context ... are available) [28, 50]: Partition the equations set in groups ofsve and solve each group with a global convergent technique for systems of polynomial equations, like homotopy continuation methods [=-=35, 42]-=-. Each system will give a set of solutions and the solution common to all of them is chosen. This method { presented in [28] { has the great advantage of global convergence, but is computationally exp... |

8 |
Can multiple views make up for lack of camera registration
- Trivedi
- 1988
(Show Context)
Citation Context ...ondition is automatically satised, since det(F) = 0, but the second condition can be decomposed [29] in two independent polynomial relations that are equivalent to the two equations found by Trivedi [=-=47]-=-. This is an algebraic interpretation of the so-called rigidity constraint, namely the fact that for any fundamental matrix F there exist two intrinsic parameters matrix A and A 0 and a rigid motion r... |

5 |
Sequential update of projective and ane structure from motion
- Beardsley, Zisserman, et al.
- 1997
(Show Context)
Citation Context |

4 |
Ecient model library access by projectively invariant indexing functions
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- 1992
(Show Context)
Citation Context ...~ T. A projective reconstruction can be computed starting from points correspondences only, without any a-priori knowledge [17, 18, 45, 44, 19, 3, 2, 48]. Despite it conveys some useful in formations =-=[40, 39-=-], we would like to obtain a Euclidean reconstruction, a very special one that diers from the true reconstruction by an unknown similarity transformation. This is composed by a rigid displacement (due... |

3 |
Strati of 3-D Vision: Projective, Ane, and Metric Representations
- Faugeras
- 1995
(Show Context)
Citation Context ...d the direction of translation t are obtained directly from the factorization (12) of E. In [28] direct and iterative methods are compared. Recently, new approaches based on the idea of stratication [=-=30, 7]-=- have been introduced. Starting from a projective reconstruction, which can be computed from the set of correspondences f ~ m i j g only, the problem is computing the proper ~ T that upgrades it to a ... |