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 ....

[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.

....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.

....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 ....

[Article contains additional citation context not shown here]

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.

No context found.

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.

No context found.

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.

No context found.

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.

No context found.

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.

No context found.

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.

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