A. Shashua. Trilinearity in visual recognition by alignment. In Proc. 3rd European Conf. on Computer Vision, Stockholm, volume 1, pages 479--484, May 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is rank deficient if and only if a i b b i a j = 0 for all i, j = 1, n. Applying this For a vector u u denotes the associated skew symmetric matrix such that uw = u w for all w . to the multiple view matrix M , we obtain the wellknown trilinear constraints [12]: x i (T i x 1 R R i x 1 T j )# x j = 0 (7) for all i, j = 2, m. In addition to those, we also obtain extra constraints from minors of M that include the last row, which are due to the planar condition: x i T i # x 1 x i R i x 1 # = 0, i = 2, m. 8) If the ....

A. Shashua. Trilinearity in visual recognition by alignment. In Proceedings of ECCV, pages 479--484, 1994.

....be done up to an arbitrary projective transformation, i.e. if M i and P j are solutions of (1) so are M i Q and Q Gamma1 P j for any nonsingular 4 Theta 4 matrix Q. Several effective techniques for computing a projective scene representation from multiple images have been proposed (e.g. [5, 12, 15, 21, 27, 28, 35]) As in the affine case, the projective reconstruction can be upgraded to a full metric model [7] by exploiting a priori knowledge of camera calibration parameters (e.g. 8, 9, 13, 20, 23, 24] or scene geometry (e.g. 1] Now, although the internal parameters of a camera may certainly be ....

A. Shashua. Trilinearity in visual recognition by alignment. In J.-O. Eklundh, editor, Proc. European Conf. Comp. Vision, volume 800 of Lecture Notes in Computer Science, pages 479--484. Springer-Verlag, 1994.

.... error may be measured without re estimating image point locations by simply finding the intersection of two epipolar lines in an image [24] This is equivalent to the point transfer function used to evaluate how well an estimated trifocal tensor matches a set of matching point triplets [36, 18]. However, since we are dealing with imagery in which many points that are matched across two views are not matched in three views, we choose not to rely on the reprojection error over point triplets. Oliensis [33] and Zhang [54] give comprehensive analyses of the 79 conditions under which the ....

....The general problem of detecting model degeneracy is an important issue in multiple view camera analysis and is discussed briefly in Chapter 6. In order to obtain a complete projective representation of a collinear camera set, it is necessary to use the trilinear constraints on point triplets [36, 13]. Because of these scenarios, the problems of degeneracy detection and multiple model hypothesis selection are important issues in projective camera modeling. Torr presents a case based algorithm for using information criteria to choose the simplest projective model that is not degenerate for a ....

A. Shashua. Trilinearity in Visual Recognition by Alignment. In European Conference on Computer Vision, LNCS 800, pages 479--484, 1994.

....activity. We have advanced that position by recovering model free structure in real time whilst tracking, using a unified representation. Closest relations, essential differences. There are several others tracking in real time ( 22, 7, 13] and several others recovering affine structure ([33, 23, 27, 28, 29]) The essential difference here is that we are combining the two. The convex hull of objects has also been calculated in a projective frame ( 26] but not in an affine frame and not in the context of dynamic scenes. Usefulness. The overall utility has we trust been demonstrated in the paper. We ....

A. Shashua. Trilinearity in visual recognition by alignment. In Proc. 3rd European Conf. on Computer Vision, Stockholm, volume 1, pages 479--484, May 1994.

....utility is its paucity of assumptions, i.e. the generality of the setting in which the algorithm is applicable. Some algorithms assume special camera configurations in order to ease the work of matching or tracking points across frames. Examples include closely spaced stereo pairs or triples [17, 34], linear gantries [8] or (in small scale laboratory settings) circular camera paths arranged by placing the subject of acquisition on a turntable [20, 30] Other methods assume a smooth image sequence (e.g, from a slowly moving video camera) with the camera manually reoriented so as to keep a ....

A. Shashua. Trilinearity in visual recognition by alignment. In ECCV9g, pages A:479 484, 1994.

....registered case using (14) 15) to give k IJ e k = 0, 22) a previously unpublished relation. The set of four independent incidence relations for points [5] I ( 33 J x K i ) 0 (23) for j, k 2 , corresponding to the trilinearity relationships [18] specialize to I IK e K = 0 , j, k 2 . 24) It is easily verified that these are the four affine trifocal constraints derived in [21] and in [15] generalizing well known orthographic relations [26] As shown in section 6.1, only three of the four relations are linearly ....

A. Shashua. Trilinearity in visual recognition by alignment. In Proc. ECCV'94, LNCS 800, pages 479--484, Springer.

....utility is its paucity of assumptions, i.e. the generality of the setting in which the algorithm is applicable. Some algorithms assume special camera configurations in order to ease the work of matching or tracking points across frames. Examples include closely spaced stereo pairs or triples [17, 34], linear gantries [8] or (in small scale laboratory settings) circular camera paths arranged by placing the subject of acquisition on a turntable [20, 30] Other methods assume a smooth image sequence (e.g, from a slowly moving video camera) with the camera manually reoriented so as to keep a ....

A. Shashua. Trilinearity in visual recognition by alignment. In ECCV94, pages A:479--484, 1994.

....transformations) unless additional scene constraints are taken into account. Related methods have been proposed by several authors in both the ane and projective cases [3, 56, 95, 73] For three images taken by a calibrated pinhole camera, Avidan and Shashua [3] have used trifocal tensors [100] to constrain the reprojection of the points in these images. In other words, the three images of a scene pointobeycertainmultilinear matching constraints captured by the trifocal tensor that can be constructed from three images in a linear fashion. This tensor can then be used to construct a ....

A. Shashua. Trilinearity in visual recognition by alignment. In J.-O. Eklundh, editor, Proc. European Conf. Comp. Vision,volume 800 of Lecture Notes in Computer Science, pages 479-484. Springer-Verlag, 1994.

....SFM solution is needed for each decision. Solving SFM has been approached in a number of ways, from non linear least squares bundle adjustment [10, 12] using Levenburg Marquadt (LM) to recursive methods using the extended Kalman filtering (EKF) 6, 11, 1, 3, 5] to linear methods on a few frames [4, 2, 9]. Bundle adjustment is the process of iteratively adjusting the camera poses and point positions in order to move toward the optimal least squares answer. As with all gradient descent methods, there is the possibility of getting stuck in local minima. If these local minima are avoided, the minimum ....

A. Shashua. Trilinearity in visual recognition by alignment. In ECCV94, pages A:479--484, 1994.

....geometry, represented by the fundamental matrix [3] This provides a mapping from points in one image to lines in the other, and consequently is not a suitable mapping for determining fixed entities directly. However, between three views the fundamental geometric relation is the trifocal tensor [6, 14, 15], which provides a mapping of points to points, and lines to lines. It is therefore possible to solve directly for fixed image entities as fixed points and lines under transfer by the trifocal tensor. In the following we obtain these fixed image entities, and thence the camera calibration, from a ....

Shashua, A. Trilinearity in visual recognition by alignment. In Proc. ECCV, 1994.

....in this topic. First of all, we do not yet have a clear picture about the relationship between multilinear constraints and the (statistical) optimality of motion and structure estimates. Although we have understood very well the geometric (or algebraic) relationship among multilinear constraints [5, 7, 12, 18] (which will be brie y reviewed in Section 3) when it comes to using them for designing motion or structure recovery algorithms, they are usually used as objectives, rather than constraints. Many researchers believe that multilinear tensors should be recovered rst and, from them, motion and ....

....from motion. In doing so, one will be able to see what roles multilinear constraints play in the design of optimal algorithms. In addition, the clari cation of the relationship between geometric and algebraic dependency among multilinear constraints is an important complement to the results in [5, 7, 12, 18]. Our results, especially the normalized epipolar constraint, may also help improve existing recursive methods such as those in [8, 14] if the lter objective function is modi ed to the one given by us. Moreover, studying the Hessian of such an objective will allow to extend existing sensitivity ....

A. Shashua, Trilinearity in visual recognition by alignment. In Proceedings of ECCV, Volume I, pp. 479-484. Springer-Verlag, 1994.

....has become a main stream of the research. The trilinear constraints are now considered the fundamental equations for motion analysis from three views. The associated tensor (called trifocal tensor) has already been extensively studied and its properties are described in a whole body of literature [9, 11, 6, 2, 14, 8]. However, as algebraic equations, the trilinear constraints do not have obvious geometrical interpretations. In a noiseless case, the algebraic equations are exactly satisfied for all the scene points. In presence of noise however, the residual error generated by the constraint equations may vary ....

A. Shashua. Trilinearity in visual recognition by alignment. Proc. 3 rd Europ. Conf. Comput. Vision, J.-O. Eklundh (Ed.), LNCS-Series Vol. 800-801, SpringerVerlag, I:479--484, 1994.

....in this topic. First of all, we do not yet have a clear picture about the relationship between multilinear constraints and the (statistical) optimality of motion and structure estimates. Although we have understood very well the geometric (or algebraic) relationship among multilinear constraints [5, 7, 12, 18] (which will be brie y reviewed in Section 3) when it comes to using them for designing motion or structure recovery algorithms, they are usually used as objectives, rather than constraints. Many researchers believe that multilinear tensors should be recovered rst and, from them, motion and ....

....In doing so, one will be able to see what roles multilinear constraints play in the design of optimal algorithms. In addition, the revelation of the statistical relationship between bilinear and trilinear constraints is an important complement to the well known algebraic or geometric results [5, 7, 12, 18]. Our results, especially the normalized epipolar constraint, may also help improve existing recursive methods such as those in [8, 14] if the lter objective function is modi ed to the one given by us. Moreover, studying the Hessian of such an objective will allow to extend existing sensitivity ....

A. Shashua, Trilinearity in visual recognition by alignment. In Proceedings of ECCV, Volume I, pp. 479-484. Springer-Verlag, 1994.

....the epipolar intersection method (cf. 26,6,11] orby using projective structure instead of the relativeaffine structure [34, 36] In all the above methods the epipolar geometry plays a key and preconditioned role. More direct methods, that do not require the epipolar geometry can be found in [35, 37]. 3.4 Application III: Image Coding The re projection paradigm, described in the previous section, can serve as a principle for model based image compression. In a sender receiver mode, the sender computes the relative affine structure between twoextreme views of a sequence, and sends the first ....

....the level of accuracy one can obtain when all the information is being used. In practice wewould liketousemuch fewer points from the re projected view, and therefore, re projection methods that avoid the epipoles all together would be preferred an example of such a method can be found in [35,37]. For the image coding paradigm (see Section 3.4) relative affine structure of the 34 sample points were computed between the first and last frame of the ten frame sequence (displays (a) and (d) in Fig. 5) Display (a) in Fig. 9 shows a graph of the average re projection error for all the ....

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A. Shashua. Trilinearity in visual recognition by alignment. In Proceedings of the European Conferenceon Computer Vision,Stockholm, Sweden, May 1994.

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A. Shashua. Trilinearity in visual recognition by alignment. In Proc. 3rd European Conf. on Computer Vision, Stockholm, volume 1, pages 479--484, May 1994.

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A. Shashua. Trilinearity in visual recognition by alignment. In Proc. 4th European Conference on Computer Vision, pages 479--484, 1994.

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A. Shashua. Trilinearity in visual recognition by alignment. In Proc. 3rd European Conf. on Computer Vision, Stockholm, volume 1, pages 479--484, May 1994.

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A. Shashua. Trilinearity in visual recognition by alignment. ECCV'94.

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Shashua, A. Trilinearity in visual recognition by alignment. In Proc. ECCV, 1994.

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A. Shashua. Trilinearity in visual recognition by alignment. In J.O. Eklundh, editor, Proceedings of the 3rd European Conference on Computer Vision, Stockholm, Sweden, pages 479-484. Springer-Verlag, May 1994.

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A. Shashua, "Trilinearity in visual recognition by alignment," in Proc. 3rd Eur. Conf. Computer Vision, 1994, pp. 479--484.

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Shashua, A. Trilinearity in visual recognition by alignment. In Proc. ECCV, 1994.

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A. Shashua. Trilinearity in visual recognition by alignment. In the Proceedings of ECCV, Volume I, pages 479-484. SpringerVerlag, 1994.

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A. Shashua. Trilinearity in visual recognition by alignment. In European Conference on Computer Vision, pages 479--484, 1994.

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A. Shashua. Trilinearity in visual recognition by alignment. In [Ekl94], pages 479484.

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