## On The Geometry Of Visual Correspondence (1994)

Venue: | International Journal of Computer Vision |

Citations: | 19 - 12 self |

### BibTeX

@ARTICLE{Fermüller94onthe,

author = {Cornelia Fermüller and Yiannis Aloimonos},

title = {On The Geometry Of Visual Correspondence},

journal = {International Journal of Computer Vision},

year = {1994},

volume = {21},

pages = {223--247}

}

### OpenURL

### Abstract

Image displacement fields---optical flow fields, stereo disparity fields, normal flow fields---due to rigid motion possess a global geometric structure which is independent of the scene in view. Motion vectors of certain lengths and directions are constrained to lie on the imaging surface at particular loci whose location and form depends solely on the 3D motion parameters. If optical flow fields or stereo disparity fields are considered, then equal vectors are shown to lie on conic sections. Similarly, for normal motion fields, equal vectors lie within regions whose boundaries also constitute conics. By studying various properties of these curves and regions and their relationships, a characterization of the structure of rigid motion fields is given. The goal of this paper is to introduce a concept underlying the global structure of image displacement fields. This concept gives rise to various constraints that could form the basis of algorithms for the recovery of visual information f...

### Citations

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(Show Context)
Citation Context ...s, motion disparity measurements, motion fields, partial disparity fields such as those along the x- or y-axes, or normal motion fields (the projections of motion fields along the gradient direction) =-=[3, 16, 18, 19, 28]-=-. Second, knowledge of the model of the geometric transformation between the multiple views provides constraints on the local descriptors; these constraints are used to relate image measurements to th... |

453 | Three Dimensional Computer Vision - Faugeras - 1993 |

410 |
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Citation Context ...asks. Indeed, the problems of stereo, 3D motion estimation, calibration, obstacle detection, pose estimation for recognition, etc. can be regarded as special instances of the general recovery problem =-=[6--10, 14, 20, 24, 25, 30, 32, 33]-=-. Approaches to various aspects of this problem that have appeared in the literature seek a solution in two computational steps. First, a description that relates local measurements in multiple views ... |

332 | Active perception - Bajcsy - 1988 |

313 |
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(Show Context)
Citation Context ... of the geometric transformation between the multiple views provides constraints on the local descriptors; these constraints are used to relate image measurements to the 3D scene and viewing geometry =-=[11, 17, 20, 21, 23, 26, 29, 31]-=-. This paper deals with the case where the transformation between the views is described by a rigid motion. The rigid motion model imposes constraints on the local image measurements (disparity measur... |

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Citation Context ...enote r 1 t by u and r 2 t by v, t = (U; V; W ), ! = (ff; fi; fl). If we introduce new coordinates for the direction of translation (x 0 ; y 0 ) = ( Uf W ; V f W ), we obtain the well-known equations =-=[22]-=- u = u trans + u rot = = (\Gammax 0 + x) W Z + ff xy f \Gamma fi( x 2 f + f) + fly (13) v = v trans + v rot = = (\Gammay 0 + y) W Z + ff( y 2 f + f) \Gamma fi xy f \Gamma flx (14) The projection of a ... |

292 |
Affine structure from motion
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Citation Context ...asks. Indeed, the problems of stereo, 3D motion estimation, calibration, obstacle detection, pose estimation for recognition, etc. can be regarded as special instances of the general recovery problem =-=[6--10, 14, 20, 24, 25, 30, 32, 33]-=-. Approaches to various aspects of this problem that have appeared in the literature seek a solution in two computational steps. First, a description that relates local measurements in multiple views ... |

157 |
Theory of reconstruction from image motion
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(Show Context)
Citation Context ... of the geometric transformation between the multiple views provides constraints on the local descriptors; these constraints are used to relate image measurements to the 3D scene and viewing geometry =-=[11, 17, 20, 21, 23, 26, 29, 31]-=-. This paper deals with the case where the transformation between the views is described by a rigid motion. The rigid motion model imposes constraints on the local image measurements (disparity measur... |

142 |
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(Show Context)
Citation Context ...heir translational motion components. 4 Normal motion constraints The vector field which represents the components of the motion field perpendicular to edges is referred to as the normal motion field =-=[1, 34]-=-. It is uniquely defined by local image measurements and can be derived without confronting the aperture problem. Although a normal motion field seems to contain less information than the exact motion... |

132 |
Motion from point matches: Multiplicity of solutions
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(Show Context)
Citation Context ... of the geometric transformation between the multiple views provides constraints on the local descriptors; these constraints are used to relate image measurements to the 3D scene and viewing geometry =-=[11, 17, 20, 21, 23, 26, 29, 31]-=-. This paper deals with the case where the transformation between the views is described by a rigid motion. The rigid motion model imposes constraints on the local image measurements (disparity measur... |

130 | Relative orientation
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Citation Context |

111 |
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Citation Context ...s, motion disparity measurements, motion fields, partial disparity fields such as those along the x- or y-axes, or normal motion fields (the projections of motion fields along the gradient direction) =-=[3, 16, 18, 19, 28]-=-. Second, knowledge of the model of the geometric transformation between the multiple views provides constraints on the local descriptors; these constraints are used to relate image measurements to th... |

101 | Motion field and optical flow:qualitative properties
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(Show Context)
Citation Context ...heir translational motion components. 4 Normal motion constraints The vector field which represents the components of the motion field perpendicular to edges is referred to as the normal motion field =-=[1, 34]-=-. It is uniquely defined by local image measurements and can be derived without confronting the aperture problem. Although a normal motion field seems to contain less information than the exact motion... |

91 |
Handbuch der physiologischen Optik
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Citation Context ...d due to binocular disparity measurements obtained by a stereo system fixating at a point deserves some separate discussion. This configuration has often been studied in the psychophysical literature =-=[15]-=-. In particular, a concept has been investigated that can be regarded as a special case of the iso-motion contours: The locus of points in space that yield zero disparity, the so-called horopter . For... |

90 |
Computations underlying the measurement of visual motion
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Citation Context ...s, motion disparity measurements, motion fields, partial disparity fields such as those along the x- or y-axes, or normal motion fields (the projections of motion fields along the gradient direction) =-=[3, 16, 18, 19, 28]-=-. Second, knowledge of the model of the geometric transformation between the multiple views provides constraints on the local descriptors; these constraints are used to relate image measurements to th... |

44 |
Regularization of discontinuous flow fields
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Citation Context |

43 |
Dynamic aspects in active vision
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Citation Context ...asks. Indeed, the problems of stereo, 3D motion estimation, calibration, obstacle detection, pose estimation for recognition, etc. can be regarded as special instances of the general recovery problem =-=[6--10, 14, 20, 24, 25, 30, 32, 33]-=-. Approaches to various aspects of this problem that have appeared in the literature seek a solution in two computational steps. First, a description that relates local measurements in multiple views ... |

42 |
The Interpretation of Structure from Motion
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(Show Context)
Citation Context |

41 |
Egomotion and relative depth map from optical flow
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(Show Context)
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34 | Analytical results on error sensitivity of motion estimation from two views - Danilidis, Nagel - 1990 |

28 |
Estimation of rigid body motion using straight line correspondences
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(Show Context)
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21 | On the Error Sensitivity in the Recovery of Object Descriptions - Daniilidis - 1992 |

21 |
The role of fixation in visual motion analysis
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(Show Context)
Citation Context ... In particular, the normal motion vectors of length zero are bounded by curves which can be described by only three parameters. Thus a simple search technique in a three-dimensional space, such as in =-=[12, 13]-=-, could be employed to find the 3D motion parameters. The iso-normal motion areas, however, also allow to obtain bounds on the solutions for the motion parameters in very simple ways. Computations of ... |

21 |
Optimal computing of structure from motion using point correspondences in two frames
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- 1988
(Show Context)
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19 | Active Perception - Aloimonos - 1993 |

18 | Stereopsis, vertical disparity and relief transformations
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(Show Context)
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17 |
Finding motion parameters from spherical flow fields (or the advantages of having eyes in the back of your head
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- 1988
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16 |
Navigatinal preliminaries
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- 1993
(Show Context)
Citation Context ...d of considering vectors of the same value, we examine various classes of vectorvalued functions. In particular, we investigate the coaxis- and copoint vectors, which we described in an earlier paper =-=[12]-=-. 24 FOE AOR n 1 n 2 (a) (b) FOE (c) (d) AOR (e) (f) Figure 22: (a) Normal motion field due to translation and rotation. Superimposed on the vector field are the boundaries of the two normal motion ar... |

14 |
Introducing a smoothness constraint in a matching approach for the computation of optical flow fields
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(Show Context)
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14 | The critical sets of lines for camera displacement estimation:a mixed euclideanprojective and constructive approach
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13 | Motion and structure from motion from point and line matching. ICCV - Lustman, Faugeras, et al. - 1987 |

9 | Motion from flow along contours: A note on robustness and ambiguous cases - Bergholm - 1988 |

1 | Disparity Space under Binocular Fixation - Baratoff |