A. Shashua and N. Navab. Relative Affine Structure: Canonical Model for 3D from 2D Geometry and Applications. IEEE Trans. on PAM I , 18(9):873--883, Sep. 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....then the generation of extrapolated views is described (Sec. 5.2) 2. Background In this section we review some background notions needed to understand the paper. A complete discussion and formulation of the relative affine structure theory, and its close relative plane parallax, can be found in [16, 17]. A more general reference on the geometry of multiple views is [4] Two views of a planar set of points are related via a homography, i.e, a non singular linear transformation of the projective plane into itself. The most general homography is represented by a non singular 3 3 matrix H . If m ....

....3 View synthesis algorithm A very important property is that the relative affine structure is independent of the choice of the second view. Therefore, arbitrary second views can be synthesized by specifying a plane homography and the epipole. This leads to the following view synthesis algorithm [17]: 1. given a set of conjugate pairs (m i ; m i ) i = 0 : n; 2. recover the epipole e and the homography H ; 3. choose a point m 0 and scale H to satisfy 0 v H m 0 e The scale factor is computed with a formula analogous to Eq. 8) 4. compute the relative affine ....

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A. Shashua and N. Navab. Relative affine structure: Canonical model for 3D from 2D geometry and applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18(9):873--883, September 1996.

....can be interpreted as the homography induced by a very special plane, the infinity plane, as can be seen by letting d # in Eq. 5) In the general case (full 3D scene and arbitrary camera motion) the relationship between the two views can be cast in terms of a homography plus a parallax term [23], depending on the scene structure and camera translation. If the depth range of the scene is small compared to the distance from the camera, or the translation is small, then the parallax can be neglected. 3. HOMOGRAPHY COMPUTATION Let us suppose that we are given an image sequence with a ....

Shashua A, Navab N. Relative affine structure: Canonical model for 3D from 2D geometry and applications. IEEE Transactions on Pattern Analysis and Machine Intelligence 1996; 18(9): 873--883

....of each camera s optical center on the retinal plane of the other. The epipoles encode information related to the relative position of the two cameras and have been employed in applications such as stereo rectification [33] self calibration [27, 45, 24] projective invariants estimation [13, 36] and point features matching [9] Although it is generally known that the epipoles are hard to estimate accurately 1 [25] the uncertainty pertaining to their estimates is rarely quantified. Here, a simple method is presented that permits the estimation of the epipoles covariance matrices based ....

A. Shashua and N. Navab. Relative affine structure - canonical model for 3D from 2D geometry and applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18:873-- 883, 1996.

....the 3D geometry without modeling. However, they require knowledge of the correspondence between features in the different images. Correspondence is a difficult and generally unsolved problem, leading to a high dimensional search space. Among such methods are those using the trilinear tensor [22, 2, 18]. They require finding a substantial number of corresponding points and lines in at least three images. This can be accomplished only for very small disparities. Much of the multiple image work is intended for camera calibration, e.g [9, 17, 29] rather than object recognition. Techniques for ....

A. Shashua and N. Navab. Relative Affine Structure: Canonical Model for 3D from 2D Geometry and Applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18:873--883, 1996.

.... has shifted to non metric reconstruction from uncalibrated cameras [9] by computing the fundamental matrix (two views) 12] and the trilinear tensor (three views) 16] Also, different camera models were assumed; i.e. orthographic [20, 23] perspective projection [11, 25] or a unified model [1, 15]. Structure and motion algorithms typically assume given correspondences between features in successive frames. Finding such correspondences in a reliable way is a problem that still occupies researchers in the field. The most common approaches used to solve this problem are methods based on ....

A. Shashua and N. Navab. Relative affine structure: Canonical model for 3D from 2D geometry and applications. PAMI, 18(9), 1996.

.... has shifted to non metric reconstruction from uncalibrated cameras [25] by computing the fundamental matrix (two views) 28] and the trilinear tensor (three views) 42] Also, different camera models were assumed; i.e. orthographic [49, 53] perspective projection [27, 54] or a unified model [4, 41]. Determining the geometric relationship between various views of the environment and its 3D structure is a key component in a myriad of practical applications: reverse engineering, virtual reality, visualization, surgical planning, movie special effects, computer aided design, non tactile ....

A. Shashua and N. Navab. Relative affine structure: Canonical model for 3D from 2D geometry and applications. PAMI, 18(9), 1996.

....Figure (3d) shows the RMS reprojection error using the images Fig. 2a and Fig. 2b. The minimum error is obtained near the value of K1 found using the distortion calibration. 6.4. Experiment 3: euclidean reconstruction Projective reconstruction from the three images was performed according to [11]. Transformation to Euclidean 3D coordinates then required 5 control points. Three points were chosen from the planar surface and two other points were chosen such that the five were in general position, no 4 points coplanar. The world coordinate system was chosen such that the planar surface was ....

A. Shashuaand N. Navab. Relative Affine Structure: Canonical Model for 3D from 2D Geometry and Applications. IEEE Trans. PAMI (PAMI) Vol. 18(9), pp. 873--883, (1996).

.... has shifted to non metric reconstruction from uncalibrated cameras [9] by computing the fundamental matrix (two views) 12] and the trilinear tensor (three views) 16] Also, different camera models were assumed; i.e. orthographic [20, 23] perspective projection [11, 25] or a unified model [1, 15]. Structure and motion algorithms typically assume given correspondences between features in successive frames. Finding such correspondences in a reliable way is a problem that still occupies researchers in the field. The most common approaches used to solve this problem are methods based on ....

A. Shashua and N. Navab. Relative affine structure: Canonical model for 3D from 2D geometry and applications. PAMI, 18(9), 1996.

.... has shifted to non metric reconstruction from uncalibrated cameras [25] by computing the fundamental matrix (two views) 28] and the trilinear tensor (three views) 42] Also, different camera models were assumed; i.e. orthographic [49, 53] perspective projection [27, 54] or a unified model [4, 41]. Determining the geometric relationship between various views of the environment and its 3D structure is a key component in a myriad of practical applications: reverse engineering, virtual reality, visualization, surgical planning, movie special effects, computer aided design, non tactile ....

A. Shashua and N. Navab. Relative affine structure: Canonical model for 3D from 2D geometry and applications. PAMI, 18(9), 1996.

....a plane which may be used to register the views. The plane plus points configuration has received significant attention in the past, not least because it arises frequently in everyday scenes. The image motion is decomposed into a planar homographic transfer plus a residual image parallax vector [12, 14, 21, 23]. This decomposition has the advantage that it partially factors out dependence on the camera relative rotation and internal parameters. In this paper we show that similar advantages apply in the plane plus smooth object case. Furthermore there are additional advantages in that the multiple view ....

A. Shashua and N. Navab. Relative affine structure: Canonical model for 3d from 2d geometry and applications. IEEE Transaction on PAMI, 18(9):873--883, 1996.

....computed. The partial camera displacement can be also estimated from an homography matrix related to a reference plane on the target [12, 35] The homography matrix can be estimated jointly to the epipole using, for example, the algorithms presented in [3, 16] or after the epipole has been found [29] (if more than two views are available, see [19, 31] It will be shown in this paper that the motion parameters estimation is more robust from an homography matrix than from the fundamental matrix, especially when the epipole is not defined in the image (for example, if the motion is a pure ....

A. Shashua and N. Navab. Relative affine structure: canonical model for 3d from 2d geometry and applications. IEEE Trans. on PAMI, 18(9):873--883, September 1996.

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A. Shashua and N. Navab. Relative affine structure: Canonical model for 3D from 2D geometry and applications. IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI), 18(9):873--883, 1996.

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A. Shashua and N. Navab. Relative Affine Structure: Canonical Model for 3D from 2D Geometry and Applications. IEEE 18(9):873--883, 1996.

....set of operators, that applies uniformly to pairs or triplets of views. In other words, the unification efforts that have appeared so far in the literature focus on the transformation groups (projective, affine and Euclidean) represented by the camera matrix, leading to a canonical framework [3, 15, 9] for the geometry of two views. Given the recent progress on multi linear tensorial constraints across more than two views, there is a need to make a similar unification attempt but now across the temporal axis (number of views) rather than on the spatial axis (transformation groups) The paper ....

A. Shashua and N. Navab. Relative affine structure: Canonical model for 3D from 2D geometry and applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18(9):873-- 883, 1996.

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A. Shashua and N. Navab. Relative affine structure: Canonical model for 3D from 2D geometry and applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18(9):873--883, 1996.

....to a reference plane : p 0 i = A p i ae i v 0 = A ; v 0 ]P i where A is the homography matrix mapping view 1 onto view 2 due to the plane . The scalar ae i represents the relative deviation of the point P i from the plane and is called the relative affine structure [21]. The choice of the plane determines the projective representation of object space. For purposes of visualization, it is useful to choose such that it is situated in between the space points making it possible to treat ae i as simple depth variable. In other words, let A = P j ff j H j ....

A. Shashua and N. Navab. Relative affine structure: Canonical model for 3D from 2D geometry and applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18(9):873--883, 1996.

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A. Shashua and N. Navab. Relative Affine Structure: Canonical Model for 3D from 2D Geometry and Applications. IEEE Trans. on PAM I , 18(9):873--883, Sep. 1996.

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A. Shashua and N. Navab. Relative affine structure: Canonical model for 3d from 2d geometry and applications. IEEEPAM I , 18(9):873--883, 1996.

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A. Shashua and N. Navab. Relative Affine Structure - Canonical Model for 3D From 2D Geometry and Applications. IEEE Trans. on PAMI, PAMI-18(9):873--883, Sep. 1996.

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A. Shashua and N. Navab. Relative Affine Structure: Canonical Model for 3D from 2D Geometry and Applications. IEEE Trans. on PAM I , 18(9):873--883, Sep. 1996.

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A. Shashua and N. Navab. Relative affine structure: Canonical model for 3D from 2D geometry and applications. PAMI, 18(9), 1996.

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A. Shashua and N. Navab. Relative affine structure: Canonical model for 3d from 2d geometry and applications. Pattern Analysis and Machine Intelligence, 18(9):873--883, Sept. 1996.

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A. Shashua and N. Navab. Relative affine structure: Canonical model for 3D from 2D geometry and applications. PAMI, 18(9), 1996.

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A. Shashua and N. Navab. Relative Affine Structure: Canonical Model for 3D from 2D Geometry and Applications. IEEE Trans. on PAMI, 18(9):873--883, Sep. 1996.

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SHASHUA, A., AND NAVAB, N. Relative affine structure: Canonical model for 3D from 2D geometry and applications. IEEE Transactions on Pattern Analysis and Machine Intelligence 18, 9 (September 1996), 873--883. 5

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