## Structure and Motion from Uncalibrated Catadioptric Views (2001)

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Venue: | In Proc. CVPR |

Citations: | 49 - 5 self |

### BibTeX

@INPROCEEDINGS{Geyer01structureand,

author = {Christopher Geyer and Kostas Daniilidis},

title = {Structure and Motion from Uncalibrated Catadioptric Views},

booktitle = {In Proc. CVPR},

year = {2001},

pages = {279--286}

}

### Years of Citing Articles

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### Abstract

In this paper we present a new algorithm for structure from motion from point correspondences in images taken from uncalibrated catadioptric cameras with parabolic mirrors. We assume that the unknown intrinsic parameters are three: the combined focal length of the mirror and lens and the intersection of the optical axis with the image. We introduce a new representation for images of points and lines in catadioptric images which we call the circle space. This circle space includes imaginary circles, one of which is the image of the absolute conic. We formulate the epipolar constraint in this space and establish a new 4 × 4 catadioptric fundamental matrix. We show that the image of the absolute conic belongs to the kernel of this matrix. This enables us to prove that Euclidean reconstruction is feasible from two views with constant parameters and from three views with varying parameters. In both cases, it is one less than the number of views necessary with perspective cameras.

### Citations

468 |
Multiple View Geometry
- Hartley, Zisserman
- 2000
(Show Context)
Citation Context ...correspondences. This is considered now a thoroughly understood problem. Solutions and insights gained from these studies boosted applications in video processing and image based rendering. Two books =-=[10]-=- and [5] contain comprehensive treatments of the subject. In the meantime, computer vision researchers realized that perspective cameras are just one modality among many. Motivated by the need for a p... |

292 |
A theory of self-calibration of a moving camera
- Maybank, Faugeras
- 1992
(Show Context)
Citation Context ... camera parameters. In both cases, it is one view less, than in the case of perspective views with the same unknowns (focal length and image center): Three views are necessary for constant parameters =-=[14, 13]-=- and four views are necessary for varying parameters [11]. In particular the fundamental matrix has seven degrees of freedom whereas the intrinsics have three and the motion has five for a total of ei... |

282 | Catadioptric Omnidirectional Camera
- Nayar
- 1997
(Show Context)
Citation Context .... [17] studied several configurations of unknown and varying parameters. In the omnidirectional vision literature, there are very few approaches dealing with structure from motion. Gluckman and Nayar =-=[9]-=- studied ego-motion estimation by mapping the catadioptric image to the sphere. Svoboda et al [20] first established the epipolar geometry for all central catadioptric systems. Kang [12] proposed a di... |

158 | Self-calibration and metric reconstruction in spite of varying and unknown internal camera parameters
- Pollefeys, Gool
- 1998
(Show Context)
Citation Context ...erate configurations for the same assumption. Heyden and Astrom [11] proved that four views suffice for unknown varying focal length and image center but known aspect ratio and skew. Pollefeys et al. =-=[17]-=- studied several configurations of unknown and varying parameters. In the omnidirectional vision literature, there are very few approaches dealing with structure from motion. Gluckman and Nayar [9] st... |

141 | A unifying theory for central panoramic systems
- Geyer, Daniilidis
- 2000
(Show Context)
Citation Context ...dioptric, attracted special attention due to their elegant and useful geometric properties. Several authors have studied the properties of central catadioptric cameras and the image formation in them =-=[15, 20, 3, 22, 12, 8]-=-. Kang [12] proposed a single view approach from the image of the circular mirror boundary of a paraboloid mirror. Geyer and Daniilidis showed [7, 8] how calibration of a parabolic catadioptric system... |

106 | Euclidean Reconstruction from Image Sequences with Varying and
- Heyden, Aström
- 1997
(Show Context)
Citation Context ...in the case of perspective views with the same unknowns (focal length and image center): Three views are necessary for constant parameters [14, 13] and four views are necessary for varying parameters =-=[11]-=-. In particular the fundamental matrix has seven degrees of freedom whereas the intrinsics have three and the motion has five for a total of eight. In the three view case the trifocal tensor has 19 fr... |

96 | Catadioptric Camera Calibration
- Geyer, Daniilidis
- 1999
(Show Context)
Citation Context ...as and the image formation in them [15, 20, 3, 22, 12, 8]. Kang [12] proposed a single view approach from the image of the circular mirror boundary of a paraboloid mirror. Geyer and Daniilidis showed =-=[7, 8]-=- how calibration of a parabolic catadioptric system can be achieved from a single view of three lines in space or from a single view of two sets of parallel lines. In this paper, we study the recovery... |

71 | Automatic recovery of relative camera rotations for urban scenes
- Antone, Teller
- 2000
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Citation Context ...ystems. Kang [12] proposed a direct selfcalibration by minimizing the epipolar constraint. Fermueller and Aloimonos [6] proved the superiority of the sphere over the plane regarding stability. Teller =-=[1]-=- showed how to compute ego-motion from spherical mosaics. Multiple view algorithms for the perspective case which assume piecewise planar environments are simpler when modified for catadioptric imager... |

68 |
The Geometry of Multiple Images: The Laws that Govern the Formation of Images of a Scene
- Faugeras, Luong, et al.
- 2001
(Show Context)
Citation Context ...dences. This is considered now a thoroughly understood problem. Solutions and insights gained from these studies boosted applications in video processing and image based rendering. Two books [10] and =-=[5]-=- contain comprehensive treatments of the subject. In the meantime, computer vision researchers realized that perspective cameras are just one modality among many. Motivated by the need for a panoramic... |

63 | Epipolar geometry for panoramic cameras
- Svoboda, Pajdla, et al.
- 1998
(Show Context)
Citation Context ...dioptric, attracted special attention due to their elegant and useful geometric properties. Several authors have studied the properties of central catadioptric cameras and the image formation in them =-=[15, 20, 3, 22, 12, 8]-=-. Kang [12] proposed a single view approach from the image of the circular mirror boundary of a paraboloid mirror. Geyer and Daniilidis showed [7, 8] how calibration of a parabolic catadioptric system... |

61 | Catadioptric self-calibration
- Kang
- 2000
(Show Context)
Citation Context ...dioptric, attracted special attention due to their elegant and useful geometric properties. Several authors have studied the properties of central catadioptric cameras and the image formation in them =-=[15, 20, 3, 22, 12, 8]-=-. Kang [12] proposed a single view approach from the image of the circular mirror boundary of a paraboloid mirror. Geyer and Daniilidis showed [7, 8] how calibration of a parabolic catadioptric system... |

49 |
Real-time omnidirectional image sensor (COPIS) for vision-guided navigation
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- 1994
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45 | Critical motion sequences for the self-calibration of cameras and stereo systems with variable focal length, Image Vision Comput
- Sturm
- 2002
(Show Context)
Citation Context ... but constant. The results still hold for known aspect ratio and skew. Hartley [10] showed that a varying focal length can be recovered from two views with all other intrinsic parameters fixed. Sturm =-=[18]-=- studied the degenerate configurations for the same assumption. Heyden and Astrom [11] proved that four views suffice for unknown varying focal length and image center but known aspect ratio and skew.... |

36 |
Geometry, a comprehensive course
- Pedoe
- 1988
(Show Context)
Citation Context ...elds the intrinsic parameters. 2.2. Parabolic Circle Space In the next few paragraphs we consider an abstract paraboloid which is different from the physical paraboloid of the mirror. Following Pedoe =-=[16], -=-we use this surface to describe a correspondence between points in space and circles in the plane. Lines in this circle space correspond � to one parameter systems of coaxial circles. Planes in the ... |

33 |
Panoramic Vision
- Benosman, Kang
- 2000
(Show Context)
Citation Context ...a panoramic field of view, catadioptric cameras have been designed and can be already purchased off-the-shelf. For an extensive coverage the reader is referred to the recent book by Benosman and Kang =-=[2] -=-and the proceedings of the Workshop for Omnidirectional Vision [4]. Among several designs, the catadioptric systems with a single effective viewpoint, called central £ This work has been supported by... |

20 |
Omniview cameras with curved surface mirrors
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- 2000
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19 | Ambiguity in structure from motion: Sphere versus plane
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- 1998
(Show Context)
Citation Context ...da et al [20] first established the epipolar geometry for all central catadioptric systems. Kang [12] proposed a direct selfcalibration by minimizing the epipolar constraint. Fermueller and Aloimonos =-=[6]-=- proved the superiority of the sphere over the plane regarding stability. Teller [1] showed how to compute ego-motion from spherical mosaics. Multiple view algorithms for the perspective case which as... |

19 | Euclidean reconstruction and reprojection up to subgroups
- Ma, Soatto, et al.
- 2000
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Citation Context ... camera parameters. In both cases, it is one view less, than in the case of perspective views with the same unknowns (focal length and image center): Three views are necessary for constant parameters =-=[14, 13]-=- and four views are necessary for varying parameters [11]. In particular the fundamental matrix has seven degrees of freedom whereas the intrinsics have three and the motion has five for a total of ei... |

16 | A method for 3D-reconstruction of piecewise planar objects from single panoramic images
- Sturm
- 2000
(Show Context)
Citation Context ...owed how to compute ego-motion from spherical mosaics. Multiple view algorithms for the perspective case which assume piecewise planar environments are simpler when modified for catadioptric imagery. =-=[21, 19]-=-. In the next section we mention introductory facts about catadioptric geometry. We introduce the notion of circle space and we find the image of the absolute conic on that space. We finish the second... |

6 |
Video Plus
- Taylor
- 2000
(Show Context)
Citation Context ...owed how to compute ego-motion from spherical mosaics. Multiple view algorithms for the perspective case which assume piecewise planar environments are simpler when modified for catadioptric imagery. =-=[21, 19]-=-. In the next section we mention introductory facts about catadioptric geometry. We introduce the notion of circle space and we find the image of the absolute conic on that space. We finish the second... |