A. Shashua. Algebraic Functions for Recognition. IEEE Tran. Pattern Anal. Machine Intelligence, 16:778--790, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....with real images is achieved. For two images [2] the uncalibrated epipolar geometry, represented by the fundamental matrix [9] encodes the rigidity between two views. Torr [3] uses uncalibrated three view relations by means of the trifocal tensor. It encodes the rigidity among three views [10], 11] It is well known the problem of overparameterisation. In fact, in [12] it is reported the importance of model selection in order to successfully compute automatically matches and motion from image sequences. For an indoor mobile robot with a fixed camera, the number of parameters of the ....

A. Shashua. "Algebraic functions for recognition." IEEE Trans. Pattern Analysis and Machine Intelligence, 17(8):779-789, 1885.

....for stereo rigs, see e.g. 16] 7] The rst step in the scheme outlined, the structure and motion problem, is central for computer vision. Recovery of 3D structure by means of a sequence of 2D images is often accomplished by epipolar geometry and multilinear constraints, cf. 1] 2] 3] 4] [8], 15] Here we will use an alternative approach, based on the notions of aOEne shape and depth, developed in a series of papers [9] 10] 11] 12] In the case of uncalibrated cameras it is well known that only projective reconstruction is possible, cf. 1] 10] As will be seen below, the ....

A. Shashua, Algebraic functions for recognition. IEEE Trans. Pattern Anal. Machine Intell. 17:8

.... consistency checks we use are based on the conditions present in separate, bottom up structure and motion estimation under 2 D to 2 D line correspondences in the Euclidean calibrated case ( 10, 15, 28] A later body of work for calibrated, uncalibrated, and projective reconstruction cases ([22, 21, 8] addresses trilinear constraints between points and lines in 3 views, and presents linear, bottom up, and model free rigid reconstruction methods (see [9] for a comprehensive review) However, unlike these approaches, we assume an incomplete deformable and Euclidean model and we don t attempt to ....

A.Shashua. Algebraic functions for recognition, PAMI, 17(8):779-789, 1995.

....Section III shows the experimental results obtained using our new rectification methods. Section IV gives concluding remarks. II. RECTIFYING UNCALIBRATED TRINOCULAR IMAGES A. Obtaining Fundamental Matrices In the case of three cameras or three images taken by one camera, it has been shown in [11] that the correspondence constraint is expressed by the trilinear tensor. The tensor can be recovered linearly from at least seven corresponding points or lines across the three views. It is also shown in [11] that the concatenation of epipolar geometries across three views fails in cases where ....

....In the case of three cameras or three images taken by one camera, it has been shown in [11] that the correspondence constraint is expressed by the trilinear tensor. The tensor can be recovered linearly from at least seven corresponding points or lines across the three views. It is also shown in [11] that the concatenation of epipolar geometries across three views fails in cases where trilinearities do not. The trilinearities use all the three views together, rather than in pairs as in the case of the fundamental matrix, thereby gaining additional numerical stability. Introduce a Cartesian ....

[Article contains additional citation context not shown here]

A. Shashua, "Algebraic functions for recognition," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 17, no. 8, pp. 779--789, 1995.

....Analysisofmultipleviewsofthesamesceneisanareaof active research in computer vision. The study of the structure of points and lines in two views received much attention in the eighties and early nineties [2, 4, 8] Studies on the constraints existent in three and more views have followed since then [3, 10, 11, 12]. These multiview studies have concentrated on how geometric primitives like points, lines and planes are related across views. Specifically, the algebraic constraints satisfied by the projections of such primitives in different views have been the focus of intense studies. An important issue in ....

A. Shashua. Algebraic Functions for Recognition. IEEE Tran. Pattern Anal. Machine Intelligence, 16:778--790, 1995.

....3 for a combination of one paracatadioptric and two perspective views, and so forth. Studying the properties of these tensors in more detail is beyond the scope of this paper though. As for trifocal tensors between triplets of cameras of the same type, the perspective case has been treated e.g. in [15] and the affine case in [18] To our knowledge, no existing publication deals with the trifocal tensor for three para catadioptric views or for the mixed configurations considered here. Four views. In this case, the joint matrix is of size 12 8. Its rank deficiency implies that the ....

A. Shashua, "Algebraic Functions for Recognition," IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 779--789, Vol. 17, No. 8, 1995.

....because once the trifocal tensors are known, they can be used for a variety of useful tasks [BGP93, AZH96] even in the cases where the use of the fundamental matrix is impossible. This question has received little attention except for the obvious application of quadratic least squares methods [Har94a, Sha95]. What makes the use of these methods very questionable is the fact that the trinocular tensor is very constrained: it has been shown that the trifocal tensor depends only upon 18 independent parameters in general. Therefore its 27 components must satisfy a number of algebraic constraints, some of ....

....a new estimate T i that is close to T i 0 and satises all the trilinear constraints. Once this is done, the resulting tensor can be parameterized as shown in section 3.5 and non linear minimization can be used to rene it with respect to the image data. 4. 1 Initial estimate It is well known [Har94a, Sha95] that an approximation to the trilinear tensor can be estimated linearly. Doing so, the 27 coeOEcients of the tensor are taken as unknowns so that the trilinear constraints are not satised. But, as soon as at least 14 line matches are available, the trifocal tensor can be estimated using the ....

A. Shashua. Algebraic functions for recognition. PAMI, 17(8):779789, 1995.

....present in the third dimension, popularly referred to as the depth or the z dimension. It is easy to see that a plurality of projections can compensate for this loss more than a single view can. The multiview relations have found many applications in view generation [1] object recognition [2], video stabilization [3] etc. The primary application of multiview constraints is in reconstructing the third dimension from a set of projections. The simplest example is classical stereo vision [4, 5] The algebraic relations among the projections of a point onto multiple cameras have been ....

....application of multiview constraints is in reconstructing the third dimension from a set of projections. The simplest example is classical stereo vision [4, 5] The algebraic relations among the projections of a point onto multiple cameras have been studied extensively in Computer Vi sion [5, 6, 2, 7]. Multiple independent views of a dynamic event can be obtained using multiple video cameras. The multiview algebraic relations are then satisfied between the corresponding points of the views of the same time instant, provided the videos are synchronized to a common video signal. Using a ....

[Article contains additional citation context not shown here]

A. Shashua, "Algebraic functions for recogni- tion," IEEE Tran. Pattern Anal. Machine Intelligence, vol. 16, pp. 778-790, 1995.

....it requires estimation of a dense structure map. In this paper, we propose a trifocal motion model, which captures scene depth and camera motion without explicit 3 D motion and structure estimation, for digital video compression and mosaic generation based on the trifocal tensor representation [7] [9] The model requires a trifocal tensor that 1051 8215 98 10.00 1998 IEEE 668 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 8, NO. 5, SEPTEMBER 1998 is computed by matching image features across three views and dense correspondence between two of the three views. We ....

....by rotation of a camera about its center of projection (for arbitrary 3 D scenes) III. TRIFOCAL MOTION MODEL This section develops trifocal motion model for motion compensation of digital video based on the recently developed SUN AND TEKALP: TRIFOCAL MOTION MODELING 669 trilinear constraints [7] [9] across three perspective views of a rigidly moving 3 D scene that is captured by an uncalibrated camera. Section III A reviews the basics of the trifocal tensor representation. Robust parameter estimation of the trifocal tensor is addressed in Section III B. Finally, forward, backward, and ....

[Article contains additional citation context not shown here]

A. Shashua, "Algebraic functions for recognition," IEEE Trans. Pattern Anal. Machine Intell., vol. 17, pp. 779--789, Aug. 1995.

....but satisfying a set of algebraic and geometric constraints. These constraints allow to parameterize the tensors with minimal 18 parameters [3, 8, 14, 4] Also, M 0 ; M 00 belong to a family of homographies spanned with 3 degree of freedom and cannot be uniquely determined from the tensors [10, 13], therefore we have to enforce certain constraints to retrieve the projection matrices M 0 ; M 00 . The properties of these quantities and details of these constraints are not the main concern in this paper, we recommend the readers to check the references if interested. Here we generally ....

....of c 1 ; c 2 is not as significant as f , however the closer they are to zero, the better the results will be. This is also a reason why usually only 4 trilinearities i = 1; 2) l = 1; 2) j = m = 3 out of the total 9 constraints are used for 3 views motion estimation with uncalibrated scenes [10]. 8 Conclusion This paper gives a new geometrical interpretation of the algebraic trilinear constraints. It is shown that minimizing the trilinear algebraic equations in a least squares sense is equivalent to minimizing an error in Euclidean space with appropriate weighting applied on every ....

A. Shashua. Algebraic functions for recognition. IEEE Trans. Pattern Anal. Mach. Intell., 17(8):779--789, Aug 1995.

.... computing the trilinear tensor, and our intent for future releases of the Projective Vision Toolkit(PVT) Keywords : Projective Vision, Modeling, Virtual Reality, Computer Vision 1 Introduction Uncalibrated computer vision has been a topic that has gathered much interest in that last decade [1,2], with a lot of work being done in the last four years. The goal is to produce information about a scene without the aid of calibrated sensors, ultimately to produce a valid 3D model. There have been several systems implemented [3,4,5] that claim to automatically produce these models from an ....

....2 , 1] T , and m 3 = x 3 , y 3 , 1] T In addition, in a slight abuse of notation, we define m i as the i th element of m 1 . It has been shown that there is a 27 element quantity called the trifocal tensor relating the pixel coordinates of the projection of this 3D point in the three images [1]. Individual elements of are labeled ijk, where the subscripts vary in the range of 1 to 3. If the three 2D co ordinates (m 1 , m 2 , and m 3 ) truly correspond to the same 3D point, then the following four trilinear constraints hold m 3 i13 m i m 3 m 2 i33 m i m 2 i31 m i i11 m i ....

[Article contains additional citation context not shown here]

A. Shashua, "Algebraic functions for recognition" IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 17, no. 8, pp. 779-789, 1995.

....models, and they handle geometric ill conditioning, distortion and non rigidities gracefully. As far as we know the idea of basing image correspondence on joint distributions is new, but there are several related threads in the literature. The theory of multi image geometric matching constraints [14, 16, 3, 4, 7, 9, 21, 19] is both the starting point of this work and our focus for generalization, although the joint distribution correspondence principle goes well beyond them. Explicit model selection [12, 13, 18] is the approach that we are reacting against [15] and offering an alternative to. Plane parallax [10, 11, ....

....of this paper is theoretical, but 5 briefly discusses implementation details. 6 summarizes, and an appendix sketches mathematical properties of the tensor joint image. We assume familiarity with the affine camera, tensorial projective geometry and affine and projective matching constraints [8, 2, 14, 16, 3, 4, 7, 9, 21, 19]. Notation: Bold italic x, y denotes 2 component inhomo1 p(y) x 2 p(y x ) p(x) 1 p(y x ) p(x,y) x x y 2 1 Figure 1: The basic principle of feature correspondence using a joint feature distribution. geneous image vectors, bold upright x, y 3 component homogeneous ones, X,Y 3D ones. ....

[Article contains additional citation context not shown here]

A. Shashua. Algebraic functions for recognition. IEEE Trans. Pattern Analysis & Machine Intelligence, 17(8):779-- 89, 1995.

.... development (Tomasi Kanade 1992, Morita Kanade 1997, Quan Kanade 1997, Sturm Triggs 1996) These directions of research have recently been combined with the ideas behind the fundamental matrix (Longuet Higgins 1981, Faugeras 1992, Xu Zhang 1997) and have lead to the trilinear tensor (Shashua 1995, Hartley 1995, Heyden 1995) as a uni ed model for point and line correspondences for three cameras, with interesting applications (Beardsley et al. 1996) as well as a deeper understanding of the relations between point features and line features over multiple views (Faugeras Mourrain 1995, ....

....the projections of an unknown con guration of lines in 3 D, it is necessary to have at least three independent views. A canonical model for describing the geometric relationships between point correspondences and line correspondences over three perspective views is provided by the trilinear tensor (Shashua 1995, Shashua 1997, Hartley 1995, Heyden et al. 1997) For ane cameras, a compact model of point correspondences over multiple frames can be obtained by factorizing a matrix with image measurements to the product of two matrices of rank 3, one representing motion, and the other one representing ....

[Article contains additional citation context not shown here]

Shashua, A. (1995), `Algebraic functions for recognition', IEEE-PAMI 17(8), 779-789.

....of the feature points to minimize this error. The error function is in general well behaved and the distortion parameters can easily be found by nonlinear search techniques. 1.2. The three image method Corresponding points in three images are related by 4 independent trilinear equations. [9]. These equations have 27 parameters which can be found in a linear manner given at least 7 point correspondences. These parameters allow us to reproject corresponding points given in two of the images into the third image. Due to noise and lens distortion, the reprojection is not perfect. We ....

....will be denoted P j . Using homogeneous coordinates the perspective projection can be written as: ae ij m ij = P j M i (1) for points i = 1: n in images j = 1: m. ae ij is an unknown scale factor which is different for each point and each image. 3.1. The trilinear tensor tonstraint Shashua [9] shows that given a set of 3D points there exists a set of trilinear equations between the projections of those points into any three perspective views. In total there exist 9 such equations for each point with at most 4 being independent. Four of the nine equations are as follows: x ff T ....

A. Shashua, A., Algebraic Functions for Recognition. IEEE Trans. PAMI 17, 779-789, (1995)

....from uncalibrated views. We are interested in euclidean reconstruction. Many algorithms have been proposed, differing e.g. on the assumptions concerning the calibration parameters and or motion [5] Studies of the precision of the estimation of the fundamental matrix [21] and trifocal tensor [15], which represent multilinear constraints that tracked 2D features This work has been supported by projects INCO COPERNICUS Proj. 960174 VIRTUOUS and PRAXIS 2 2.1 TPAR 2074 95 1 must verify can be found in [3, 18, 9] A study of critical (pathological) cases for self calibration can be found ....

A. Shashua. Algebraic functions for recognition. IEEE Trans. PAMI, 17(8):779--789, August 1994.

....point line correspondences (Spetsakis and Aloimonos 1989) iTrilinearj means that the constraints are linear in the point line RR n Sigma2927 44 Zhengyou Zhang coordinates of each image, and the epipolar constraint (5) is a bilinear relation. The trilinear constraints have been rediscovered in (Shashua 1995) in the context of uncalibrated images. Similar to the fundamental matrix for two images, the constraints between three images can be described by a 3 Theta 3 Theta 3 matrix dened up to a scale factor (Spetsakis and Aloimonos 1989, Hartley 1994) There exist at most 4 linear independent ....

.... between three images can be described by a 3 Theta 3 Theta 3 matrix dened up to a scale factor (Spetsakis and Aloimonos 1989, Hartley 1994) There exist at most 4 linear independent constraints in the elements of the above matrix, and 7 point matches are required to have a linear solution (Shashua 1995). However, the 27 elements are not algebraically independent. There are only 18 parameters to describe the geometry between three uncalibrated images (Faugeras and Robert 1994) and we have three algebraically independent constraints. Therefore, we need at least 6 point matches to determine the ....

[Article contains additional citation context not shown here]

Shashua, A.: 1995, Algebraic functions for recognition, IEEE Transactions on Pattern Ana lysis and Machine Intelligence 17(8), 779789.

....from uncalibrated views. We are interested in euclidean reconstruction. Many algorithms have been proposed, differing, e.g. on the assumptions concerning the calibration parameters and or motion [6] Studies of the precision of the estimation of the fundamental matrix [7] and trifocal tensor [8], which represent multilinear constraints that tracked 2D features must verify can be found in Refs. 9 11] A study of critical (pathological) cases for self calibration can be found in Ref. 12] and the achievable precision in the calibrated case is addressed in Ref. 13] In this paper, we ....

A. Shashua, Algebraic functions for recognition, IEEE Transactions on PAMI 17 (8) (1994) 779--789.

....calibration is difficult owing to uncertain camera motions, changes in internal parameters (focus, zooming) or the use of several cameras. In response to these needs, there has recently been a significant amount of theoretical work on the structure of multi image projection and reconstruction [8, 7, 14, 13, 17, 1, 16, 6, 11, 12, 23, 20, 21, 2]. The problem turns out to have a surprisingly rich mathematical structure, and several complimentary approaches exist. The field is developing rapidly and there is no space for a complete survey here, so I will only mention a few isolated results. The epipolar constraint (the geometry of stereo ....

....and several complimentary approaches exist. The field is developing rapidly and there is no space for a complete survey here, so I will only mention a few isolated results. The epipolar constraint (the geometry of stereo pairs) has been understood for quite a while (e.g. c.f. 4] Shashua [16] and Hartley [8] developed the theory of the trivalent tensor (three view constraint) Faugeras and Mourrain [6] and I [21] systematically studied the This work was supported by an EC HCM grant and INRIA Rhone Alpes. British Machine Vision Conference complete family of multi image constraints ....

[Article contains additional citation context not shown here]

A. Shashua. Algebraic functions for recognition. IEEE Trans. Pattern Analysis & Machine Intelligence, 1995.

....accurate tracking results. Keywords : Feature tracking, affine invariant, perspective invariant, Hough transform 1 Introduction Recently, a number of papers have discussed the problem of computing the information of an image using the information from two or more other images of the same scene [1, 2, 3]. This problem is strongly related to the tracking problem. Many of the methods proposed assume the knowledge of the epipolar geometry or the cameras are weakly calibrated [1, 8] However, without camera calibration, some important results [7, 6, 3] have been obtained by a number of researchers ....

....from two or more other images of the same scene [1, 2, 3] This problem is strongly related to the tracking problem. Many of the methods proposed assume the knowledge of the epipolar geometry or the cameras are weakly calibrated [1, 8] However, without camera calibration, some important results [7, 6, 3] have been obtained by a number of researchers using geometric invariance to map image features from two reference images to a third image. Shashua [3] shows the of a trilinear function between three perspective views and that the coefficients of the function can be recovered linearly without ....

[Article contains additional citation context not shown here]

A. Shashua, "Algebraic Functions for Recognition, " IEEE Trans on PAMI, vol. 17, August 1995.

....A1 FB1C3 (A1 B1) 0 ffl A3B3C3 T A2 0 A3 A1 T B2B3 B1 e C3 1 (A1 B1) 0 where as usual T 2 0 3 1 T 23 1 T Delta23 1 . But this whole approach seems over complicated. Given that T 23 1 is actually linear in P 2 , we might as well just find a homographyepipole decomposition [7, 11] T 23 1 = H 2 1 Omega e 3 1 Gamma e 2 1 Omega H 3 1 Gamma T 23 1 Delta x 1 Delta = Gamma H 2 1 j e 2 1 Delta i 0 x1 Gammax 1 0 j Gamma H 3 1 j e 3 1 Delta and work directly in terms of P i = Gamma H i 1 j e i 1 Delta for i = 1; 2; 2 0 ; 3. As ....

A. Shashua. Algebraic functions for recognition. IEEE Trans. Pattern Analysis & Machine Intelligence, 17(8):779-- 89, 1995.

....various motivations for this strategy. As discussed in Section 2.2.7, under some conditions projective optimization may be a good approach even for calibrated sequences, but its usefulness is probably quite limited. Section 2. 3 critiques non optimal invariants based algorithms such as in [33, 35] [13, 9, 8] Though strictly these are not projective algorithms (Section 2.2.5) the projective approach is often believed to be responsible for their virtues of speed and simplicity. 2.2.3 Self Calibration One of the advantages claimed for the projective approach is that it simultaneously ....

....is: if a multi frame algorithm is simple and does not represent the structure explicitly, it is likely to be far from optimal. Thus, for general sequences of three or more images, one must choose between a slow optimization approach or a simpler, faster, but far from optimal algorithm such as [33, 35] [13, 9, 8] Since recent experimental work [23] 46] has shown that two frame optimization is quite robust and accurate as well as fast, it may actually be preferable to these three frame algorithms. The algorithms [33, 35] 13, 9, 8] may actually be more fragile than two frame optimization on ....

[Article contains additional citation context not shown here]

A. Shashua, "Algebraic functions for recognition," PAMI 17, 779-789, 1995.

....we consider the 3D and 2D directions of lines as 2D and 1D projective points. This means that the affine reconstruction of lines with a two dimensional affine camera is equivalent to the projective reconstruction of points with a one dimensional projective camera There have been many recent works [3, 5, 24, 13, 4, 6, 21, 22] on projective reconstruction and the geometry of multi views of two dimensional uncalibrated cameras. Particularly, the tensorial formalism developed by Triggs [24] is very interesting and powerful. We are now extending this study to the case of the one dimensional camera. 3. Uncalibrated ....

A. Shashua. Algebraic functions for recognition. IEEE TPAMI, 1994. in press.

....reconstruction. Thus, if one wants to run a 3 frame algorithm on a general sequence, one must decide between an approach that is optimal but slow, since it requires optimizing over the structure (at least implicitly) as well as the motion, or a non optimal and potentially unreliable one as in [48, 47] [16, 14, 11] 4 . Since there exists a two frame algorithm that is both fast and near optimal, using it may often be preferable to using 3 frame algorithms 5 6 . For example, fusing near optimal two frame reconstructions may turn out to give better multi frame estimates than fusing the ....

....algorithm that is both fast and near optimal, using it may often be preferable to using 3 frame algorithms 5 6 . For example, fusing near optimal two frame reconstructions may turn out to give better multi frame estimates than fusing the far from optimal three frame results of [47] [11] Studying these issues experimentally is an important topic for future research. Lastly, we show that a well known problem with the Tomasi Kanade algorithm is often not a significant one. In this paper we assume no occlusion. The algorithm can be extended to handle occlusion using methods ....

[Article contains additional citation context not shown here]

A. Shashua, "Algebraic functions for recognition," PAMI 17, 779-789, 1995.

....occlusions are allowed. 1 Introduction The structure and motion problem is central for computer vision, dealing with the analysis of a 3D scene by means of a sequence of 2D images. It is often studied by epipolar geometry and multilinear constraints, cf. 2] 3] 4] 5] 6] 7] 10] [11], 19] 21] The present paper uses an alternative approach, based on the notions of aOEne shape and depth, developed in a series of papers [12] 13] 14] 15] 16] 17] 18] Depending on the apriori information available, the structure and motion problem can be treated on dioeerent ....

Shashua, A.: Algebraic functions for recognition. IEEE Trans. Pattern Anal. Machine Intell. 17:8 (1995)

....is in general well behaved and the distortion parameters can easily be found by nonlinear search techniques. 1. 2 The three image method: Given a set of corresponding points in three images, there exist 4 independent trilinear equations that relate location of the points in the three images [9]. These equations have 27 parameters and given at least 7 point correspondences can be found in a linear manner. These parameters allow us to reproject points given in two of the images into the third image. Again, because of noise and lens distortion, the reprojection is not perfect. We define ....

....equation of that line is given by F jk m i;k . Given 8 or more point correspondences the Fundamental Matrix F can be determined up to a scale factor using the eight point algorithm which is described in [6] with many important implementation details. 3. 2 The Trilinear Tensor Constraint: Shashua [9] shows that given a set of 3D points there exists a set of trilinear equations between the projections of those points into any three perspective views. In total there exist 9 such equations for each point with at most 4 being independent. Four of the nine equations are as follows: x ff T ....

Shashua, A. "Algebraic Functions for Recognition", IEEE Trans. Pattern Anal. Machine Intell. 17, 779-789, (1995)

....take into account as many as possible of the above properties, and can be used as input to nonlinear methods if more precision is required. Partly in response to this, there has recently been a significant amount of work on the theoretical foundations of multi image projection and reconstruction [11, 10, 19, 18, 23, 2, 22, 8, 15, 16, 31, 27, 28, 3]. The problem turns out to have a surprisingly rich mathematical structure and several complementary approaches exist. The field is developing rapidly and there is no space for a survey here, so I will only mention a few isolated results. The epipolar constraint (the geometry of stereo pairs) is ....

....rich mathematical structure and several complementary approaches exist. The field is developing rapidly and there is no space for a survey here, so I will only mention a few isolated results. The epipolar constraint (the geometry of stereo pairs) is now well understood (e.g. 5] Shashua [22] and Hartley [11] developed the theory of the trivalent tensor (three view constraint) Faugeras and Mourrain [8] and I [28] systematically studied the complete family of multi image constraints (only one was unknown: a quadrilinear one) As a means to this, I developed a tensorial approach to ....

[Article contains additional citation context not shown here]

A. Shashua. Algebraic functions for recognition. IEEE Trans. Pattern Analysis & Machine Intelligence, 1995.

....that generate the algebraic constraints between corresponding tokens in different images. For two images we have the 3 Theta 3 fundamental matrix F and its calibrated cousin the essential matrix E [23, 5] while for three images there are three distinct 3 Theta 3 Theta 3 trifocal tensors G [24, 14]. Besides their use in image matching, matching tensors are important because they characterize the camera geometry and generate an implicit 3D scene reconstruction [26, 32] However they are an over parametrization. For m images the camera geometry has 6m Gamma 7 parameters for calibrated ....

A. Shashua. Algebraic functions for recognition. IEEE Trans. Pattern Analysis & Machine Intelligence, 17(8):779-- 89, 1995.

....A1 FB1C3 (A1 B1) 0 ffl A3B3C3 T A2 0 A3 A1 T B2B3 B1 e C3 1 (A1 B1) 0 where as usual T 2 0 3 1 T 23 1 T Delta23 1 . But this whole approach seems over complicated. Given that T 23 1 is actually linear in P 2 , we might as well just find a homographyepipole decomposition [7, 11] T 23 1 = H 2 1 Omega e 3 1 Gamma e 2 1 Omega H 3 1 Gamma T 23 1 Delta x 1 Delta = Gamma H 2 1 j e 2 1 Delta i 0 x1 Gammax 1 0 j Gamma H 3 1 j e 3 1 Delta and work directly in terms of P i = Gamma H i 1 j e i 1 Delta for i = 1; 2; 2 0 ; 3. As ....

A. Shashua. Algebraic functions for recognition. IEEE Trans. Pattern Analysis & Machine Intelligence, 17(8):779-- 89, 1995.

....for line factorization will be a two dimensional projective reconstruction from onedimensional projective spaces. This projective reconstruction will allows us to rescale properly the image directions for further submitting them to factorization. This part is largely inspired by many recent works [24, 22, 6, 7, 20, 21] on the geometry of multi views of two dimensional perspective camera, especially the approaches taken by Triggs and Sturm [24, 22] We extend these ideas to one dimensional camera. It turns out some interesting properties which were absent for 2 dimensional camera. 4.1 Matching constraints of ....

A. Shashua. Algebraic functions for recognition. Ieee Transactions on PAMI, 1994. in press.

....the second reference view is used but only half of it (i.e. only the x coordinates) Of course, 3) and (4) can be rewritten using the y coordinates of the second reference view instead. The extension of algebraic functions of views in the case of perspective projection was carried out in [20 23]. In particular, it was shown that three perspective views of an object satisfy a trilinear function. Moreover, it was shown that a simpler and more practical pair of algebraic functions exist when the reference views are orthographic [20, 21] This is useful for realistic object recognition ....

....the case of perspective projection was carried out in [20 23] In particular, it was shown that three perspective views of an object satisfy a trilinear function. Moreover, it was shown that a simpler and more practical pair of algebraic functions exist when the reference views are orthographic [20, 21]. This is useful for realistic object recognition applications. In this paper, we consider the case of orthographic projection assuming 3D linear transformations only. 3. A FRAMEWORK FOR INDEXING USING ALGEBRAIC FUNCTIONS OF VIEWS Algebraic functions of views can be used to predict the image ....

[Article contains additional citation context not shown here]

A. Shashua, Algebraic functions for recognition, IEEE Trans. Pattern Anal. Mach. Intelligence 17(8), 1995, 779--789.

....the unknown projective scale factors of the image measurements must be recovered before factorization becomes possible. In the affine case these are constant, so they can be directly eliminated from the problem) As part of the current blossoming of interest in multiimage reconstruction, Shashua [13] recently extended the well known two image epipolar constraint to a trilinear constraint between matching points in three images. Hartley [6] showed that this constraint also applies to lines in three images, and Faugeras Mourrain [4] and I [16, 17] completed that corner of the puzzle by ....

....very quickly (within 2 3 iterations) 3 Line Reconstruction Scenes containing 3D lines can also be reconstructed using the above techniques. We will only give a brief sketch of the theory here as a full discussion requires consideration of the trilinear three image matching constraint for lines [6, 13, 4, 17, 16]. A 3D line L can be defined by any two distinct points lying on it, say Y and Z. In each image i, L projects to some image line l i and Y and Z project to image points y i and z i lying on l i : l i Delta y i = 0, l i Delta z i = 0. The points y i , i = 1; m are in epipolar ....

A. Shashua. Algebraic functions for recognition. IEEE Trans. Pattern Analysis & Machine Intelligence, 1995.

....image 6. Map textures (colors) from the original images to the new images The most crucial and difficult part is Step 1 and Step 2. Step 1 will be reviewed in Appendix A. The second step consists in estimating the fundamental matrix between two images [39] the trifocal tensor between three images [40], or the P matrices (camera projection matrices) between N images [41] A complete review of techniques for estimating the fundamental matrix and projective reconstruction is done by Zhang [42] A good technique for estimating the trifocal tensor is developed in [43] The PhD thesis of Laveau [44] ....

A. Shashua, "Algebraic functions for recognition," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 17, no. 8, pp. 779--789, 1995.

.... in P 2 (projective plane) to a point u = u 1 ; u 2 ) T in P 1 (projective line) This projection may be described by a 2 Theta 3 homogeneous matrix M as u = M 2 Theta3 x: We now examine the geometric constraints available for points seen in multiple views similar to the 2D camera case [17, 18, 9, 21, 7]. There is a constraint only in the case of 3 views, as there is no any constraint for 2 views (two projective lines always intersect in a point in a projective plane) Let the three views of the same point x be given as follows: 8 : u = Mx; 0 u 0 = M 0 x; 00 u 00 = M 00 x: ....

A. Shashua. Algebraic functions for recognition. Trans. PAMI, 17(8):779--789, 1995.

....are needed, however it has not been clear how to find the unknown projective scale factors of the image measurements that are required for this. In the affine case the scales are constant and can be eliminated) As part of the current blossoming of interest in multiimage reconstruction, Shashua [14] recently extended the wellknown two image epipolar constraint to a trilinear constraint between matching points in three images. Hartley [6] showed that this constraint also applies to lines in three images, and Faugeras Mourrain [4] and I [18, 19] completed that corner of the puzzle by ....

....instances of equation (2) to find the corresponding, correctly scaled via points in the other images. The required fundamental matrices can not be found directly from line matches, but they can be estimated from point matches, or from the trilinear line matching constraints (trivalent tensor) [6, 14, 4, 19, 18]. Alternatively, the trivalent tensor can be used directly: in tensorial notation [18] the trivalent via point transfer equation is l B k GC j A i B k y C j = l Bk e Bk j )y A i . As with points, redundant equations may be included if and only if a self consistent normalization is chosen ....

A. Shashua. Algebraic functions for recognition. IEEE Trans. Pattern Analysis & Machine Intelligence, 1995.

....to the projective reconstruction of points with a one dimensional projective camera. One of the major remaining efforts will be concerned with 2D projective reconstruction from the points in P 1 . There have been many recent works [30] 31] 32] 33] 34] 35] 36] 37] 38] 39] [40], 41] 10] 42] 43] on projective reconstruction and the geometry of multi views of two dimensional uncalibrated projective cameras. Particularly, the tensorial formalism developed by Triggs [36] is very interesting and powerful. We now extend this study to the case of the one dimensional ....

A. Shashua, "Algebraic functions for recognition", ieee Transactions on Pattern Analysis and Machine Intelligence, vol. 17, no. 8, pp. 779--789, August 1995.

.... In the case of uncalibrated cameras it is known that reconstruction is possible only up to projective transformations, cf. 16] 2] 7] For three or more views, much recent research has been directed on tri , quadri , and general multilinear constraints for point matches, cf. e.g. 20] [13], 6] 22] 5] 9] 10] A The work has been supported by the Swedish Research Council for Engineering Sciences, TFR, project 95 64 222. duality relation between reconstruction and positioning has been discovered in [1] The present paper follows another line of development, where ....

A. Shashua. Algebraic functions for recognition. IEEE Trans. Pattern Anal. Machine Intell., 17(8), 1995.

No context found.

# A. Shashua, "Algebraic Functions for Recognition," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 8, pp. 779-789, Aug. 1995.

.... 1, 2, we have four trilinear equations (unique up to linear combinations) If we choose the standard form where s # (and r # ) represent vertical and horizontal scan lines, i.e. s x y j = 1 3 2 4 6 5 10 01 then the four trilinear forms, referred to as trilinearities [10], have the following explicit form: xpxxpxp p ypyxpxp p xpxypyp p ypyypyp p i i i i i i i i i i i i i i i i i i UUUU UUUU UUUU UUUU 13 33 31 11 13 33 32 12 23 33 31 21 23 33 32 22 0 0 0 0 These constraints were first derived in ....

.... have the following explicit form: xpxxpxp p ypyxpxp p xpxypyp p ypyypyp p i i i i i i i i i i i i i i i i i i UUUU UUUU UUUU UUUU 13 33 31 11 13 33 32 12 23 33 31 21 23 33 32 22 0 0 0 0 These constraints were first derived in [10]; the tensorial derivation leading to (4) and (5) was first derived in [12] 13] The trilinear tensor has been well known in disguise in the context of Euclidean line correspondences and was not identified at the time as a tensor but as a collection of three matrices (a particular contraction of ....

[Article contains additional citation context not shown here]

# A. Shashua, "Algebraic Functions for Recognition," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 8, pp. 779-- 789, Aug. 1995.

....These methods are dubbed direct methods because they do not require prior computation of optical flow. As with other gradient methods we assume small image motions on the order of a few pixels. Applying the constant brightness constraint [6] to the trilinear tensor of Shashua and Werman [12, 15] results in an equation relating camera motion and calibration parameters to the image gradients (first order only) We get one equation for each point in the image and we have a fixed number of parameters which results in a highly over constrained set of equations. Starting with the general ....

....= 0 (2) where s j are any two lines (s 1 j and s 2 j ) intersecting at p 0 , and r ae k are any two lines intersecting p 00 . Since the free indices are ; ae each in the range 1,2, we have 4 trilinear equations (which are unique up to linear combinations) More details can be found in [4, 12, 15, 13]. Geometrically, a trilinear matching constraint is produced by contracting the tensor with the point p of image 0, some line coincident with p 0 in image 1, and some line coincident with p 00 in image 2. In particular, we may use the tangent to the iso brightness contour at p 0 and p 00 ....

Shashua, A. "Algebraic functions for recognition ", IEEE Transactions on Pattern Analysis and Machine Intelligence, 17(8):779--789, 1995.

....pairs of images has been studied in great detail over the past 20 years [3] 4] 8] Much less is known about reconstruction from three or more images. General frameworks for reconstruction from multiple images are described in [5] 6] 7] 13] and more detailed results are given in [9] [11]. Experimental results are reported in [1] 6] 9] 11] 12] It is shown in [9] that reconstruction from three images of six points is subject to a three way ambiguity. Our main result is the surprising fact that the three way ambiguity is preserved as long as the optical centre of the camera ....

....the past 20 years [3] 4] 8] Much less is known about reconstruction from three or more images. General frameworks for reconstruction from multiple images are described in [5] 6] 7] 13] and more detailed results are given in [9] 11] Experimental results are reported in [1] 6] 9] [11], 12] It is shown in [9] that reconstruction from three images of six points is subject to a three way ambiguity. Our main result is the surprising fact that the three way ambiguity is preserved as long as the optical centre of the camera remains on a certain quadric surface ; the ambiguity is ....

A. Shashua. "Algebraic functions for recognition. " IEEE Trans. Pattern Recognition and Machine Intelligence, Vol. 17, pp. 779-789, 1995.

No context found.

A. Shashua. Algebraic Functions for Recognition. IEEE Tran. Pattern Anal. Machine Intelligence, 16:778--790, 1995.

No context found.

A. Shashua. Algebraic functions for recognition. IEEE Tran. Pattern Anal. Machine Intelligence, 16:778--790, 1995.

No context found.

A. Shashua, "Algebraic functions for recognition,", IEEE Tran. Pattern Anal. Machine Intelligence, vol. 16, pp. 778--790, 1995.

No context found.

A. Shashua, "Algebraic Functions for Recognition," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 8, pp. 779-789, Aug. 1995.

No context found.

A. Shashua. Algebraic functions for recognition. IEEE Trans. PAMI, 17(8):779--789, August 1995.

No context found.

A. Shashua. Algebraic functions for recognition. IEEE Trans. Pattern Anal. Machine Intell., 17:779--780, 1995.

No context found.

A. Shashua, "Algebraic functions for recognition," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 17, no. 8, pp. 779--789, Aug. 1995.

No context found.

A. Shashua, "Algebraic functions for recognition," IEEE Trans. Pattern Anal. Machine Intell., vol. 17, pp. 779--789, Aug. 1995.

No context found.

A. Shashua. Algebraic functions for recognition. IEEE Trans. Pattern Anal. Machine Intell., 17:779--780, 1995.

No context found.

A. Shashua, "Algebraic functions for recognition," IEEE Transactions on PAMI, vol. 17, no. 8, pp. 779--789, Aug. 1995.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC