D. Weinshall and C. Tomasi, \Linear and incremental acquisition of invariant shape models from image sequences", in Proceedings of the 4th International Conference on Computer Vision, Berlin, Germany. May 1993, pp. 675-682, ieee.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....symmetric. As a test case, we describe in this chapter, the reconstruction of 3D mirror symmetric connected configurations from their noisy 2D projections. Reconstruction of general 3D structures from 2D projections, or the problem of structure from motion, is widely studied in computer vision [102, 95, 101, 110]. However, this topic is outside the scope of this dissertation and we will describe here only the enhancement in performance that can be obtained, using existing structure from 2D methods, when the reconstructed object is known to be mirror symmetric. 6.1 Previous Work When dealing with 3D ....

....and reducing number of frames in structure from motion problems [79, 67] See also Section 2.5. However none of these studies deal with exploiting symmetry for improving the input data for structure from motion algorithms. 85 In this chapter we combine the invariant reconstruction algorithm [110] (reviewed in Section 6.6.1) with the method dealing with inexact symmetries described in previous chapters, in order to pre process the input and post process the results of the structure reconstruction from several views. 6.2 Reconstruction of 3D Mirror Symmetric Structures We assume in this ....

[Article contains additional citation context not shown here]

D. Weinshall and C. Tomasi. Linear and incremental acquisition of invariant shape models from image sequences. In International Conference on Computer Vision, pages 675--682, Berlin, Germany, 1993.

....method. The factorization method [1] seeks the optimal 3 D shape and motion parameters under orthographic pro jection, by singular value decomposition of a given measurement matrix subject to rank 3 constraint. It was later extended to more general camera models, such as weak perspective [2], para perspective [3] affine [4] and projective factorization [5] It was also extended to in clude uncertainty handling [6] 8] Sequential or recursive methods [9] 10] 11] compute and filter 3 D shape and motion estimates by Kalman filtering such that the reprojections 2 have the ....

D. Weinshall and C. Tomasi, "Linear and incremental acquisition of invariant shape models fi'om image se- quences," IEEE Trans. Pattern Analysis Machine Intelligence, vol. 17, pp. 512-517, May 1995.

....the largest singular values of S. This construction relies on the well known fact that the closest rank 3 approximation to a given matrix in the sense of the Frobenius form is obtained by zeroing its three smallest singular values, and it has been used in various contexts in computer vision (e.g. [14, 32, 34]) Unfortunatey, in the presence of noise, S is not guaranteed (and in fact is unlikely) to be positive, and the above method does not apply (see, for example, 25, 34] for a discussion of this problem) To tackle this difficulty, we will use an elementary property of symmetric matrices: let us ....

.... is obtained by zeroing its three smallest singular values, and it has been used in various contexts in computer vision (e.g. 14, 32, 34] Unfortunatey, in the presence of noise, S is not guaranteed (and in fact is unlikely) to be positive, and the above method does not apply (see, for example, [25, 34] for a discussion of this problem) To tackle this difficulty, we will use an elementary property of symmetric matrices: let us consider an arbitrary n Theta n symmetric matrix S with real coefficients, and diagonalize this matrix in an orthonormal basis as S = UDU , where D is the diagonal ....

D. Weinshall and C. Tomasi. Linear and incremental acquisition of invariant shape models from image sequences. IEEE Trans. Patt. Anal. Mach. Intell., 17(5), May 1995.

....is not required, more freedom in picture taking is allowed such as taking pictures of pictures 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) [17, 2, 30, 12,46, 47, 13, 26,7,32, 34, 36, 25, 45, 29,8,10, 23, 31,16,15,48] and relevant to this paper are the work described in [17,7,13,34,36] The material introduced so far in the literature, concerning 3D geometry from multiple views, focuses on the projective framework [7, 13, 36] or the affine framework. The latter requires either assuming parallel projection ....

D. Weinshall and C. Tomasi. Linear and incremental acquisition of invariant shape models from image sequences. In Proceedings of the International Conference on Computer Vision, pages 675--682, Berlin, Germany, May 1993.

.... u ### m : u #f# # : u #f# m v ### # : v ### m : v #f# # : v #f# m # # # # # # # # # # # # # # has rank 3 [108, 85] note that # #= # # ) and that the image trajectories of a scene point are linear combinations of the trajectories of three reference points [114]. Let us now consider a line parameterized by its direction# and the vector D joining the origin of the world coordinate system to its projection onto (see Figure 5.2) We can parameterize the image of onto the image plane by the image vector d that joins the origin 50 p d u v ....

D. Weinshall and C. Tomasi. Linear and incremental acquisition of invariant shape models from image sequences. IEEE Trans. Patt. Anal. Mach. Intell., 17(5), May 1995.

.... 3 and can be decomposed as 3 D shape and rotation (after cancellation of the translation terms) by singular value decomposition (SVD) The factorization method for the case of orthographic projection [6] was followed by a series of extensions to more general camera models, from weak perspective [8], para perspective [9] affine [10] to perspective projection [11, 12] It was also extended to sequential computation [13] using line correspondences (instead of point correspondences) 14] and uncertainty handling [15 17] To the best of our knowledge, no quantitative sensitivity analysis or ....

D. Weinshall and C. Tomasi, Linear and incremental acquisition of invariant shape models from image sequences, IEEE Trans. Pattern Anal. Machine Intell. 17, 1995, 512--517.

....representation [21, 23 25] in which the coordinates of vertices on a virtual object are relative to an affine reference frame defined by the fiducial points (Fig. 2) Affine object representations have been a topic of active research in computer vision in the context of 3D reconstruction [21, 24, 26] and recognition [27] While our results draw heavily from this research, the use of affine object models in the context of augmented reality has not been previously studied. Here we show that placement of affine virtual objects as well as visible surface rendering can be performed efficiently ....

....are three points from the collection that are not coplanar with the origin; and the affine coordinates of the points p 1 ; p n , expressing the points p i ; i = 1; n in terms of the origin and affine basis points. We use the following two properties of affine point representations [21, 24, 26] (Fig. 4) Property 1 (Re Projection Property) When the projection of the origin and basis points is known in an image I m , we can compute the projection of a point p from its affine coordinates: 2 4 u m p v m p 3 5 = 2 4 u m b1 Gamma u m po u m b2 Gamma u m po u m ....

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D. Weinshall and C. Tomasi, "Linear and incremental acquisition of invariant shape models from image sequences," in Proc. 4th Int. Conf. on Computer Vision, pp. 675--682, 1993.

.... is twofold: i) it extends the iterative pose determination algorithms described in [7] and in [13] to deal with the problem of shape and motion from multiple views and (ii) it is a generalization to perspective of the factorization methods [23] 15] and of the affine invariant methods [27] [28]. More precisely, the affine iterative reconstruction methods that we propose here have a number of interesting features: ffl They solve the sign (or reversal) ambiguity that is inherent with affine reconstruction; ffl They are fast because they converge in a few iterations (typically 3 to 5 ....

.... Euclidean shape (P 1 : P n ) and Euclidean motion, i.e. k matrices of the form: 0 B B B B i T j t x j j T j t y j k T j t z j 0 0 0 1 1 C C C C A Various methods are available for estimating Euclidean structure from affine structure either with a calibrated camera [23] [28] (weak perspective) 15] paraperspective) or with an uncalibrated camera [16, 17] Based on the parameters of the Euclidean shape and motion thus computed one can estimate ij for all i and for all j using eq. 30) step 4. The first iteration of the algorithm performs a 3 D reconstruction ....

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D. Weinshall and C. Tomasi. Linear and incremental acquisition of invariant shape models from image sequences. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17(5):512--517, May 1995. 32

....b. 5.3. Relation to Previous Work There already exist a number of algorithms for the recovery of aOEne structure from two aOEne images. They can be divided into two categories. The rst relies on use of a local coordinate frame by choosing four non coplanar points to form the aOEne basis [15, 4, 21, 31]. One drawback is that the error in the basis points directly aoeects the precision of the entire solution. The second category is characterized by the work of Shapiro [24] Inspired by the work of Tomasi and Kanade [29] for a long image sequence under orthography, Shapiro uses the singular value ....

D. Weinshall and C. Tomasi. Linear and incremental acquisition of invariant shape models from image sequences. In Proceedings of the 4th International Conference on Computer Vision, pages 675682, Berlin, Germany, May 1993. IEEE Computer Society Press.

....components that are introduced as intermediate parameters are still subject to 9 complicated algebraic constraints. The algorithm can hardly be stable. A subsequent nonlinear optimization step is almost unavoidable to refine the solution [2, 11, 22, 7] In parallel, there has been a lot of work [23, 26, 20, 16, 17, 9, 10, 8, 14, 25] on structure from motion with simplified camera models varing from orthographic projections via weak and para perspective to affine cameras, almost exclusively for point features. These simplified camera models provide a good approximation to perpsective projection when the depth of the object is ....

D. Weinshall and C. Tomasi. Linear and incremental acquisition of invariant shape models from image sequences. In ICCV. 1993.

....arbitrary camera motion and only requires that the scene is static. However, many SFM techniques, such as the Factorization Method and its derivatives, assume a non perspective projection model and therefore are only accurate for specific camera motions and scene structures [59] 5] 42] 46] [70], 60] This thesis describes a new SFM technique called Projected Error 2 Refinement that recovers the positions of feature points and the locations of the cameras, and avoids some of the additional constraints imposed by other techniques on the inverse projection problem. 1.1 Problem ....

.... efficient by exploiting invariants of the projection equations for a small number of images [66] 44] 36] 17] 28] 64] 71] 67] 29] 74] 26] 50] 51] More recently, the reliability of these methods to noise has been the focus [76] 10] 54] 40] 9] 48] 2] 42] 63] [70], 59] Nevertheless, SFM has yet to be used extensively in real applications because other practical requirements still remain; in particular, dealing with outliers, occlusion and scalability over multiple images. The requirements of a general purpose SFM technique can be summarized as: 1. Fast ....

[Article contains additional citation context not shown here]

D. Weinshall and C. Tomasi, "Linear and incremental acquisition of invariant shape models from image sequences," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 17, no. 5, 1995, pp. 512517.

....is not required, more freedom in picture taking is allowed such as taking pictures of pictures 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) [17, 2, 30, 12, 46, 47, 13, 26, 7, 32, 34, 36, 25, 45, 29, 8, 10, 23, 31, 16, 15, 48] and relevant to this paper are the work described in [17, 7, 13, 34, 36] The material introduced so far in the literature, concerning 3D geometry from multiple views, focuses on the projective framework [7, 13, 36] or the affine framework. The latter requires either assuming parallel ....

D. Weinshall and C. Tomasi. Linear and incremental acquisition of invariant shape models from image sequences. In Proceedings of the International Conference on Computer Vision, pages 675--682, Berlin, Germany, May 1993.

....gives robust reconstruction in situations commonly encountered in outdoor robot navigation when perspective effects are large. It is valid for general motion 1 . There is no restriction on the depth variations of 3D feature points, in contrast with the approach due to Tomasi and its extensions [25, 27, 17, 2]. Since the reconstructions are designed to approximate maximum likelihood estimates (MLE) they are quite accurate; in typical experiments they are within a factor of 1.5 of the MLE. The true MLE is computed by a standard Levenberg Marquardt approach using the ground truth for a starting guess. ....

....converged result. The basic approach reported here (though not the details of the general motion algorithm) was previously described in [16] and [15] the current paper is partly based upon these previous ones. 2 Background and Motivation The approach most similar in spirit to ours is Tomasi s [25, 27, 17, 2]. Like ours, his algorithm and its extensions are based on using all data to compute a good approximation to the MLE. However, the domain of validity of his approximation and algorithm differs from that of our algorithm. Tomasi s approach is usually described as being based on an orthographic (or ....

D. Weinshall and C. Tomasi, "Linear and incremental acquisition of invariant shape models from image sequences," ICCV 675-682, Berlin 1993.

....(Lee and Huang, 1990) Quan, 1992) and (Tomasi and Kanade, 1992) that affine shape can be recovered from affine cameras. One solution to this problem is the elegant factorization method proposed by Tomasi and Kanade (1992) in the orthographic projection case, and extended to weak perspective by Weinshall and Tomasi (1993) and para perspective case by Poelman and Kanade (1994) Following Tomasi and Kanade (1992) suppose n points are tracked over v distinct views, we can write 0 B B B B B B u 11 : u 1n v 11 : v 1n : u v1 : u vn v v1 : v vn ....

....= M 2v Theta3 D and S 0 = D Gamma1 S represent the real affine camera matrix and Euclidean shape. To determine D, the so called metric constraints were used. The different solutions for different special cases have been proposed by Tomasi and Kanade (1992) Poelman and Kanade (1994) Weinshall and Tomasi (1993), Weishall (1993) Shapiro et al. 1994) Ullman and Basri (1991) These methods will be assembled into the same framework in terms of calibration matrix in Section 7. All of these methods require the knowledge of the intrinsic parameters of the camera, though there may be only one, the aspect ....

[Article contains additional citation context not shown here]

D. Weinshall and C. Tomasi. Linear and incremental acquisition of invariant shape models from image sequences. In Proceedings of the 4th International Conference on Computer Vision, Berlin, Germany. Ieee, 1993.

....algorithms proposed in [11, 21, 8] were based on a heavy overparametrization which still lead to unstable solutions. In the last few years, a family of linear algorithms for structure from motion using highly redundant image sequences called the factorization method has been extensively studied [23, 26, 19, 16, 18] (the works [25, 9, 17, 10, 13, 25] are also closely related) for point features from orthographic projections to affine cameras. This kind of algorithm decomposes directly the feature points of the image stream into object shape and camera motion. Using simplified camera models from orthographic ....

D. Weinshall and C. Tomasi. Linear and incremental acquisition of invariant shape models from image sequences. In Proceedings of the 4th International Conference on Computer Vision, Berlin, Germany. Ieee, 1993.

....5.4. The choice of basis points Often there exist more than four matched points in the three frames so a choice of basis vectors is available. Demey et al. 1992) point out that this choice affects the stability of the computed transfer and recommend that the points be chosen to span the object. Weinshall and Tomasi (1993) give an heuristic for choosing a stable basis set based on procedure described in (Golub and Van Loan 1989) which minimizes the change in coordinates of a point for basis point perturbations. In the planar case a rapid stable choice can be made by automatically choosing the left and rightmost ....

D. Weinshall and C. Tomasi. Linear and incremental acquisition of invariant shape models from image sequences. In Proc. 4th Int'l Conf. on Computer Vision, Berlin, pages 675--682, Los Alamitos, CA, 1993. IEEE Computer Society Press.

....bas Q (31) Wenow describe B in terms of Q and B bas . Substituting Eq (31) into the definition of B in Eq (10) and using Eq (30) we obtain B bas Q The linear and incremental computation of the matrices Q and B bas from at least three images of the object points is described in [Weinshall and Tomasi, 1992]. 26 C Eliminating translation In this appendix we show that translation can be ignored if we set the centroids of both model and image points to be the origin. To show this, we prove that the best rigid and affine transformations maps the model centroid to the image centroid. We begin ....

Weinshall, D. and Tomasi, C. (1992). Linear and incremental acquisition of invariant shape models from image sequences. ResearchReportRC 18549 (81133), IBM T. J. Watson ResearchCenter.

....Q and B bas . Substituting Eq (36) into the definition of B in Eq (15) and using Eq (35) we obtain B = P ) T P = Q ) T Delta B bas Delta Q The linear and incremental computation of the matrices Q and B bas from at least three images of the object points is described in [WT92]. 29 C Eliminating translation In this appendix we show that translation can be ignored if we set the centroids of both model and image points to be the origin. To show this, we prove that the best rigid and affine transformations maps the model centroid to the image centroid. We begin by ....

Weinshall, D. and Tomasi, C. (1992). Linear and incremental acquisition of invariant shape models from image sequences. Research Report RC 18549 (81133), IBM T. J. Watson Research Center.

.... (a 3D graph structure composed of one or more connected components) We are given several noisy 2D projections of such an object, where the projection is approximately weak perspective (scaled orthographic) In this work we combine the invariant reconstruction algorithm described in [3] with the method dealing with inexact symmetries suggested in [4] for improving the input and output data in the structure reconstruction from several views. Previous work on exploiting symmetry is described in [4] We employ two approaches to exploit the fact that the 3D structure to be ....

....which 3D mirror symmetric connected configurations are reconstructed from noisy 2D perspective projections. We use the two approaches of correction for symmetry which were described in Section 1. The reconstruction method used in the simulations is the invariant reconstruction method described in [3]. The correction procedures were the following: 1. The invariant reconstruction method was applied directly to the 2D data with no symmetry assumption. Following the reconstruction, correction for symmetry was applied to the 3D reconstruction by finding the closest 3D mirror symmetric ....

D. Weinshall and C. Tomasi. Linear and incremental acquisition of invariant shape models from image sequences. In International Conference on Computer Vision, pages 675--682, Berlin, Germany, 1993.

....Q and B bas . Substituting Eq (31) into the definition of B in Eq (10) and using Eq (30) we obtain B = P ) T P = Q ) T Delta B bas Delta Q The linear and incremental computation of the matrices Q and B bas from at least three images of the object points is described in [Weinshall and Tomasi, 1992]. C Eliminating translation In this appendix we show that translation can be ignored if we set the centroids of both model and image points to be the origin. To show this, we prove that the best rigid and affine transformations maps the model centroid to the image centroid. We begin by showing ....

Weinshall, D. and Tomasi, C. (1992). Linear and incremental acquisition of invariant shape models from image sequences. Research Report RC 18549 (81133), IBM T. J. Watson Research Center.

....offers robustness and stability in building the 3D description of objects from multiple images. In Section 4.1 we use this property to outline a new structure from motion algorithm. In Section 4.2 we show results with simulated and real data. A more complete and efficient algorithm is described in [25], with additional tests and a quantitative comparison to other algorithms. 4.1 Linear structure from motion algorithm given correspondence We have shown that the rigid representation D rig can be computed from as few as three images of four points with a linear algorithm. We have also shown in ....

D. Weinshall and C. Tomasi. Linear and incremental acquisition of invariant shape models from image sequences. In Proceedings of the 4th International Conference on Computer Vision, pages 675--682, Berlin, Germany, 1993. IEEE, Washington, DC.

....these equations are completely symmetrical with respect to M i and p j : if we interchange the 2 vectors, we will get exactly the same image point. With k frames and n 4 points, we get the 2k Theta n measurements matrix W whose ji element is x ji for j k, and y (j Gammak)i for k j 2k (cf. [11, 14]) Now if we read W by columns, the i th column gives us multi camera geometry; if we read it by rows, the j th and (j k) Gamma th rows give us multi point geometry in a single image. We use this symmetry to obtain dual relations to those obtained for multicamera geometry, by reading the ....

....Z 2 : Zn W 2 : Wn 3 Thus W is of rank 4. This result gives us the following algorithm for the reconstruction of shape using many views and many points, for an object with n 6 points: 1. Choose a subset of 6 good points (an algorithm on how to choose good basis points in described in [14]) 2. For every additional point M i , i 7: Using all available frames (but at least 3) compute the shape vector V7 i of the set of 7 points 1; 2; 3; 4; 5; 6; 6 i 3. Define the 11 Theta n matrix W, whose i th column is the shape vector V7 i . From Result 1, the rank of W is 4. Let ....

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D. Weinshall and C. Tomasi. Linear and incremental acquisition of invariant shape models from image sequences. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17(5):512--517, 1995.

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D. Weinshall and C. Tomasi, \Linear and incremental acquisition of invariant shape models from image sequences", in Proceedings of the 4th International Conference on Computer Vision, Berlin, Germany. May 1993, pp. 675-682, ieee.

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D. Weinshall and C. Tomasi, "Linear and Incremental Acquisition of Invariant Shape Models from Image Sequences," Proc. Fourth Int'l Conf. Computer Vision, Berlin, Germany, pp. 675-682,

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D. Weinshall and C. Tomasi. Linear and incremental acquisition of invariant shape model from image sequences. IEEE PAMI, 17(5):512--517, 1995.

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