## Relative Affine Structure: Canonical Model for 3D from 2D Geometry and Applications (1996)

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Venue: | IEEE Transactions on Pattern Analysis and Machine Intelligence |

Citations: | 57 - 9 self |

### BibTeX

@ARTICLE{Shashua96relativeaffine,

author = {Amnon Shashua and Nassir Navab},

title = {Relative Affine Structure: Canonical Model for 3D from 2D Geometry and Applications},

journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},

year = {1996},

volume = {18},

pages = {88--3}

}

### Years of Citing Articles

### OpenURL

### Abstract

We propose an affine framework for perspective views, captured by a single extremely simple equation based on a viewer-centered invariant we call relative affine structure. Via a number of corollaries of our main results we show that our framework unifies previous work -- including Euclidean, projective and affine -- in a natural and simple way, and introduces new, extremely simple, algorithms for the tasks of reconstruction from multiple views, recognition by alignment, and certain image coding applications.

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Citation Context ...sed in this experiment (corresponding points are marked by overlapping squares in Fig. 5). The correspondence process is based on an implementation of a coarse-to-fine optical-flow algorithm based on =-=[26]-=- and described in [3]. The surface was reconstructed with respect to the ground plane (that served as a reference plane) and transformed to a camera coordinate system by assuming that the ground plane... |

635 |
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Citation Context ...particular case, the product [v 0 ]R remains fixed if we add to R a element that vanishes as a product with [v 0 ] --- an observation that was made previously by [16]. Thirdly, the "essential&quo=-=t; matrix [25], E -=-= [v 0 ]R, is shown to be a specialization of F in the case �� is at infinity with respect to the world coordinate frame and the cameras are internally calibrated as M = M 0 = I. 4 RELATIVE AFFINE... |

582 | H.: Hierarchical model-based motion estimation
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Citation Context ... (corresponding points are marked by overlapping squares in Fig. 5). The correspondence process is based on an implementation of a coarse-to-fine optical-flow algorithm based on [26] and described in =-=[3]-=-. The surface was reconstructed with respect to the ground plane (that served as a reference plane) and transformed to a camera coordinate system by assuming that the ground plane is parallel to the i... |

502 |
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Citation Context ...es of objects --- and there is no need to make a distinction between orthographic and perspective projections. The list of contributions to this framework include (though not intended to be complete) =-=[21, 2, 13, 45, 46, 16, 32, 7, 39, 31, 44, 36, 9, 11, 29, 20, 19, 47]-=- --- and relevant to this paper are the work described in [21, 8, 14, 39]. The material introduced so far in the literature, concerning 3D geometry from multiple views, focuses on the projective frame... |

411 |
Recognition by linear combinations of models
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Citation Context ...es of objects --- and there is no need to make a distinction between orthographic and perspective projections. The list of contributions to this framework include (though not intended to be complete) =-=[21, 2, 13, 45, 46, 16, 32, 7, 39, 31, 44, 36, 9, 11, 29, 20, 19, 47]-=- --- and relevant to this paper are the work described in [21, 8, 14, 39]. The material introduced so far in the literature, concerning 3D geometry from multiple views, focuses on the projective frame... |

362 | Camera self-calibration: theory and experiments
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Citation Context ...es of objects --- and there is no need to make a distinction between orthographic and perspective projections. The list of contributions to this framework include (though not intended to be complete) =-=[21, 2, 13, 45, 46, 16, 32, 7, 39, 31, 44, 36, 9, 11, 29, 20, 19, 47]-=- --- and relevant to this paper are the work described in [21, 8, 14, 39]. The material introduced so far in the literature, concerning 3D geometry from multiple views, focuses on the projective frame... |

315 |
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Citation Context ...ma1 p = Ap: In other words, the homography matrix associated with �� is A �� = M 0 (R + Tn ? d �� )M \Gamma1 ; (2) which is the generalization of the classical motion of planes in the cali=-=brated case [10, 42]. 4 RELA-=-TIVE AFFINE STRUCTURE 8 let v 0 = M 0 T , then for a general point P we have: p 0 �� = M 0 RM \Gamma1p + 1 z v 0 = Ap \Gamma 1 d �� v 0 n ? M \Gamma1 p + 1 z v 0 = Ap + 1 z \Gamma n ? (M \Gamm... |

294 |
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Citation Context |

257 | Recursive estimation of motion, structure, and focal length
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Citation Context ...at the University of Massachusetts at Amherst. Hence, one can measure the performance of reconstruction at a quantitative level. Reconstruction results on this set by other approaches can be found in =-=[6, 34, 23, 46, 1]-=-. On the sneaker images, a set of points were manually selected on one of the frames, referred to as the first frame, and their correspondences were automatically obtained along all other frames used ... |

254 |
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Citation Context ...amera transformation matrix is [A; v 0 ]. 4.2 Application II: Recognition by Alignment The relative affine invariance relation, captured by Theorem 1, can be used for visual recognition by alignment (=-=[43, 17], and references the-=-rein). In other words, the invariance of k can be used to "re-project" the object onto any third view p 00 , as follows. Given two "model" views in full correspondence p i /! p 0 i... |

239 |
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192 | Recovering 3d shape and motion from image streams using non-linear least squares - Szeliski, Kang - 1994 |

176 | Canonic representations for the geometries of multiple projective views
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Citation Context ...camera motion parameters, the parameters of ��, and the internal camera parameters of both cameras. This result is a natural extension of the classical motion of planes found in [10, 42], and also=-= in [28]. Eqn. 2-=- also implies that any homography matrix A can be described by some other homography matrix, the epipole and four parameters: A = A �� + 1 d v 0 n ? ; where A �� is a homography matrix associa... |

163 |
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Citation Context ...amera transformation matrix is [A; v 0 ]. 4.2 Application II: Recognition by Alignment The relative affine invariance relation, captured by Theorem 1, can be used for visual recognition by alignment (=-=[43, 17], and references the-=-rein). In other words, the invariance of k can be used to "re-project" the object onto any third view p 00 , as follows. Given two "model" views in full correspondence p i /! p 0 i... |

148 | Algebraic functions for recognition - Shashua - 1995 |

134 |
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133 |
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107 | What can two images tell us about a third one
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Citation Context ...v 00 . Recognition is achieved if the distance between p 00 i andsp 00 i , i = 7; :::; n, is sufficiently small. Other methods for achieving reprojection include the epipolar intersection method (cf. =-=[32, 5, 12]-=-), or by using projective structure instead of the relative affine structure [39]. 4.3 Application III: Image Coding The re-projection paradigm, described in the previous section, can serve as a princ... |

105 | Trilinearity in visual recognition by alignment - Shashua - 1994 |

101 | Example based image analysis and synthesis
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- 1993
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Citation Context ...periments can be found later in the text, and more detailed experiments concerning the use of optical flow in full registration of images for purposes of model-based image compression can be found in =-=[4]-=-. 5 Experimental Results The following experiments were conducted to illustrate the applications that arise from the relative affine framework (reconstruction, recognition by alignment, and image codi... |

91 | On determining the Fundamental matrix: analysis of different methods and experimental results
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- 1993
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Citation Context ...responding points are needed for a linear solution, and a least-squares solution is possible if more points are available. In practice the best results were obtained using the non-linear algorithm of =-=[27]. Th-=-e epipoles follow by F j v j = 0 and F ? v 0 j = 0 [7]. The latter readily follows from Corollary 4 as [v 0 j ]A j v j �� = [v 0 j ]v 0 j = 0 and A ? j [v 0 j ] ? v 0 j = \GammaA ? j [v 0 j ]v 0 j... |

91 | Estimating the Kinematics and Structure of a Rigid Object from a Sequence of Monocular Images - Broida, Chellappa |

89 | Relative 3D reconstruction using multiple uncalibrated images
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89 | Robust recovery of the epipolar geometry for an uncalibrated stereo rig - Deriche, Zhang, et al. - 1994 |

78 | Projective reconstruction and invariants from multiple images
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Citation Context ...ributions to this framework include (though not intended to be complete) [21, 2, 13, 45, 46, 16, 32, 7, 39, 31, 44, 36, 9, 11, 29, 20, 19, 47] --- and relevant to this paper are the work described in =-=[21, 8, 14, 39]-=-. The material introduced so far in the literature, concerning 3D geometry from multiple views, focuses on the projective framework [8, 14, 39], or the affine framework. The latter requires either ass... |

76 | Direct recovery of shape from multiple views: A parallax based approach
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- 1994
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Citation Context ... very simple an intuitive way, and produces a simple method for computing invariants from perspective views. This approach has also independently evolved in various forms and notations in the work of =-=[41, 18, 22, 37]. (0-=-,0,0,1) (1,1,1,1) (1,0,0,0) (0,1,0,0) (0,0,1,0) (a,b,g,0) (a,b,g,0) ' ' ' P P P 1 2 3 O O' P (1,1,1,0) P v v' p p' Fig. 2. See proof of Theorem 1. Theorem 1 (Relative Affine Structure) Let �� be s... |

73 | Geometry and photometry in 3D visual recognition - Shashua - 1992 |

72 | Stratification of Three Dimensional Vision: Projective, Affine and Metric Representations - Faugeras - 1995 |

63 | Recovery of ego-motion using image stabilization
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Citation Context ... very simple an intuitive way, and produces a simple method for computing invariants from perspective views. This approach has also independently evolved in various forms and notations in the work of =-=[41, 18, 22, 37]. (0-=-,0,0,1) (1,1,1,1) (1,0,0,0) (0,1,0,0) (0,0,1,0) (a,b,g,0) (a,b,g,0) ' ' ' P P P 1 2 3 O O' P (1,1,1,0) P v v' p p' Fig. 2. See proof of Theorem 1. Theorem 1 (Relative Affine Structure) Let �� be s... |

63 |
Geometric Invariants and Object Recognition
- Weiss
- 1993
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Citation Context |

62 | Projective structure from uncalibrated images: Structure from motion and recognition
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- 1994
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Citation Context |

61 | Hierarchical motion-based frame rate conversion - Bergen, Hingorani - 854 |

59 |
3D Geometry from Planar Parallax
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- 1994
(Show Context)
Citation Context ... very simple an intuitive way, and produces a simple method for computing invariants from perspective views. This approach has also independently evolved in various forms and notations in the work of =-=[41, 18, 22, 37]. (0-=-,0,0,1) (1,1,1,1) (1,0,0,0) (0,1,0,0) (0,0,1,0) (a,b,g,0) (a,b,g,0) ' ' ' P P P 1 2 3 O O' P (1,1,1,0) P v v' p p' Fig. 2. See proof of Theorem 1. Theorem 1 (Relative Affine Structure) Let �� be s... |

58 | Relative affine structure: Theory and application to 3D reconstruction from perspective views
- Shashua, Navab
- 1994
(Show Context)
Citation Context ...all the available degrees of freedom are captured by four points and one projective transformation, was also recently presented in [41] using different notations and tools than those used here and in =-=[38, 40]. This &qu-=-ot;middle ground" approach has several advantages. First, the results are sharper than a full projective reconstruction approach ([7, 16]) where five scene points are needed. The increased sharpn... |

57 | Recursive estimation of structure and motion using relative orientation constraints - Azarbayejani, Horowitz, et al. - 1993 |

54 |
Relative 3d positioning and 3d convex hull computation from a weakly calibrated stereo pair
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- 1995
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52 | and incremental acquisition of invariant shape models from image sequences,” Pattern Analysis and Machine Intelligence
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49 |
Affine reconstruction from perspective image pairs with a relative object-camera translation in between
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- 1996
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Citation Context ...of course, we cannot recover the homography due to the plane at infinity unless we are given prior information on the nature of the scene structure [35], or the camera motion is purely translational (=-=[30]-=-. Thus in the general case, we can realize either the relative affine framework or the projective framework. Corollary 3 The projective structure of the scene can be described as the ratio of two rela... |

44 | Factoring image sequences into shape and motion - Tomasi, Kanade - 1992 |

41 | Computing matched-epipolar projections
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- 1993
(Show Context)
Citation Context ...; where A �� is a homography matrix associated with some plane ��, and n; d are the parameters describing the plane associated with the homography A. A similar form of this equation was shown =-=also in [15]-=-. The relative affine structure k is described as a product of the affine structure under parallel projection (d=d o ) and a term that contains the location of the camera center of 4 RELATIVE AFFINE S... |

41 | Projective depth: A geometric invariant for 3D reconstruction from two perspective/orthographic views and for visual recognition - Shashua - 1993 |

41 | Motion from point matches using affine epipolar geometry - Shapiro, Zisserman, et al. - 1994 |

34 | A common framework for kinetic depth, reconstruction and motion for deformable objects
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- 1994
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34 | A ne Structure from Motion - Koenderink, Doorn - 1992 |

31 | Affine and Projective Structure from Motion
- Demey, Zisserman, et al.
- 1992
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Citation Context ...v 00 . Recognition is achieved if the distance between p 00 i andsp 00 i , i = 7; :::; n, is sufficiently small. Other methods for achieving reprojection include the epipolar intersection method (cf. =-=[32, 5, 12]-=-), or by using projective structure instead of the relative affine structure [39]. 4.3 Application III: Image Coding The re-projection paradigm, described in the previous section, can serve as a princ... |

31 | Projective structure from two uncalibrated images: Structure from motion and recognition - Shashua - 1992 |

28 | Space efficient 3D model indexing
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- 1992
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28 |
Incorporating motion error in multi-frame structure from motion
- Oliensis, Thomas
- 1991
(Show Context)
Citation Context ...at the University of Massachusetts at Amherst. Hence, one can measure the performance of reconstruction at a quantitative level. Reconstruction results on this set by other approaches can be found in =-=[6, 34, 23, 46, 1]-=-. On the sneaker images, a set of points were manually selected on one of the frames, referred to as the first frame, and their correspondences were automatically obtained along all other frames used ... |

27 | Projective geometry for machine vision
- Mundy, Zisserman
- 1992
(Show Context)
Citation Context |

23 | On Geometric and Algebraic Aspects of 3D Affine and Projective Structures from Perspective 2D Views
- Shashua
- 1994
(Show Context)
Citation Context ... the projective framework, and is the only next general framework after projective when working with perspective views. Parts of this work, as it evolved, have been presented in the meetings found in =-=[38, 40]-=-, and in [33]. 2 Notation We consider object space to be the three-dimensional projective space P 3 , and image space to be the two-dimensional projective space P 2 . An object (or scene) is modeled b... |