## The Geometry of Projective Reconstruction I: Matching Constraints and the Joint Image (1995)

Venue: | International Journal of Computer Vision |

Citations: | 70 - 9 self |

### BibTeX

@INPROCEEDINGS{Triggs95thegeometry,

author = {Bill Triggs},

title = {The Geometry of Projective Reconstruction I: Matching Constraints and the Joint Image},

booktitle = {International Journal of Computer Vision},

year = {1995},

pages = {338--343}

}

### OpenURL

### Abstract

. This paper studies the geometry of perspective projection into multiple images and the matching constraints that this induces between the images. The combined projections produce a 3D subspace of the space of combined image coordinates called the joint image. This is a complete projective replica of the 3D world defined entirely in terms of image coordinates, up to an arbitrary choice of certain scale factors. Projective reconstruction is a canonical process in the joint image requiring only the rescaling of image coordinates. The matching constraints tell whether a set of image points is the projection of a single world point. In 3D there are only three types of matching constraint: the fundamental matrix, Shashua's trilinear tensor, and a new quadrilinear 4 image tensor. All of these fit into a single geometric object, the joint image Grassmannian tensor. This encodes exactly the information needed for reconstruction: the location of the joint image in the space of combined image ...

### Citations

1344 |
Three-Dimensional Computer Vision: a Geometric Viewpoint
- Faugeras
- 1993
(Show Context)
Citation Context ...situation was immediately evoked by those papers, although the details took several years to gel. In that time there has been a substantial amount of work on projective reconstruction. Faugeras' book =-=[4]-=- is an excellent general introduction and Maybank [15] provides a more mathematically oriented synthesis. Alternative approaches to projective reconstruction appear in Hartley et.al. [9] and Mohr et.a... |

675 |
A computer algorithm for reconstructing a scene from two projections
- Longuet-Higgins
- 1981
(Show Context)
Citation Context ...ically highly dependent. It is straightforward to provide linear minimumeigenvector methods to estimate: the 9-component fundamental matrix from at least 8 pairs of corresponding points in two images =-=[12]-=-, [13]; each of the three linearly independent 27-component trilinear tensors from at least 7 triples of points in three images; and the 81-component quadrilinear tensor from at least 6 quadruples of ... |

520 |
What can be seen in three dimensions with an uncalibrated stereo rig
- Faugeras
- 1992
(Show Context)
Citation Context ...erience effort spent learning the tensorial notation is amply repaid by increased clarity of thought. In origin this work dates from the initial projective reconstruction papers of Faugeras & Maybank =-=[3]-=-, [6], [5]. The underlying geometry of the situation was immediately evoked by those papers, although the details took several years to gel. In that time there has been a substantial amount of work on... |

373 | Camera self-calibration: theory and experiments
- Faugeras, Luong, et al.
- 1992
(Show Context)
Citation Context ...fort spent learning the tensorial notation is amply repaid by increased clarity of thought. In origin this work dates from the initial projective reconstruction papers of Faugeras & Maybank [3], [6], =-=[5]-=-. The underlying geometry of the situation was immediately evoked by those papers, although the details took several years to gel. In that time there has been a substantial amount of work on projectiv... |

305 |
Faugeras. A theory of self-calibration of a moving camera
- Maybank, D
- 1992
(Show Context)
Citation Context ...ce effort spent learning the tensorial notation is amply repaid by increased clarity of thought. In origin this work dates from the initial projective reconstruction papers of Faugeras & Maybank [3], =-=[6]-=-, [5]. The underlying geometry of the situation was immediately evoked by those papers, although the details took several years to gel. In that time there has been a substantial amount of work on proj... |

251 |
Stereo from uncalibrated cameras
- Hartley, Gupta, et al.
- 1992
(Show Context)
Citation Context ...augeras' book [4] is an excellent general introduction and Maybank [15] provides a more mathematically oriented synthesis. Alternative approaches to projective reconstruction appear in Hartley et.al. =-=[9]-=- and Mohr et.al. [17]. Luong&Vi' eville [14] have studied `canonic decompositions' of projection matrices for multiple views. Shashua [19] has developed the theory of the trilinear matching constraint... |

229 |
Structure of space-time
- Penrose
- 1968
(Show Context)
Citation Context ... (but not the `big picture ' --- the full joint image geometry). However the deepest debt of the current paper is to time spent in the Oxford mathematical physics research group lead by Roger Penrose =-=[18]-=-, whose notation I have `borrowed' and whose penetrating synthesis of the geometric and algebraic points of view has been a powerful tool and a constant source of inspiration. 2. Conventions and Notat... |

179 | Canonic representations for the geometries of multiple projective views
- Luong, Vieville
- 1993
(Show Context)
Citation Context ...ntroduction and Maybank [15] provides a more mathematically oriented synthesis. Alternative approaches to projective reconstruction appear in Hartley et.al. [9] and Mohr et.al. [17]. Luong&Vi' eville =-=[14]-=- have studied `canonic decompositions' of projection matrices for multiple views. Shashua [19] has developed the theory of the trilinear matching constraints, with input from Hartley [8]. A brief summ... |

164 |
Theory of reconstruction From Image Motion
- Maybank
- 1992
(Show Context)
Citation Context ...hough the details took several years to gel. In that time there has been a substantial amount of work on projective reconstruction. Faugeras' book [4] is an excellent general introduction and Maybank =-=[15]-=- provides a more mathematically oriented synthesis. Alternative approaches to projective reconstruction appear in Hartley et.al. [9] and Mohr et.al. [17]. Luong&Vi' eville [14] have studied `canonic d... |

152 | On the geometry and algebra of the point and line correspondence between n images
- Faugeras, Mourrain
- 1994
(Show Context)
Citation Context ...rilinear matching constraints, with input from Hartley [8]. A brief summary of the present paper appears in [20]. In parallel with the current work, both Werman & Shashua [22] and Faugeras & Mourrain =-=[7]-=- independently discovered the quadrilinear constraint and some of the related structure (but not the `big picture ' --- the full joint image geometry). However the deepest debt of the current paper is... |

150 | Algebraic Functions for Recognition - Shashua - 1995 |

118 |
The Hitchhiker’s Guide to the Galaxy
- Adams
- 1995
(Show Context)
Citation Context ...s about image projection geometry. Tensors have seldom been used in vision and many people appear to be rather tensorphobic, so it seems appropriate to say a few words in their favour: "Don't pan=-=ic!" [1]-=-. First of all, what is a tensor? --- It is a collection (a multidimensional array) of components that represent a single geometric object with respect to some system of coordinates, and that are inte... |

92 | On determining the fundamental matrix: analysis of different methods and experimental results,” INRIA, Tech. Rep. Raport de Recherche 1894
- Luong, Deriche, et al.
- 1993
(Show Context)
Citation Context ... highly dependent. It is straightforward to provide linear minimumeigenvector methods to estimate: the 9-component fundamental matrix from at least 8 pairs of corresponding points in two images [12], =-=[13]-=-; each of the three linearly independent 27-component trilinear tensors from at least 7 triples of points in three images; and the 81-component quadrilinear tensor from at least 6 quadruples of corres... |

77 | 3-D scene representation as a collection of images and fundamental matrices
- LAVEAU, FAUGERAS
- 1994
(Show Context)
Citation Context ...age given its location in two others, in a hypothesize-andtest framework. Image synthesis (e.g. image sequence compression and interpolation) is likely to be another important application of transfer =-=[11]-=-. Fundamental matrices can also be used for these applications, but because the higher order constraints `holistically' combine data from several images and there is built-in redundancy in the constra... |

76 | A unifying framework for structure and motion recovery from image sequences
- McLauchlan, Murray
- 1995
(Show Context)
Citation Context ... other hand,the simplicity of the joint image picture makes incremental recursive reconstruction techniques that correctly handle the measurement errors and constraint geometry seem more likely (c.f. =-=[16]-=-). 11.2. Tensors vs. the Rest This paper is as much about the use of tensors as a vehicle for mathematical vision as it is about image projection geometry. Tensors have seldom been used in vision and ... |

30 | Lines and points in three views – an integrated approach
- Hartley
- 1994
(Show Context)
Citation Context ...g&Vi' eville [14] have studied `canonic decompositions' of projection matrices for multiple views. Shashua [19] has developed the theory of the trilinear matching constraints, with input from Hartley =-=[8]-=-. A brief summary of the present paper appears in [20]. In parallel with the current work, both Werman & Shashua [22] and Faugeras & Mourrain [7] independently discovered the quadrilinear constraint a... |

26 |
Multiple image invariance using the double algebra
- Carlsson
- 1994
(Show Context)
Citation Context ...ortunately, one might also suggest that very little geometric insight is gained: the output is a complex set of equations with no very clearly articulated structure. The Grassmann-Cayley algebra [7], =-=[2]-=- is spiritually much closer to the tensorial point of view. Indeed, it can be viewed as a specialized index-free notation for manipulating completely antisymmetric covariant and contravariant tensors.... |

7 |
Elimination: An approach to the study of 3D-from-2D
- Werman, Shashua
- 1995
(Show Context)
Citation Context ...developed the theory of the trilinear matching constraints, with input from Hartley [8]. A brief summary of the present paper appears in [20]. In parallel with the current work, both Werman & Shashua =-=[22]-=- and Faugeras & Mourrain [7] independently discovered the quadrilinear constraint and some of the related structure (but not the `big picture ' --- the full joint image geometry). However the deepest ... |

5 |
Methods of Algebraic Geometry: Volume 1
- Hodge, Pedoe
- 1994
(Show Context)
Citation Context ...ent of projective space are based on Roger Penrose's approach to relativistic spinors and twistors [18], which partly derives (I believe) from Hodge & Pedoe's excellent treatise on algebraic geometry =-=[10]-=-. Without these powerful tools this work would not have been possible. Geometry of Projective Reconstruction 25 Appendix A Mathematical Background This appendix provides a very brief overview of the l... |

3 |
Accurate relative positioning from multiple images
- Mohr, Boufama, et al.
- 1993
(Show Context)
Citation Context ... an excellent general introduction and Maybank [15] provides a more mathematically oriented synthesis. Alternative approaches to projective reconstruction appear in Hartley et.al. [9] and Mohr et.al. =-=[17]-=-. Luong&Vi' eville [14] have studied `canonic decompositions' of projection matrices for multiple views. Shashua [19] has developed the theory of the trilinear matching constraints, with input from Ha... |

2 |
Least squares estimation in projective spaces
- Triggs
- 1995
(Show Context)
Citation Context ...reconstruction seems relatively straightforward once image tokens have been put into correspondence, although much remains to be done on the practical aspects, particularly on error models [17], [4], =-=[21]-=- and the recovery of Euclidean structure [17]. Given the complexity and algebraic redundancy of the trilinear and quadrilinear constraints it is certainly legitimate to ask whether they are actually l... |

1 |
The Grassmannian geometry of projective reconstruction
- Triggs
- 1995
(Show Context)
Citation Context ...s' of projection matrices for multiple views. Shashua [19] has developed the theory of the trilinear matching constraints, with input from Hartley [8]. A brief summary of the present paper appears in =-=[20]-=-. In parallel with the current work, both Werman & Shashua [22] and Faugeras & Mourrain [7] independently discovered the quadrilinear constraint and some of the related structure (but not the `big pic... |