A. Shashua, "Projective Structure from Uncalibrated Images: Structure from Motion and Recognition," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, pp. 778-790, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....The possible utility of these multiview constraints for improving homography estimation and for detecting nonrigid motions are also discussed. Index Terms Homographies, homologies, motion estimation, multiview analysis. I INTRODUCTION OMOGRAPHY estimation is used for 3D analysis [18] 10] [21], 25] 11] 7] 14] 17] 16] mosaicing [13] camera calibration [26] 31] and more. The induced homography between a pair of views depends on the intrinsic and extrinsic camera parameters and on the 3D plane parameters [10] While camera parameters vary across different views, the plane ....

....restricted camera motion, linear subspace constraints apply also to the homographies of a single plane across multiple views (Section 6) Different video related applications can benefit from such multiview constraints. For example, many algorithms based on planar homographies (e.g. 18] [21], 11] 28] 17] or on planar homologies (e.g. 23] 4] rely on accurate precomputation of these homographies (or homologies) However, the image region corresponding to a planar surface may be small. In such cases, the homography estimation tends to be highly inaccurate [25] i.e. when ....

A. Shashua, "Projective Structure from Uncalibrated Images: Structure from Motion and Recognition," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, pp. 778-790, 1994.

....can be improved by enforcing the multi view subspace constraints. b) Violations of these multiview constraints can be used as a cue for moving object detection. All the results derived in this paper are true for uncalibratedcameras. 1 Introduction Homography estimation is used for 3D analysis [8,9,11,4,2,6, 7], mosaicing [5] camera calibration [12] and more. The induced homography between a pair of views depends on the camera intrinsic and extrinsic parameters, and on the 3D plane parameters [1] While camera parameters vary across differentviews, the plane geometry remains the same. In this paper ....

.... over multiple ( 4) views (Section 3) This constraint is then extended to a constraint on homographies of multiple planes across multiple views (Section 4) Algorithms for 3D analysis whicharebasedonthe use of multiple homographies (in scenes with multiple planes)have been suggested (e.g. [8, 9,13,7]) Most of these algorithms rely on accurate precomputation of the homographies. However, in scenes containing multiple planes, the image region corresponding to each plane may be small. In such cases, the homography estimation tends to be highly inaccurate [11] i.e, when applied to small image ....

A. Shashua. Projective structure from uncalibrated images: Structure from motion and recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16:778--790, 1994.

....which are referred to as Triplet 2 and 3 . With each data sets, we have used several dioeerent methods to estimate the three trifocal tensors T 1 , T 2 and T 3 that are associated to the triplet of views: Lin i : One of the trifocal tensors T i ; i = 1: 3 is computed using the linear algorithm [Sha94a, Har94b] and the change of view is used to obtain the two other trifocal tensors. Cst i : The second case is similar to the previous one but the constraints were enforced as described in section 4 before applying the change of view. Data: In the third case, we estimate the three trifocal Lin1 14:6 4:9 ....

A. Shashua. Projective structure from uncalibrated images: structure from motion and recognition. PAMI, 16(8):778790, 1994.

....a challenging sequence, since the translational components of motion along the optical (i.e. OZ) axis are much smaller compared to the components that are parallel to the retinal plane. This implies that the epipoles are outside the images, making their accurate computation very difficult [42]. The depths of the 3D points were uniformly distributed in the range [20, 100] measured in focal length units. The standard deviation of the Gaussian noise added to the retinal projections of the simulated 3D points ranged from 0 to 4.0 pixels. To ensure that the recovered intrinsic calibration ....

Amnon Shashua. Projective structure from uncalibrated images: structure from motion and recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(8):778--790, 1994.

....rotate, we have a direct solution of relative camera translation parameters and scene depths (Eq. 1) or (2) 5) General movement. If camera moves in arbitrary directions, there is no direct linear solution of these motion parameters. This is the classic structure from motion (SfM) problem [7,8,9] and will not be addressed here. Case (1) 2) and (3) cover the most common camera motions: dolly (facing front) tracking (facing side) panning tilting rolling (rotation) and zooming movement. We proposed a 2.5D inter frame motion model by introducing a depth related parameter for each point. ....

A. Shashua, "Projective structure from uncalibrated images: structure from motion and recognition", IEEE Trans Pattern Recognition and Machine Intelligence, 1994, 16(8):778-790.

....techniques in the context of virtual teleconferencing system. 2 Views Synthesis 2.1 An efficient approach for real image reconstruction We propose an algorithm for real view reconstruction from uncalibrated 2D points of view of a 3D scene based on trilinear tensors, first modeled by A. Shashua [1, 2, 3, 4] to understand the geometry of correspondences between three initial images. These relations generalize well known bilinearities, called epipolar constraints [5, 6] and allow us to reconstruct an existing view from two other neighboring views without explicit calibration stage. Following are the ....

....views without explicit calibration stage. Following are the steps: ffl an analysis step, using more than seven corresponding points in the three original uncalibrated views, to estimate the eighteen parameters of a trilinear form, for more details about trilinear parameters definition see [7, 1, 2] and [8] ffl a synthesis step, using corresponding points of the external images and the estimated parameters (ff i ) i=1: 18 to reconstruct the cen tral view, by the following system: x 0 (ff 1 x 00 ff 2 y 00 ff 3 ) x 0 x(ff 4 x 00 ff 5 y 00 ff 6 ) x(ff 7 x 00 ....

[Article contains additional citation context not shown here]

A Shashua. "Projective structure from uncalibrated images: structure from motion and recognition". IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(8):778--790, August 1994.

....can be improved by enforcing the multi view subspace constraints. b) Violations of these multiview constraints can be used as a cue for moving object detection. All the results derived in this paper are true for uncalibrated cameras. 1 Introduction Homography estimation is used for 3D analysis [8, 9, 11, 4, 2, 6, 7], mosaicing [5] camera calibration [12] and more. The induced homography between a pair of views depends on the camera intrinsic and extrinsic parameters, and on the 3D plane parameters [1] While camera parameters vary across different views, the plane geometry remains the same. In this paper ....

.... over multiple ( 4) views (Section 3) This constraint is then extended to a constraint on homographies of multiple planes across multiple views (Section 4) Algorithms for 3D analysis which are based on the use of multiple homographies (in scenes with multiple planes) have been suggested (e.g. [8, 9, 13, 7]) Most of these algorithms rely on accurate precomputation of the homographies. However, in scenes containing multiple planes, the image region corresponding to each plane may be small. In such cases, the homography estimation tends to be highly inaccurate [11] i.e, when applied to small image ....

A. Shashua. Projective structure from uncalibrated images: Structure from motion and recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16:778--790, 1994.

....and its potential domains of application in the concluding section. 2. VIRTUALIZED NAVIGATION IN A REAL SCENE: PREVIOUS WORK 2.1. Trilinearity to Synthesize Points of View We propose an algorithm for view synthesis from uncalibrated 2D views of a real 3D scene based on the trilinear tensors [1], extending the stereovision concepts [2, 3] on three different perspective views of the same scene. If we consider a triplet of views (i Gamma 1, i and i 1) extracted from n original views of the 3D scene, a new point of view i 0 can therefore be generated from its two neighboring initial ....

.... steps: ffl Analysis: Using seven or more corresponding points in three original uncalibrated views denoted i Gamma 1, i and i 1 (modelled in a discrete approach by the reference texture i 1 mapped on three meshes m i Gamma1 , m i and m i 1 [4] eighteen trilinear parameters are estimated [1, 5]. ffl Synthesis: An unknown intermediate view i 0 is generated, using all the corresponding mesh nodes of the images i Gamma 1 and i 1 (i.e m i Gamma1 and m i 1 ) and the estimated parameters, algebraically manipulated to simulate a change of the focal length or a 3D displacement of the ....

A. Shashua. Projective Structure from Uncalibrated Images: Structure from Motion and Recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(8):778--790, August 1994.

....parameters from orthographic and or paraperspective views. 20] 18] 19] 17] 14. Structure from Motion: Multi body Factorization from orthographic and or paraperspective views. 20] 18] 7] 4 15. Structure from Motion: Three orthographic perspective views using the tri linear tensor. [1], 2] 3] ....

Shashua, A. Projective Structure from Uncalibrated Images: Structure from Motion and Recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence (T-PAMI), Vol. 16(8), pp. 778--790, 1994.

....do not make sense. However, a projective structure still contains rich information, such as coplanarity, collinearity, and cross ratios (ratio of ratios of distances) which is sometimes suOEcient for articial systems, such as robots, to perform tasks such as navigation and object recogni tion (Shashua 1994, Zeller and Faugeras 1994, Beardsley, Zisserman and Murray 1994) In many applications such as the reconstruction of the environment from a sequence of video images where the parameters of the video lens is submitted to continuous modi cation, camera calibration in the classical sense is not ....

Shashua, A.: 1994, Projective structure from uncalibrated images: structure from motion and recognition, IEEE Transactions on Pattern Analysis and Machine Intelligence 16(8), 778790.

....not rotate, we have a direct solution of relative camera translation parameters and scene depths (Eq. 1) or (2) 5) General movement. If camera moves in arbitrary directions, there is no direct linear solution of these motion parameters. This is the classic structure from motion (SfM) problem [7,8,9] and will not be addressed here. Cases (1) 2) and (3) cover the most common camera motions: dolly (facing front) tracking (facing side) panning tilting rolling (rotation) and zooming movement. We proposed a 2.5D inter frame motion model by introducing a depth related parameter for each point. ....

A. Shashua, "Projective structure from uncalibrated images: structure from motion and recognition", IEEE Trans Pattern Recognition and Machine Intelligence, 1994, 16(8):778-790.

.... complete, the main fact being that correspondences between points in two images are completely described by the epipolar geometry which can itself be summarized algebraically in a 3 Theta 3 matrix of rank 2, the fundamental matrix [17, 18, 19] Moreover, it has been shown by several authors [9, 13, 26] that once the fundamental matrix is known, and correspondences between points established, the 3 D scene imaged by the two cameras can be reconstructed up to a projective transformation. The case of three images or more has not been studied as extensively as the case of two. Faugeras and Robert ....

Amnon Shashua. Projective structure from uncalibrated images: structure from motion and recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(8):778790, 1994.

....parameters from orthographic and or paraperspective views. 20] 18] 19] 17] 12. Structure from Motion: Multi body Factorization from orthographic and or paraperspective views. 20] 18] 7] 13. Structure from Motion: Three orthographic perspective views using the tri linear tensor. [1], 2] 3] 4 ....

Shashua, A. Projective Structure from Uncalibrated Images: Structure from Motion and Recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence (T-PAMI), Vol. 16(8), pp. 778--790, 1994.

.... two other neighboring views without any explicit calibration stage as follows: ffl Analysis: Using seven or more corresponding points in three original uncalibrated views, eighteen parameters of a trilinear form can be estimated (for more details about the definition of trilinear parameters see [16, 17, 18] and [4] ffl Synthesis: The central view is reconstructed using all corresponding points of the external images (i.e left and right) and the estimated parameters from the analysis, as shown in figure 1. view 1 view 2 view 3 view 1 view 2 view 3 mesh nodes mesh nodes mesh nodes 18 trilinear ....

....synthesis image texture warping on image 2 mesh Delaunay triangulation necessary inputs necessary inputs view 3 ref. texture ANALYSIS SYNTHESIS Figure 1. Regeneration process of a real view Contrary to the image based rendering methods for view synthesis typically found in the literature [1, 17, 18, 2] and resorting to dense correspondences, one of our contributions is to use a mesh oriented approach. We represent the three original images as a reference texture, mapped on three associated meshes, defined using the Delaunay triangulation [21] on homologous points from the three initial ....

A. Shashua. Projective Structure from Uncalibrated Images: Structure from Motion and Recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(8):778-- 790, August 1994.

.... view regeneration from two other neighboring images, as follows: i) an analysis step, using corresponding points (equivalent to the nodes of the meshes) in the three original uncalibrated views, estimates the eighteen parameters of a trilinear form, derived from the trinocular vision theory (see [7] for more details about trilinear parameters definition) ii) a synthesis step, using the meshes nodes of the external images and the estimated parameters, reconstructs the mesh of the central image, on which the reference texture of the initial triplet of views is mapped (figure 3 (a) The ....

A. Shashua. Projective Structure from Uncalibrated Images: Structure from Motion and Recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(8):778--790, August 1994.

....d eterminer pr ecis ement la place du correspondant dans la troisi eme vue d un couple de points homologues des deux autres vues [OF88] Celleci justifie le fait qu il existe un syst eme de deux relations trilin eaires, mis en evidence par A. SHASHUA, relatif a trois vues d une meme sc ene 3D [AS94] : x 0 (ff 1 x 00 ff 2 y 00 ff 3 ) x 0 x(ff 4 x 00 ff 5 y 00 ff 6 ) x(ff 7 x 00 ff 8 y 00 ff 9 ) ff 10 x 00 ff 11 y 00 ff 12 = 0 y 0 (fi 1 x 00 fi 2 y 00 fi 3 ) y 0 x(fi 4 x 00 fi 5 y 00 fi 6 ) x(fi 7 x 00 fi 8 y 00 fi ....

Amnon SHASHUA, Projective structure from uncalibrated images: structure from motion and recognition, IEEE Transactions on Pattern Analysis and Machine Intelligence, 1994.

....3D projective invariants provided we have some estimate of F . A more detailed discussion on this issue you can find in [22] 6 Projective Structure Using n Uncalibrated Cameras In this section we present the application of cross ratio [20] for computing the projective depth discovered by Sashua [26]. This can be easily calculated using the cross ratio of projected points lying on an epipolar line of any of the n cameras. This relation remains constant also for the ratio of the segments of an optical ray delimited by a tetrahedron or reference frame as is depicted in the Figure 2. Fig. 2. ....

.... and the general form for the i camera and j camera ae = i (f R ij (P i )E ij )I Gamma1 2 (P j E ij )I Gamma1 2 ji (P j f S ij (P i ) I Gamma1 2 (f R ij (P i )f S ij (P i ) I Gamma1 2 j : 53) The term in the right bracket is termed projective depth in [26]. If we have a number of views available then, in this framework, a more robust estimate of k would be given by k = 1 n X (i6=j) P j f S ij (P i ) I Gamma1 2 (f R ij (P i )f R ij (P i ) I Gamma1 2 ; 54) where n is the number of estimates used. Finally according ....

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Shashua, A. 1994. Projective structure from uncalibrated images: structure from motion and recognition PAMI, 16(8), 778:790.

....a whole range of new problems, like occlusion, calibration, correspondence, and representational issues. Whereas the two image problem has been thoroughly studied in computer vision, theories of multi image projective geometry, calibration, and correspondence have only recently begun to emerge [Sha94, Har94, LV94, Tri95, FM95, HA95] Furthermore, the view synthesis problem, as presently formulated, raises a number of unique challenges that push the limits of existing multi image techniques. 59 In this chapter, we describe an approach for view synthesis from multiple basis views that seeks ....

Amnon Shashua. Projective structure from uncalibrated images: Structure from motion and recognition. IEEE Trans. on Pattern Analysis and Machine Intelligence, 16(8):778--790, 1994.

....to use projective or affine framework to formulate the camera projection and image transformation. Under these frameworks, difficulties such as camera calibration encountered in the metric SFM methods can be avoided without losing much fidelity of the structure recovery from multiple images [1, 6, 11, 12, 13]. Remarkably, the inherent linear mathematicasis of the projective and affine models permits the applications of compact matrix manipulation schemes to the interpretations of the mechanisms of scene s imaging and image matching over the multiple views. Even more importantly, the projective and ....

.... algebraic invariants inherently existing in the projective transformations to different vision tasks including the difficult SFM problems [9] For these reasons, investigating SFM methods modulo a projective or affine ambiguity of the structure recovery has been attracting more and more attentions [7, 2, 8, 3, 4, 11, 12]. Amnon Shashua and Nassir Navab have investigated the problem of shape recovery of 3D objects under the projective and affine frameworks [11, 12] They formulated the transformation between two views in projective space by deriving an intermediate parameter termed relative affine structure, from ....

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A. Shashua, "Projective structure from uncalibrated images: Structure from motion and recognition," IEEE Trans. PAMI, Vol.16, No.8, 1994.

....Non Metric Vision, 3D Reconstruction, Fundamental Matrix. 1. Introduction Since the work of Koenderink and van Doorn [15] on aOEne structure from motion and that of Forsyth et al. 12] on invariant description, the development of non metric vision has attracted quite a number of researchers [5, 13, 26, 17] (to cite a few) We can nd a range of applications: object recognition [12] 3D reconstruction of scenes [15, 27, 9] image matching [35] visual navigation [3, 33] motion segmentation [20, 30] image synthesis [8] etc. This paper mainly addresses the recovery of structure and motion from two ....

....image synthesis [8] etc. This paper mainly addresses the recovery of structure and motion from two uncalibrated images of a scene under full perspective or under aOEne projection. The extension to N views is given in Appendix D. There is already a large amount of work reported in the literature [5, 7, 13, 26, 38], and it is known that the structure of the scene can only be recovered up to a projective transformation for two perspective images and up to an aOEne transformation for two aOEne images. We cannot obtain any metric information from a projective or aOEne structure: measurements of lengths and ....

A. Shashua. Projective structure from uncalibrated images: structure from motion and recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence,

....require calibrated cameras are of little value for active robots continuously changing their visual attention, since this involves altering the calibration parameters of the cameras. Even though the robots cannot reconstruct any metric information about its surroundings with uncalibrated cameras [1, 2], it is still possible to extract useful, comparative information, such as planarity and collinearity. In fact, such measures are often more relevant for navigation than detailed metric information. In door environments abound with planar surfaces [3] and consequently robot vision benefits by a ....

A. Shashua. Projective structure from uncalibrated images: Structure-from-motion and recognition. PAMI, 16(8):778--790, August 1994.

....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 ....

....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 (cf. 17, 46, 45, 30] or certain apriori assumptions on object ....

[Article contains additional citation context not shown here]

A. Shashua. Projective structure from uncalibrated images: structure from motion and recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1994. In press.

....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 material introduced so far in the literature focuses on the projective framework [4, 8, 25, 15], or the affine framework. The latter requires either assuming parallel projection (cf. 10, 32, 31, 22] or certain apriori assumptions on object structure (for determining the location of the plane at infinity [4, 19] or assuming purely translational camera motion [16] In this paper, we ....

....2 o . The ratio k 1 =k 2 removes the dependence on the projection center O (z=z o cancels out) and is therefore a projective invariant. This projective invariant is the ratio of cross ratios of the rays OP and OP o with their intersections with the two planes 1 and 2 , which was introduced in [24, 25] as projective depth . Because d i p =d i o is the affine structure under parallel projection with respect to plane i (the ratio of perpendicular distances of P and P o from i ) then the projective structure of the scene can be described (up to a uniform scale factor) as the ratio of ....

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A. Shashua. Projective structure from uncalibrated images: structure from motion and recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1994. in press.

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A. Shashua, Projective structure from uncalibrated images, IEEE Trans. Pattern Analysis and Artificial Intellingence(T-PAMI),Vol. 16 (1994), no. 8, pp. 778--790.

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A. Shashua. Projective structure from uncalibrated images: Structure from motion and recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(8):778--790, August 1994.

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