## Vanishing point calculation as a statistical inference on the unit sphere (1990)

Venue: | In Proc. ICCV |

Citations: | 66 - 7 self |

### BibTeX

@INPROCEEDINGS{Collins90vanishingpoint,

author = {Robert T. Collins and Richard S. Weiss},

title = {Vanishing point calculation as a statistical inference on the unit sphere},

booktitle = {In Proc. ICCV},

year = {1990},

pages = {400--403}

}

### Years of Citing Articles

### OpenURL

### Abstract

\Lambda In this paper vanishing point computation is characterized as a statistical estimation problem on the unit sphere; in particular as the estimation of the polar axis of an equatorial distribution. This framework facilitates the construction of confidence regions for 3D line orientation.

### Citations

230 |
Statistics of Directional Data
- MARDIA
- 1972
(Show Context)
Citation Context ...ting the true polar axis of an antipodally symmetric equatorial distribution given a random sample of size n. Posed in this way, the problem falls squarely within the domain of directional statistics =-=[7, 11]-=-. In this section two methods for estimating the pole of an equatorial distribution f (X) are examined. The first is a parametric method based on the assumption that f (X) is a Bingham distribution; t... |

107 | On the sensitivity of the Hough Transform for Object Recognition
- Grimson
- 1990
(Show Context)
Citation Context ...onal benefits of the approach. This drawback can be overcome by using a more sophisticated hierarchical apparatus, as in Quan and Mohr [13]. Yet, other problems inherent to the Hough transform remain =-=[5]-=-, namely problems occurring when the true vanishing point falls near a histogram boundary, causing candidates which should be grouped together to fall into separate buckets. In summary, while algorith... |

70 |
Interpreting perspective images
- Barnard
- 1983
(Show Context)
Citation Context ...how to cluster line segments going to a single vanishing point, and how to estimate an accurate vanishing point from a given line cluster. The former is handled elegantly by Hough transform approachs =-=[1, 8, 13]-=-. In Barnard [1], line segments are mapped onto great circles in a histogram representing the surface of a unit sphere. Potential vanishing points are detected as peaks in the histogram, corresponding... |

42 |
Determining perspective structures using hierarchical Hough transform
- Quan, Mohr
- 1989
(Show Context)
Citation Context ...how to cluster line segments going to a single vanishing point, and how to estimate an accurate vanishing point from a given line cluster. The former is handled elegantly by Hough transform approachs =-=[1, 8, 13]-=-. In Barnard [1], line segments are mapped onto great circles in a histogram representing the surface of a unit sphere. Potential vanishing points are detected as peaks in the histogram, corresponding... |

37 |
Aggarwal: Determining Vanishing Points from Perspective Images
- Magee, K
- 1984
(Show Context)
Citation Context ...and variance should be based on the line segments themselves rather than the arbitrary bucket boundaries of a histogram data structure. In contrast to the Hough transform approach, Magee and Aggarwal =-=[10]-=- compute intersections of pairs line segments directly, using cross product operations. Vanishing points are detected as clusters of intersection points on the sphere. Since computed points are mainta... |

27 |
An antipodally symmetric distribution on the sphere
- Bingham
- 1974
(Show Context)
Citation Context ...ymmetric distributions. 2.1 Case 1 : Bingham's Distribution Bingham's distribution is a standard statistical distribution for representing both bipolar and equatorial clusters of points on the sphere =-=[2, 11]-=-. Bingham's distribution describes the intersection of a trivariate normal distribution N3(x; 0; \Sigma ) with the surface of the unit sphere. The covariance matrix \Sigmascan be decomposed into U S U... |

15 |
Using Perspective Transformation in Scene Analysis
- Haralick
- 1980
(Show Context)
Citation Context .... The common point of intersection, perhaps at infinity, is called the vanishing point. Vanishing point analysis provides strong cues for inferring the 3D structure of a scene from only a single view =-=[6]-=-. Under known camera geometry, the vanishing point of a group of parallel lines determines their orientation in three-space. Two or more vanishing points from lines known to lie in a single 3D plane e... |

10 |
Shape from texture: An aggregation transform that maps a class of textures into surface orientation
- Kender
- 1979
(Show Context)
Citation Context ...how to cluster line segments going to a single vanishing point, and how to estimate an accurate vanishing point from a given line cluster. The former is handled elegantly by Hough transform approachs =-=[1, 8, 13]-=-. In Barnard [1], line segments are mapped onto great circles in a histogram representing the surface of a unit sphere. Potential vanishing points are detected as peaks in the histogram, corresponding... |

7 |
An efficient and accurate method for computing vanishing points
- Collins, Weiss
- 1989
(Show Context)
Citation Context ... of line segments yields a complexity of O(n2). The camera focal point and the endpoints p and q of an image line segment form a projection plane with unit normal OE = (p \Thetasq) = kp \Thetasqk: In =-=[3]-=- a leastsquares perpendicular error plane is fit to the projection plane normals of line segments in convergent line clusters. The normal vector of a cluster's best-fit plane provides an estimate of t... |

6 |
A unified view of the theory of directional statistics, 1975–1988
- Jupp, Mardia
- 1989
(Show Context)
Citation Context ...ting the true polar axis of an antipodally symmetric equatorial distribution given a random sample of size n. Posed in this way, the problem falls squarely within the domain of directional statistics =-=[7, 11]-=-. In this section two methods for estimating the pole of an equatorial distribution f (X) are examined. The first is a parametric method based on the assumption that f (X) is a Bingham distribution; t... |

5 |
Asymptotic Expansions for the Bingham Distribution
- Kent
- 1987
(Show Context)
Citation Context ...the maximum likelihood estimate of the orientation matrix U is the matrix of eigenvectors A. Maximum likelihood estimates of the nonzero shape parameters k1 and k2 are nontrivial functions of \Lambdas=-=[9]-=-. The polar axis of this equatorial distribution, and hence the desired 3D orientation vector, has maximum likelihood estimate ^u = ^u1 = a1. Bingham suggestssa. uniform k1 := k2 := 0 b. symmetric bip... |

5 |
A distribution-free method of interval estimation for unsigned directional data
- Prentice
- 1984
(Show Context)
Citation Context ...of the O/2 distribution with 2 degrees of freedom. 2.2 Case 2 : General Antipodal Symmetry Prentice develops a nonparametric estimation procedure for antipodally symmetric distributions on the sphere =-=[12]-=-. His argument, based on the method of moments, is that the population second moment matrix E(xxt) = U KU t can be estimated by the sample second moment matrix (1). Equating matrix factors in the obvi... |

3 |
Deriving Line and Surface Orientation by Statistical Methods
- Collins, Weiss
- 1990
(Show Context)
Citation Context ...the unit sphere, and the statistical error of the estimate is determined. Section 3 presents an investigation into the sensitivity of the estimates to the number of lines in a convergent cluster. See =-=[4]-=- for a companion paper describing an application of directional statistics to another computer vision problem, determining 3D line and plane orientations from stereo line correspondences. 2 Vanishing ... |