## Understanding the Relationship Between the Optimization Criteria in Two-View Motion Analysis (1998)

Citations: | 6 - 0 self |

### BibTeX

@MISC{Zhang98understandingthe,

author = {Zhengyou Zhang},

title = {Understanding the Relationship Between the Optimization Criteria in Two-View Motion Analysis},

year = {1998}

}

### OpenURL

### Abstract

The three best known criteria in two-view motion analysis ' are based, respectively, on the distances between points' and their corresponding epipolar lines, on the gradientweighted epipolar errors', and on the distances between points' and the reprojections of their reconstructed points'. The last one has a better statistical interpretation, but is', however, much slower than the first two. In this paper, we show that the last two criteria are equivalent when the epipoles are at infinity, and differ from each other only a little even when the epipoles are in the image. The first two criteria are equivalent only when the epipoles are at infinity and when the observed object has the same scale in the two images. This suggests' that the second criterion is sufficient in practice because of its' computational efficiency. The result is valid for both calibrated and uncalibrated images.

### Citations

1263 |
Three-dimensional computer vision, a geometric viewpoint
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- 1993
(Show Context)
Citation Context ...when uncalibrated full perspective projection model is assumed). Table 1: Sample references which use the three criteria considered in this paper (see text). criterion Euclidean Affine Projective J 1 =-=[2]-=- [10] [8] J 2 [6] [10] [8] J 3 [13] [11] [3] In this paper, the relationship of the three criteria is studied. Uncalibrated images are considered, but the results are equally valid for calibrated imag... |

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Citation Context ...ection model is assumed). Table 1: Sample references which use the three criteria considered in this paper (see text). criterion Euclidean Affine Projective J 1 [2] [10] [8] J 2 [6] [10] [8] J 3 [13] =-=[11]-=- [3] In this paper, the relationship of the three criteria is studied. Uncalibrated images are considered, but the results are equally valid for calibrated images. Analytical analysis is carried out, ... |

630 |
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Citation Context ... The two domains (calibrated or uncalibrated) share the same mathematical basis. If lens distortion can be ignored, two images are related by a 3 \Theta 3 matrix, which is called the essential matrix =-=[7, 12]-=- when image are calibrated and normalized image coordinates are used, and is called the fundamental matrix [8] when pixel image coordinates are used. The corresponding points between the two images mu... |

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Citation Context ...ation between two images with known intrinsic parameters (we say they are calibrated). Recently, analysis of uncalibrated images has attracted quite a number of researchers since the work of Faugeras =-=[1]-=- and Hartley [5]. There are several reasons: the calibration is fastidious and not very stable; it is impossible in many applications such as video sequence analysis; etc. The reader is referred to [1... |

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Citation Context ...fficiently large set of point correspondences: f(m i ; m 0 i ) j i = 1; : : : ; ng, where ns7. The point correspondences between two images can be established by a technique such as that described in =-=[17]-=-. We do not assume that the camera's intrinsic parameters are known, but the following analysis is equally valid for that case. 3 Three Criteria and Their Relationship There are a number of techniques... |

322 | Determining the epipolar geometry and its uncertainty: A review
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Citation Context ...1] and Hartley [5]. There are several reasons: the calibration is fastidious and not very stable; it is impossible in many applications such as video sequence analysis; etc. The reader is referred to =-=[15]-=- for a review. The two domains (calibrated or uncalibrated) share the same mathematical basis. If lens distortion can be ignored, two images are related by a 3 \Theta 3 matrix, which is called the ess... |

313 |
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Citation Context ... The two domains (calibrated or uncalibrated) share the same mathematical basis. If lens distortion can be ignored, two images are related by a 3 \Theta 3 matrix, which is called the essential matrix =-=[7, 12]-=- when image are calibrated and normalized image coordinates are used, and is called the fundamental matrix [8] when pixel image coordinates are used. The corresponding points between the two images mu... |

242 |
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Citation Context ...o images with known intrinsic parameters (we say they are calibrated). Recently, analysis of uncalibrated images has attracted quite a number of researchers since the work of Faugeras [1] and Hartley =-=[5]-=-. There are several reasons: the calibration is fastidious and not very stable; it is impossible in many applications such as video sequence analysis; etc. The reader is referred to [15] for a review.... |

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Citation Context ... full perspective projection model is assumed). Table 1: Sample references which use the three criteria considered in this paper (see text). criterion Euclidean Affine Projective J 1 [2] [10] [8] J 2 =-=[6]-=- [10] [8] J 3 [13] [11] [3] In this paper, the relationship of the three criteria is studied. Uncalibrated images are considered, but the results are equally valid for calibrated images. Analytical an... |

234 | Euclidean reconstruction from uncalibrated views
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- 1994
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Citation Context ...n model is assumed). Table 1: Sample references which use the three criteria considered in this paper (see text). criterion Euclidean Affine Projective J 1 [2] [10] [8] J 2 [6] [10] [8] J 3 [13] [11] =-=[3]-=- In this paper, the relationship of the three criteria is studied. Uncalibrated images are considered, but the results are equally valid for calibrated images. Analytical analysis is carried out, show... |

233 | The fundamental matrix: Theory, algorithms, and stability analysis
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Citation Context ..., two images are related by a 3 \Theta 3 matrix, which is called the essential matrix [7, 12] when image are calibrated and normalized image coordinates are used, and is called the fundamental matrix =-=[8]-=- when pixel image coordinates are used. The corresponding points between the two images must satisfy the epipolar constraint to be presented below. An important step in motion analysis, after the esta... |

133 |
Epipolar Geometry in Stereo Motion and Object Recognition
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Citation Context ...e \Pi, defined by m, C and C 0 , with the second image plane I 0 . Algebraically, in order for m in the first image and m 0 in the second image to be matched, the following equation must be satisfied =-=[14]-=-: ~ m 0T F ~ m = 0 ; (2) where F = [P 0 p ? ] \Theta P 0 P + : (3) Here, P + is the pseudo-inverse of matrix P: P + = P T (PP T ) \Gamma1 , and p ? is any 4-vector that is perpendicular to all the row... |

130 | Optimal motion and structure estimation
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Citation Context ... projection model is assumed). Table 1: Sample references which use the three criteria considered in this paper (see text). criterion Euclidean Affine Projective J 1 [2] [10] [8] J 2 [6] [10] [8] J 3 =-=[13]-=- [11] [3] In this paper, the relationship of the three criteria is studied. Uncalibrated images are considered, but the results are equally valid for calibrated images. Analytical analysis is carried ... |

117 |
defense of the eight-point algorithm
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- 1997
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Citation Context ... is not physically meaningful, and the result obtained is usually not satisfactory, which is usually refined through nonlinear optimization based on some appropriate criterion. Note, however, Hartley =-=[4]-=- shows that a simple normalization of image coordinates considerably improves the performance of the eight-point algorithm. 3.1 Three Criteria In nonlinear estimation techniques, the three best known ... |

73 | Active tracking of foveated feature clusters using affine structure
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- 1996
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Citation Context ... = 2 4 0 0s0 0s3 5 : For example, when the epipoles are at infinity or when the cameras are affine, the two criteria are equivalent. The case for affine cameras has already been shown by Oxford group =-=[10, 9]-=-. 3.3 Relationship Between J 1 and J 2 Denote again the epipolar residual by r i , i.e., r i = e m 0T i F e m i . Let w i = e m 0T i FZZ T F T e m 0 i and w 0 i = e m T i F T ZZ T F e m i , then J 1 (... |

60 |
3D motion recovery via affine epipolar geometry
- Shapiro, Zisserman, et al.
- 1995
(Show Context)
Citation Context ... uncalibrated full perspective projection model is assumed). Table 1: Sample references which use the three criteria considered in this paper (see text). criterion Euclidean Affine Projective J 1 [2] =-=[10]-=- [8] J 2 [6] [10] [8] J 3 [13] [11] [3] In this paper, the relationship of the three criteria is studied. Uncalibrated images are considered, but the results are equally valid for calibrated images. A... |

22 |
On the optimization criteria used in two-view motion analysis
- Zhang
- 1998
(Show Context)
Citation Context ...s in both images are at the image center, i.e., [255; 255] T . The test images are shown in Fig. 5.1, and the result is shown in Table 5. 5.2 Bias of J 1 with respect to J 2 The reader is referred to =-=[16]-=- for the details on the bias of J 1 with respect to J 2 . 5.3 Difference between J 2 and J 3 J 2 approximates J 3 quite well if an epipolar line does not change its orientation too much in the neighbo... |