Q.T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. INRIA, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....a calibration free specification of the location of an image point. Define by m and m the 3 vectors in homogeneous coordinates derived from m and m such that m = u; v; 1) m ; 1) 6) We may now express the epipolar equation as = 0: 7) Here, F is the fundamental matrix [10, 24, 25] (except that in our case F relates corresponding points in the reverse direction) and is defined as ; 8) where T is the skew symmetric matrix formed from the baseline vector, and A contains the left camera intrinsic parameters, such that 0 Gammat z t y Gammat y t x 0 A (9) 1 0 ....

Luong, Q.-T., Deriche, R., Faugeras, O., and Papadopoulo, T. On Determining the Fundamental Matrix: Analysis of Different Methods and Experimental Results. Tech. Rep. 1894, INRIA, 1993.

....it only exploits the minimum number of point correspondences necessary to estimate the epipolar geometry and is thus unable to deal with noise. More robust approaches to weak calibration from a large number of point correspondences have been proposed recently in the computer vision community: Luong et al. 1993, 1995 ] have proposed various linear and non linear least squares methods for estimating the fundamental matrix, which captures the epipolar geometry in algebraic form. In particular, they have shown that, although Longuet Higgins eight point algorithm [ Longuet Higgins, 1981 ] generalizes to ....

....in the presence of noise. This has prompted Luong et al. to propose an iterative non linear algorithm that minimizes the distance between image points and the corresponding epipolar lines. The reliability and accuracy of this technique have been established through extensive experimentation in [ Luong et al. 1993, Luong and Faugeras, 1995 ] Recently, Hartley [ 1995 ] has shown that the poor characteristics of the eight point method can be traced to the fact that the corresponding matrices are ill conditioned, so that adding a simple preprocessing step (translating the data so it is centered at the ....

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Q.-T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: analysis of different methods and experimental results. Technical Report 1894, INRIA Sophia-Antipolis, 1993.

....it only exploits the minimum number of point correspondences necessary to estimate the epipolar geometry and is thus unable to deal with noise. More robust approaches to weak calibration from a large number of point correspondences have been proposed recently in the computer vision community: Luong et al. 1993, 1995 ] have proposed various linear and nonlinear least squares methods for estimating the fundamental matrix, which captures the epipolar geometry in algebraic form. In particular, they have shown that, although Longuet Higgins eightpoint algorithm [ Longuet Higgins, 1981 ] generalizes to the ....

....in the presence of noise. This has prompted Luong et al. to propose an iterative non linear algorithm that minimizes the distance between image points and the corresponding epipolar lines. The reliability and accuracy of this technique have been established through extensive experimentation in [ Luong et al. 1993, Luong and Faugeras, 1995 ] Recently, Hartley [ 1995 ] has shown that the poor characteristics of the eight point method can be traced to the fact that the corresponding matrices are ill conditioned, so that adding a simple preprocessing step (translating the data so it is centered at the ....

[Article contains additional citation context not shown here]

Q.-T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: analysis of different methods and experimental results. Technical Report 1894, INRIA Sophia-Antipolis, 1993.

....v j #v j using the relation p ij F j p io =0,over all i. Eight corresponding points (frame 0 and frame j) are needed for a linear solution, and a least squares solution is possible if more points are available. In practice the best results were obtained using the non linear algorithm of [21]. The epipoles followby F j v j = 0 and F j = 0 [7] The latter readily follows from Corollary 5 as [v j ]A j v j j = 0 and j ] j = A j =0. 2. Compute A j from the equations A j p io = p ij , i = 1# 2# 3, and A j v j j . This leads to a linear set of eight ....

....8 Epipoles were recovered by either one of the following two methods. First, by using the four ground points to recover the homography A, and then by Corollary 5 to compute the epipoles using all the remaining points in a least squares manner. Second, using the non linear algorithm proposed by [21]. The two methods gave rise to very similar results for reconstruction, and slightly different results for re projection (see later) In the reconstruction paradigm, we recovered relative affine structure from two views and multiple views. In the two view case we used either a small base line ....

[Article contains additional citation context not shown here]

Q.T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. Technical Report INRIA, France, 1993.

....scene geometry are applied. For synthesizing a view from a virtual camera position, the image pixels are reprojected appropriately. The geometric constraints can be of the form of known depth values at each pixel [CW93] or epipolar constraints between pairs of images are used (fundamental matrix [LDFP93], LF94] or constraints between pairs of cylindrical panoramas [MB95] It is also possible to use three images with trilinear tensors [AS98] 3 Implementation There are many possible surfaces upon which perspective projections can be mapped [GG99] The most natural one is a sphere centered ....

Q. T. Luong, Rachid Deriche, Olivier Faugeras, and Theodore Papadopoulo. On determining the fundamental matrix : analysis of different methods and experimental results. Technical Report RR-1894, Inria, May 1993.

....virtually useless for many practical applications which make use of automatic feature detection or matching, where localisation error is likely to cause problems. Consequently, a number of other methods for computing the fundamental matrix have been explored. A good review of these can be found in [7]. These methods are more complex than the eight point algorithm, involving non linear solutions. However, in a recent paper by Hartley[5] a novel technique is described, which claims to improve the performance of the eight point algorithm to a level approaching (and in some cases surpassing) that ....

....equations is in a sense the 8 point algorithm. We will now look at two different methods of solution. Solution Via Singular Value Decomposition. In order to avoid the trivial solution, F = 0, we set one of the coefficients of F to 1. The choice of coefficient is not arbitrary, as discussed in [7]. We set f 13 = 1 and use singular value decomposition [9] to solve the modified system of linear equations: 0 B B B B uu 0 vu 0 uv 0 vv 0 v 0 u v 1 : ....

Q-T. Luong, R. Deriche, O. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. Technical Report 1894, INRIA, Sophia-Antipolis, April 1993.

....to 3D coordinates via camera calibration [22] that is computing the projection matrix which relates image coordinates to a world coordinate frame. In recent years, the focus has shifted to non metric reconstruction from uncalibrated cameras [9] by computing the fundamental matrix (two views) [12], and the trilinear tensor (three views) 16] Also, different camera models were assumed; i.e. orthographic [20, 23] perspective projection [11, 25] or a unified model [1, 15] Structure and motion algorithms typically assume given correspondences between features in successive frames. Finding ....

Q.T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. INRIA, 1993.

....3D coordinates via camera calibration [51, 54] that is computing the projection matrix which relates image coordinates to a world coordinate frame. In recent years, the focus has shifted to non metric reconstruction from uncalibrated cameras [25] by computing the fundamental matrix (two views) [28], and the trilinear tensor (three views) 42] Also, different camera models were assumed; i.e. orthographic [49, 53] perspective projection [27, 54] or a unified model [4, 41] Determining the geometric relationship between various views of the environment and its 3D structure is a key ....

Q.T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. Technical report, INRIA, 1993.

....v j ; v 0 j using the relation p ij F j p io = 0, over all i. Eight corresponding points (frame 0 and frame j) are needed for a linear solution, and a least squares solution is possible if more points are available. In practice the best results were obtained using the non linear algorithm of [13]. The epipoles follow by F j v j = 0 and F v 0 j = 0 [4] The latter readily follows from Corollary 3 as [v 0 j ]A j v j = v 0 j ]v 0 j = 0 and A j [v 0 j ] v 0 j = GammaA j [v 0 j ]v 0 j = 0. 2. Compute A j from the equations A j p io = p ij , i = 1; 2; 3, and ....

....by squares) Epipoles were recovered by the following two methods. First, by using the four ground points to recover the homography A, and then by Corollary 3 to compute the epipoles using all the remaining points in a least squares manner. Second, using the non linear algorithm proposed by [13]. The two methods gave rise to very similar results for reconstruction, and slightly different results for reprojection (see later) In the reconstruction application (Section 3.1) the relative affine structure was recovered from the two extreme views (displays (a) and (d) The transformation ....

[Article contains additional citation context not shown here]

Q.T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. Technical Report INRIA, France, 1993.

....This constraint is unique to each rigid transformation and can be used to identify independently moving objects. The epipolar constraint has been used in a number of structure from motion algorithms [LH81, TH84, TK92] The epipolar constraint can be used even in the case of uncalibrated cameras [LF94, LDFP93]. This allows for segmentation without any priors on shape or scene structure. In addition, the constraint holds for each point on an object, not just at the boundaries. The optical flow can therefore be sparse at the object boundaries. As pointed out by Koenderink and van Doorn [KvD91] and ....

....motion parameter estimation was not dependent on this particular form of the geometry. If a recovery of the full perspective case was required, the same algorithm could be used. However, the calculation of the Fundamental Matrix from small displacements such as found in optical flow is not stable [WAH93, LDFP93]. 28 Figure 10: A single frame of the mobile sequence from RPI. The background consists of translating patterns while a toy train traverses the foreground. An example optical flow recovered is also shown. The labeled image is shown below. The background parts (colored grey) were identified ....

Q.-T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the Fundamental matrix: analysis of different methods and experimental results. Technical Report RR-1894, INRIA, 1993. A shorter version appeared in the Israelian Conf. on Artificial Intelligence and Computer Vision.

....to 3D coordinates via camera calibration [22] that is computing the projection matrix which relates image coordinates to a world coordinate frame. In recent years, the focus has shifted to non metric reconstruction from uncalibrated cameras [9] by computing the fundamental matrix (two views) [12], and the trilinear tensor (three views) 16] Also, different camera models were assumed; i.e. orthographic [20, 23] perspective projection [11, 25] or a unified model [1, 15] Structure and motion algorithms typically assume given correspondences between features in successive frames. Finding ....

Q.T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. INRIA, 1993.

....elements of the 3 Theta 3 matrix F it is over parametrized. This is because the matrix elements are not independent, being related by a cubic polynomial in the matrix elements, such that det[F] 0. If this constraint is not imposed then the epipolar lines do not all intersect in a single epipole [16]. Hence it is essential that this constraint is imposed. The projectivity has 9 elements and 8 degrees of freedom as these elements are only defined up to a scale. The quadratic transformation has 18 elements and 14 degrees of freedom [18] Here if the constraints between the parameters are not ....

....this is minimal whereas it is not for the quadratic transform. The non linear parametrization fixing the largest element is dubbed P5. P6 is Luong s parametrization for the fundamental matrix. This is a 7 DOF parametrization in terms of the epipoles and epipolar homography designed by Luong et al. [16], this is both minimal and consistent. After applying MLESAC, the non linear minimization is conducted using the method described in Gill and Murray [6] which is a modification of the Gauss Newton method. All the points are included in the minimization, but the effect of outliers are removed as ....

Q. T. Luong, R. Deriche, O. D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: analysis of different methods and experimental results. Technical Report 1894, INRIA (Sophia Antipolis), 1993.

....to the accuracy of the 2D measurements [5] The effect of inaccuracies in measurements is more subtle in the case of the frame to frame mapping and fundamental matrix estimation. In order to combat their undesirable influence, sophisticated nonlinear estimation procedures have been proposed [6, 10, 8, 9]. However, this widely used approach is computationally very intensive and does not always guarantee that a correct solution to the estimation problem will be found. Moreover, non linear algorithms are iterative and therefore they cannot be guaranteed to be implemented in real time. As an ....

Q. T. Luong, R. Deriche, O. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. Technical Report 1894, INRIA, Sophia-Antipolis, France, 1993.

....3D coordinates via camera calibration [51, 54] that is computing the projection matrix which relates image coordinates to a world coordinate frame. In recent years, the focus has shifted to non metric reconstruction from uncalibrated cameras [25] by computing the fundamental matrix (two views) [28], and the trilinear tensor (three views) 42] Also, different camera models were assumed; i.e. orthographic [49, 53] perspective projection [27, 54] or a unified model [4, 41] Determining the geometric relationship between various views of the environment and its 3D structure is a key ....

Q.T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. Technical report, INRIA, 1993.

.... 8 ratios and there is one cubic constraint that the determinant is zero. For the trifocal tensor the 8 constraints have not been as thoroughly explored. In the case of the fundamental matrix if the constraint is not imposed then the epipolar lines do not all intersect in a single epipole [10]. Similarly, if the British Machine Vision Conference 1. Repeat for m = 500 samplings. a) Select a random sample of the minimum number of six feature correspondences to estimate the trifocal tensor T . This provides 1 or 3 solutions. b) For each of these solutions: i. Calculate the error e ....

Q. T. Luong, R. Deriche, O. D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: analysis of different methods and experimental results. Technical Report 1894, INRIA (Sophia Antipolis), 1993.

.... h = x 1 ; x 2 ; x 3 ) such that (X; Y; Z) X 1 =X 4 ; X 2 =X 4 ; X 3 =X 4 ) and (x; y) x 1 =x 3 ; x 2 =x 3 ) The general mapping between X and x can be written in terms of a transformation matrix T = T ij ] x h = TX h (4) and the transformation matrix T can be decomposed as follows [9, 17]: T = CPG : 5) The 3 Theta3 matrix C accounts for intrinsic camera parameters: C = 2 6 4 f s o x 0 f o y 0 0 1 3 7 5 (6) where f is the camera focal length, the aspect ratio, s the skew, and (o x ; o y ) the principal point (where the optic axis intersects the image plane) The ....

Q.-T. Luong, R. Deriche, O. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. Technical Report 1894, INRIA, Sophia Antipolis, 1993.

....There are a number of different ways to obtain these: 1. Point matches: In most cases it is useful to match points as well as contours. The points can be used to estimate motion parameters with conventional methods, e.g. the linear eight point method (Longuet Higgins 1981) or nonlinear methods (Luong, Deriche, Faugeras and Papadopoulo 1993). Approximate point matches can also be obtained by matching points with high curvature in the image or by using the centroid of the matched contours. The B spline snake tracker can also be used to obtain approximate point correspondences. Individual points on the apparent contour are first ....

Luong, Q. T., Deriche, R., Faugeras, O. D. and Papadopoulo, T.: 1993, On determining the fundamental matrix: analysis of different methods and experimental results, Technical Report RR-1894, INRIA.

....it only exploits the minimum number of point correspondences necessary to estimate the epipolar geometry and is thus unable to deal with noise. More robust approaches to weak calibration from a large number of point correspondences have been proposed recently in the computer vision community: Luong et al. 1993, 1996) have proposed various linear and non linear least squares methods for estimating the fundamental matrix, which captures the epipolar geometry in algebraic form. In particular, they have shown that, although Longuet Higgins eight point algorithm (LonguetHiggins 1981) generalizes to the ....

....in the presence of noise. This has prompted Luong et al. to propose an iterative non linear algorithm that minimizes the distance between image points and the corresponding epipolar lines. The reliability and accuracy of this technique have been established through extensive experimentation in (Luong et al. 1993, Luong and Faugeras 1996) Recently, Hartley (1995) has shown that the poor characteristics of the eight point method can be traced to the fact that the corresponding matrices are ill conditioned, so that adding a simple preprocessing step (translating the data so it is centered at the origin, ....

[Article contains additional citation context not shown here]

Luong, Q.-T., Deriche, R., Faugeras, O. and Papadopoulo, T.: 1993, On determining the fundamental matrix: analysis of different methods and experimental results, Technical Report 1894, INRIA SophiaAntipolis.

....the minimum number of point correspondences necessary to estimate the epipolar geometry and is thus unable to deal with noise. More robust approaches to weak calibration from a large number of point correspondences have been proposed recently in the computer vision community: Luong et al. [15, 18] have proposed various linear and nonlinear least squares methods for estimating the fundamental matrix, which captures the epipolar geometry in algebraic form. In particular, they have shown that, although Longuet Higgins eight point algorithm [14] generalizes to the uncalibrated case and can be ....

.... subspace approach and other methods for weak calibration: our LINPACK implementation of Hartley s normalized eight point algorithm [9] Boufama s and Mohr s implementation of the virtual parallax method [1] Quan s implementation of the non linear distance minimization technique of Luong et al. [15, 18], which uses the output of the plain eight point method to initialize the minimization, and our own implementation of the distance minimization algorithm, which relies instead on the output of the linear subspace approach as initial guess. We have also tested both the linear and non linear ....

Q.-T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: analysis of different methods and experimental results. Technical Report 1894, INRIA SophiaAntipolis, 1993.

....space to be the 3D projective space P 3 , and image space to be the 2D projective space P 2 both over the field C of complex numbers. Views are denoted by i , indexed by i. The epipoles are denoted by v 2 1 and v 0 2 2 , and we assume their locations are known (for methods, see [4, 5, 27, 28, 12], for example, and briefly later in the text) The symbol = denotes equality up to scale, GLn stands for the group of n Theta n matrices, PGLn is the group defined up to scale, and SPGLn is the symmetric specialization of PGLn . 3 The Quadric Reference Surface I: Points We start with ....

Q.T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. Technical Report INRIA, France, 1993.

....it is necessary to use non linear gradient descent. This requires a parameterisation for the fundamental matrix. The particular parameterisation used is important since it defines the covariance matrix, which will be used in evaluating the performance. The parameterization is that of Luong et al. [18], where F in expressed in terms of the non homogeneous coordinates of the epipoles e = e 1 ; e 2 ) e 0 = e 0 1 ; e 0 2 ) and three of the four coefficients of the homography between the epipolar lines: F = 2 6 4 b a Gammaae 2 Gamma be 1 Gammad Gammac ce 2 de 1 de 0 2 Gamma ....

Q. T. Luong, R. Deriche, O. D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: analysis of different methods and experimental results. Technical Report 1894, INRIA (Sophia Antipolis), 1993.

....v j ; v 0 j using the relation p ij F j p io = 0, over all i. Eight corresponding points (frame 0 and frame j) are needed for a linear solution, and a least squares solution is possible if more points are available. In practice the best results were obtained using the non linear algorithm of [21]. The epipoles follow by F j v j = 0 and F v 0 j = 0 [7] The latter readily follows from Corollary 5 as [v 0 j ]A j v j = v 0 j ]v 0 j = 0 and A j [v 0 j ] v 0 j = GammaA j [v 0 j ]v 0 j = 0. 2. Compute A j from the equations A j p io = p ij , i = 1; 2; ....

....Epipoles were recovered by either one of the following two methods. First, by using the four ground points to recover the homography A, and then by Corollary 5 to compute the epipoles using all the remaining points in a least squares manner. Second, using the non linear algorithm proposed by [21]. The two methods gave rise to very similar results for reconstruction, and slightly different results for re projection (see later) In the reconstruction paradigm, we recovered relative affine structure from two views and multiple views. In the two view case we used either a small base line (the ....

[Article contains additional citation context not shown here]

Q.T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. Technical Report INRIA, France, 1993.

....to that of Faugeras et al. 7] see the Appendix for a summary of the differences. Let m and m 0 denote corresponding points, in homogeneous coordinates, in the left and right images, respectively. We may express the epipolar equation as m T Fm 0 = 0; 1) where F is the fundamental matrix [7,17], given by F = A T TRA 0 : 2) Here, R embodies the pure rotation that renders the left image parallel with the right image, T is a skew symmetric matrix formed from the baseline vector connecting the left and right optical centres, and A and A 0 are the intrinsic parameter matrices of the ....

....technique to solve the following nonlinear minimisation min F X i w i r 2 i subject to the constraints that kxk 2 = 1 and det(F) 0. Here, the residual r i = v u u t 1 l 2 i;1 l 2 i;2 1 l 02 i;1 l 02 i;2 i m T i Fm 0 i j is referred to as the epipolar distance [17] and represents the geometric distance of the points m i and m 0 i to their associated epipolar lines defined by: l i;1 ; l i;2 ; l i;3 ) Fm i ; l 0 i;1 ; l 0 i;2 ; l 0 i;3 ) F T m 0 i : The chosen weighting factor w i is that proposed by Huber [12] and later used by Luong et al. ....

[Article contains additional citation context not shown here]

Q.-T. Luong, R. Deriche, O. Faugeras and T. Papadopoulo, On determining the fundamental matrix: analysis of different methods and experimental results, Rep. No. 1894, INRIA (1993).

....orientation) parameters. We shall refer to this process as self calibration. Faugeras et al. 2] developed a model for the interior orientation of a general uncalibrated pinhole camera in terms of 5 intrinsic parameters. Moreover a generalisation of the essential matrix, the fundamental matrix [10, 11], can be defined and derived from the data that is a function of both the intrinsic and extrinsic parameters. Even if the same camera is used to take both images in a stereo pair, however, these cannot be recovered simultaneously along with the 5 extrinsic parameters from the fundamental matrix as ....

.... F is the fundamental matrix [2] embodying both extrinsic and intrinsic imaging parameters, and is given by F = A 0 GammaT BRA Gamma1 : 8) If sufficiently many corresponding points can be located in the two images, it is possible to obtain a numerical estimate, Fest , of the matrix F [11]. Note that Fest is only defined up to an arbitrary scale factor (we shall show in the next section that a sensible choice for this is to scale any estimate so that kf3k = 1 as this is likely to optimise the conditioning of the linear system defining the solution) In what follows we shall not ....

Q.-T. Luong, R. Deriche, O. Faugeras, and T. Papadopoulo, "On Determining the Fundamental Matrix: Analysis of Different Methods and Experimental Results, " Report No. 1894, INRIA, Apr 1993.

.... to recover one homography, then with the aid of at least two more points to compute the epipoles (see Appendix A) then with the epipoles and three object points we compute the second homography (Proposition 1) Second, is by recovering the fundamental matrix using the algorithm described in [24] which consists of the basic method described in Appendix A with additional nonlinear constraints (such as enforcing that the rank of the matrix is two) The first case is, thus, less general but practical in many industrial settings, and the second case is more general (but slightly less ....

....of the assumption of having four coplanar points in our set of points. In this case we must recover the epipoles first, and then use them for recovering the homographies A and E (as described in Proposition 1 and in Section III A) The epipoles were recovered using the implementation described in [24] with all the available points (thirty four points) The resulting epipoles (between first and second frames) are: v = 1135:2; 337:3; 1:0) and v 0 = 41085:4; 4825:1; 1:0) Compared to the epipoles obtained previously (using the ground plane and two additional points) the results appear quite ....

[Article contains additional citation context not shown here]

Q.T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. Technical Report INRIA, France, 1993.

....to solve the following nonlinear minimization min F X i w i r 2 i subject to the constraints that kxk 2 = 1 and det(F) 0. Here, the residual r i = s 1 l 2 i;1 l 2 i;2 1 l 02 i;1 l 02 i;2 Gamma m i Fm 0 i Delta is referred to as the epipolar distance [9] and represents the geometric distance of the points m i and m 0 i to their associated epipolar lines defined by: l i;1 ; l i;2 ; l i;3 ) Fm i ; l 0 i;1 ; l 0 i;2 ; l 0 i;3 ) F m 0 i : The chosen weighting function w i is that proposed by Huber and later used by Luong [9] and ....

.... [9] and represents the geometric distance of the points m i and m 0 i to their associated epipolar lines defined by: l i;1 ; l i;2 ; l i;3 ) Fm i ; l 0 i;1 ; l 0 i;2 ; l 0 i;3 ) F m 0 i : The chosen weighting function w i is that proposed by Huber and later used by Luong [9], and is given by w i = 1 if jr i j oe oe=jr i j if oe jr i j 3oe 0 otherwise where oe is the standard deviation of the residuals r i . The above method is different to that reported in [14] in that the rank 2 constraint is automatically imposed. We have thus described various techniques ....

Q.-T. Luong, R. Deriche, O. Faugeras, and T. Papadopoulo, "On determining the fundamental matrix: analysis of different methods and experimental results," Report No. 1894, INRIA, 1993.

....same number of parameters as the degrees of freedom (the number of independent elements) of the relation. A relation can be over parametrized, and the parametrization can still be consistent e.g. representing a homography by a 3 Theta 3 matrix. Luong s parametrization of the fundamental matrix [9] by the epipoles (4 parameters) and epipolar homography (3 parameters) is both minimal and consistent. 3 What should be minimized When computing a multiple view relation from image correspondences there are often far more correspondences than DOF in the relation. This means that the points will ....

....the transfer error has often been used as the error function. The transfer distance is different from the orthogonal distance as shown in Figure 1 for the homography. In the case of the fundamental matrix the orthogonal distance (reprojection error) is different to the error minimized by Luong [9]. Hartley and Sturm [7] show that given F then d, x 1 and x 2 may be found as the solution of a degree 6 polynomial of one variable. A computationally efficient first order approximation based on Sampson s Taubin s method is given in [17] In this paper the distance is found as part of the ....

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Q. T. Luong, R. Deriche, O. D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: analysis of different methods and experimental results. Report 1894, INRIA (Sophia Antipolis), 1993.

....available ground information for recovering of metric structure. One of the main problems on the estimation is the fact that the fundamental matrix can be very sensitive to error on the point locations. Much effort has been taken on the study of the fundamental matrix over the last few years[10, 9, 4]. Several algorithms have been proposed which try to minimise the problems due to errors on the point locations and mismatches. Any automated stereo system which is not robust to this kind of errors is doomed to fail on real world applications. This paper reports a set of implementations for the ....

....and reconstruction. In projective coordinates, the epipolar constraint [4] can be written as : u 0T Fu = 0 (1) where fu; u 0 g are the projective coordinates of two corresponding points in the two images and F is a (3x3) matrix, defined up to a scale factor, known as the Fundamental matrix [9]. From Equation (1) F can be estimated linearly, given a minimum of 8 corresponding points between two images, without involving camera calibration. The fundamental matrix has rank 2, and since it is defined up to a scale factor, there are 7 independent parameters. The null space of F and its ....

[Article contains additional citation context not shown here]

Q. Luong, R. Deriche, O. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: an analysis of different methods and experimental results. Technical Report 1894, INRIA, 1993.

....sufficient for a linear solution, in practice one would use more than eight points for recovering the fundamental matrices in a linear or non linear squares method. Since linear least squares methods are still sensitive to image noise, we used the implementation of a non linear method described in [20] which was kindly provided by T. Luong and L. Quan (these were two implementations of the method proposed in [20] in each case, the implementation that provided the better results was adopted) The first experiment is with simulation data showing that even when the epipolar geometry is ....

....matrices in a linear or non linear squares method. Since linear least squares methods are still sensitive to image noise, we used the implementation of a non linear method described in [20] which was kindly provided by T. Luong and L. Quan (these were two implementations of the method proposed in [20] in each case, the implementation that provided the better results was adopted) The first experiment is with simulation data showing that even when the epipolar geometry is recovered accurately, it is still significantly better to use the trilinear result which avoids the process of line ....

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Q.T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. Technical Report INRIA, France, 1993.

....of the camera between frame k and k 1. Let A be the intrinsic matrix of the camera: A = 2 4 k u f 0 u 0 0 k v f v 0 0 0 1 3 5 : Let q 1 and q 2 be the images of a 3D point M on the cameras. Their homogeneous coordinates are denoted q 1 and q 2 . We then have the fundamental equation [15] q 2 t A 1 t T RA 1 q 1 = 0; where T is an antisymmetric matrix such that Tx = t x for all x. F = A 1 t T RA 1 is called the fundamental matrix. Then, a simple way to improve the viewpoint computation using the interest points is to minimize minRk 1 ;t k 1 1 n n X i=1 ....

.... distance in frame k 1 between the tracked features and the projection of the model features, v i = q 1 (F q i 1 ) 2 1 (F q i 1 ) 2 2 1 (F t q i 2 ) 2 1 (F t q i 2 ) 2 2 j q i 2 t F q i 1 j measures the quality of the matching between q i and q 0 i [15], 1 and 2 are M estimators. Note that the use of a M estimator for the key points correspondences is not essential: as the key points are signi cant points in the image, false matches are unusual. The parameter controls the compromise between the closeness to the available 3D data and ....

Q.-T. Luong, R. Deriche, O. Faugeras, and T. Papadopoulo. On Determining the Fundamental Matrix: Analysis of Different Methods and Experimental Results. Rapport de recherche 1894, INRIA, 1993.

....the projected motion of the line of intersection of two planes. Keywords: fundamental matrix, plane projectivity, epipolar geometry. 1. Introduction The fundamental matrix is a key concept in vision using uncalibrated imagery. It encodes the epipolar geometry associated with the camera motion [7, 3, 6]. This may be used for motion segmentation [12, 8, 11, 10] or as the basis for recovering projective structure [6, 1, 2] The projectivity of a moving plane also encodes information about epipolar geometry only less directly as it is also a function of scene structure [4, 9] The fundamental ....

.... [12, 8, 11, 10] or as the basis for recovering projective structure [6, 1, 2] The projectivity of a moving plane also encodes information about epipolar geometry only less directly as it is also a function of scene structure [4, 9] The fundamental matrix has proven complicated to estimate [7] because enforcing the det[F] 0 constraint means that a set of non linear equations have to be solved. If the set of points being viewed by the camera lie on a plane in 3D space then the equation used to estimate F is degenerate. The routines used to estimate F may be written to test the data ....

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Q.T. Luong, R. Deriche, O. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: analysis of different methods and experimental results. Technical Report RR-1894, INRIA (Sophia Antipolis), 1993.

....least squares cost function 1 [13] ae(u) X i c t i dc i dx i i d 2 ae i dx 2 i j Gamma1 dc i dx i t Gamma1 c i fi fi fi fi (x i ;u) This is simplest for problems with scalar constraints. e.g. for the uncalibrated epipolar constraint we get the well known form [10] ae(u) X i (x t i F x 0 i ) 2 x t i F Cov(x 0 i ) F t x i x 0 i t F t Cov(x i ) F x 0 i 1 If any of the covariance matrices is singular (which happens for redundant constraints or homogeneous data x i ) the matrix inverses can be replaced with pseudo inverses. 2.3 ....

....starting from some natural redundant representation, we must either come up with some inspired nonlinear change of variables which locally removes the redundancy, or algebraically eliminate variables by brute force using consistency or gauge fixing constraints. For example, Luong et al. [10] guarantee det(F ) 0 by writing each row of the fundamental matrix as a linear combination of the other two. Each parametrization fails when its two rows are linearly dependent, but the three of them suffice to cover the whole variety. In more complicated situations, intuition fails and we have ....

Q.-T. Luong, R. Deriche, O. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. Technical Report RR-1894, INRIA, Sophia Antipolis, France, 1993.

....be the 2D projective space P 2 both over the field C of complex numbers. Views are denoted by and 0 (we will be considering only two views at a time) The epipoles are denoted by v 2 and v 0 2 0 , and we assume their locations are known (for methods, see [5] 6] 31] 33] [15], for example, and briefly later in the text) The symbol = denotes equality up to scale, GLn stands for the group of n Theta n matrices, PGLn is the group defined up to scale, and SPGLn is the symmetric specialization of PGLn . The coordinization of object space and its views is based on ....

Q.T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. Technical Report INRIA, France, 1993.

....and experimental results 4.1 Implementation of the motion module Let us now discuss the implementation of a module which takes point or token correspondences as input, and output an estimation of the affine motion parameters. Thank s to several other developments in the field such as [20, 39, 32, 24, 37] we do not have to discuss again how to implement such a module in great details but simply can base our work on previous experiences. The main features to be taken into account are the following : ffl Point or token correspondences are always defined with a certain uncertainty, often represented ....

....represented by a covariance matrix, and estimation criteria must weight their estimates using this uncertainty. ffl It is always more robust and reliable to have a criterion based on a retinal measurement error (i.e. a retinal disparity or a image related quantity) even in the uncalibrated case [20], because this quantity corresponds to the physical measure. Obviously, when the retinal disparity has been canceled, all information about the motion has been extracted. ffl It is always possible and sometimes more efficient [39, 14] to compute motion and structure at the same time, instead ....

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Q. Luong, R. Deriche, O. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: analysis of different methods and experimental results. Technical Report RR-1894, INRIA, Sophia, France, 1993.

....goal of the computation is to find the matrix which approximates at best the solution of this system by least squares according to a given criterion. RR n2308 12 Cyril ZELLER Olivier FAUGERAS A study of the computation of the fundamental matrix from image point correspondences can be found in [8]. Here, we just mention our particular implementation, which consists, on the one hand, in a direct computation considering that all the correspondences are valid and in the other hand, in a method to reject some possible outliers among the correspondences. The direct computation computes F in ....

Q.-T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the Fundamental matrix: analysis of different methods and experimental results. Technical Report RR-1894, INRIA, 1993.

....find the matrix which best approximates the solution of this system according to a given least squares criterion. INRIA Applications of non metric vision to some visually guided robotics tasks 15 A study of the computation of the fundamental matrix from image point correspondences can be found in [20]. Here, we just mention our particular implementation, which consists, on the one hand, of a direct computation considering that all the correspondences are valid and in the other hand, of a method for rejecting some possible outliers among the correspondences. The direct computation computes F ....

Q.-T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the Fundamental matrix: analysis of different methods and experimental results. Technical Report RR-1894, INRIA, 1993.

....viewed by a stereo system [8, 21] This theory make use of epipolar geometry which can be retrieved easily from point correspondences in pair of images. Since these first attempts at an uncalibrated stereovision, a lot of work has been done on the estimation of the epipolar geometry of two images [29, 26, 32, 31, 30, 22, 20, 36, 4]. Robust programs which work automatically are now publicly available. We will consider this problem as solved for the rest of this article; the interested reader is referred to the bibliography. We will use the fundamental matrix representation of the epipolar geometry. In this representation, 2 ....

Q. T. Luong, R. Deriche, O. D. Faugeras, and T. Papadopoulo. On determining the Fundamental matrix: analysis of different methods and experimental results. In Israelian Conf. on Artificial Intelligence and Computer Vision, Tel-Aviv, Israel, December 1993. A longer version is INRIA Tech Report RR-1894.

....the quantity to be minimized, which is the sum of the squared distances of the point x to the cubics f i (x) and g i (x) Thus the criterion (6) favors the points in the image, whose residual values are lower for the same distances. This observation parallels the more quantitative observation of [12] for the estimation of epipoles though the fundamental matrix using the linear algebraic expression of the epipolar constraint. Table 6: Displacement 1 and 2: results of the algebraic intersection method. Note: For each noise level the first few local minima of the sum of the squares of cubics ....

....these epipoles with and those from F 0 . We measure the error by the relative distance for each coordinate of the epipole: minf jx Gamma x 0 j min(jxj; jx 0 j) 1g The results are compared with those of another method based on non linear minimization, the classical Fundamental matrix approach [20, 2, 12, 5], because one of the goals of the paper was to investigate how direct methods to determine the epipoles compare with indirect methods that first compute the Fundamental matrix. The Table 10 summarizes 100 trials. It shows the mean value of the relative distances obtained with the different ....

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Q.-T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the Fundamental matrix: analysis of different methods and experimental results. Technical Report RR-1894, INRIA, 1993.

....linear criterion is that it leads to a non iterative computation method, however, we have found that it is quite sensitive to noise, even with numerous data points. Let us point out to the two main drawbacks of the linear criterion. A more detailed analysis of the linear criterion is performed in [43, 42], where some analytical results and numerical examples are provided. The linear criterion cannot express the rank constraint Let l 0 be an epipolar line in the second image, computed from a fundamental matrix F that was obtained by the linear criterion, and from the point m = u; v; 1) T of ....

....2 , where d is a distance. In our case, the data points are the vectors x i = u i ; v i ; u 0 i ; v 0 i ) f is one of the 7 dimensional parameterizations introduced in the previous section, and g is given by (2) We have developed a method to perform the exact computation of this distance [42, 43], based on some special properties of the surface S, but this approach is computationally very expensive. The linear criterion can be considered as a generalization of the Bookstein distance [5] for conic fitting. The straightforward idea is to approximate the true distance of the point x to the ....

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Q.-T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the Fundamental matrix: analysis of different methods and experimental results. Technical Report RR-1894, INRIA, 1993. A shorter version appeared in the Israelian Conf. on Artificial Intelligence and Computer Vision.

....on the epipolar constraint Eq. 2) i.e. m T k F ij m l = 0 we can estimate the fundamental matrices F ll , F rr , and F rl between the left images, the right images, and the left and right images, respectively, from the given point correspondences. Several methods have been proposed in [14]. We have used a linear least squares method followed by a non quadratic minimization to improve the results. Consider now the case of the left camera, i.e. F ll . For the reason of simplicity, subscript will be omitted. From the definition of F, we have E = A T FA, where E = TR is called the ....

Q.-T. Luong, R. Deriche, O. Faugeras, and T. Papadopoulo, "On determining the fundamental matrix: Analysis of different methods and experimental results," Rapport de Recherche 1894, INRIA Sophia-Antipolis, France, 1993.

....or both of the epipoles and point correspondences, and thus the recovery of epipole location is an important problem. Although the methods which we study in this paper have been already briefly mentioned elsewhere [8, 2] the only way to compute the epipole which has been studied in great detail [1, 10, 2, 6, 4] so far is to first estimate the Fundamental matrix, and then to estimate the epipoles from this matrix. However, the computation of the Fundamental matrix might be quite unstable [7] Indeed, it has been shown [7] that the difficult part in the determination of the Fundamental matrix is the ....

....We have illustrated its behavior with two different choices. The first one is the exact epipole, the second is the linear indirect method to determine the fundamental matrix. The first initialization enables us to test the stability of the minimum, the second one (which is not too accurate [6]) lets us test the convergence properties of the algorithm. method noise 0.2 pixel noise 1.0 pixel exact linear F mat exact linear F mat X ratio 0.2108 0.2969 0.3446 0.5987 X ratio C 0.1404 0.2200 0.2744 0.5496 X ratio P 0.1429 0.2305 0.2651 0.5718 X ratio P C 0.1169 0.2158 0.2344 0.5395 ....

[Article contains additional citation context not shown here]

Q.-T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the Fundamental matrix: analysis of different methods and experimental results. Technical Report RR-1894, INRIA, 1993. A shorter version appeared in the Israelian Conf. on Artificial Intelligence and Computer Vision.

....that this fundamental matrix is the only information which can be obtained from image correspondences alone. Let us review some approaches which can be used for the recovery. The problem of the robust determination of F from point correspondences (m i ; m 0 i ) has been already considered in [33, 61, 10]. The basic idea is to use the relation (9) which is linear in the entries of F. Another approach in the case of two views [35] is to first compute a homography, and then to use the relation between fundamental matrices and homographies presented in Sec. 3.3. If we have a minimum of three views, ....

Q.-T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the Fundamental matrix: analysis of different methods and experimental results. Technical Report RR-1894, INRIA, 1993.

....matrix into a rotation R and the translation t such that: E = 0 Gammat 3 t2 t3 0 Gammat 1 Gammat 2 t1 0 # z [t] Theta R is classical [11, 20, 8] Iterative minimization. The fundamental matrix is recomputed by a non linear minimization technique such as those presented in [13, 3], the difference being that the knowledge of the intrinsic parameters allows us to minimize these criteria with respect to five motion parameters, instead of the seven parameters of the epipolar transformation. We parameterize T by t1 =t3 , t2 =t3 and R by the 3D vector r whose direction is that ....

Q.-T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the Fundamental matrix: analysis of different methods and experimental results. Technical Report RR-1894, INRIA, 1993.

....This information is entirely contained in the Fundamental matrix, thus it is very important to develop precise techniques to compute it. The problem of computing the Fundamental matrix from point correspondences 1 , that are in general position has been studied recently [6] 14] 16] 2] [10]. However it is known, in the framework of motion analysis that when the points lie in a plane, the general methods will fail, and that some specific methods can be used [19] 9] 3] We show that the situation is similar for the computation of the Fundamental matrix. In the first part, after ....

....line in the second image is given by l 0 q = Fq Since the point q 0 corresponding to q belongs to the line l 0 q by definition, it follows that q 0 T Fq = 0 (1) Equation (1) is a generalization of the so called Longuet Higgins equation in motion analysis. We have studied extensively in [10] the problem of the computation of F from point correspondences using (1) This matrix, defined up to a scale factor, must be of rank two. Different parametrizations for this matrix have been proposed to take into account these important constraints and linear and non linear criteria for its ....

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Q.-T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the Fundamental matrix: analysis of different methods and experimental results. Technical Report RR, INRIA, 1993.

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Q.T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. INRIA, 1993.

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Q.T. Luong, R. Deriche, O.D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: Analysis of different methods and experimental results. INRIA, 1993.

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Q.-T. Luong, R. Deriche, O. Faugeras, and T. Papadopoulo, "On determining the fundamental matrix: analysis of different methods and experimental results." in Artificial Intelligence Journal, vol. 78, October 1995, pp. 87--119.

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Q. T. Luong, R. Deriche, O. D. Faugeras, and T. Papadopoulo. On determining the fundamental matrix: analysis of different methods and experimental results. Technical Report 1894, INRIA (Sophia Antipolis), 1993.

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Q.-T. Luong, R. Deriche, O. D. Faugeras, and T. Papadopoulo, "On determining the fundamental matrix: analysis of different methods and experimental results," Tech. Rep. 1894, Institut National de Recherche en Informatique et en Automatique, Sophia Antipolis, France, Apr. 1993.

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Q.-T. Luong, R. Deriche, O. D. Faugeras, and T. Papadopoulo, "On determining the fundamental matrix: analysis of different methods and experimental results," Technical Report No. 1894 (INRIA, Sophia Antipolis, France, 1993).

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