B. Triggs, Optimal Estimation of Matching Constraints, Proceedings of European Workshop on 3D Structure from Multiple Images of Large-scale Environments, June 1998. 6  Home/Search   Document Details and Download   Summary   Related Articles   Check

....considered constraints in the context of surface fitting and geometric modelling; few have explicitly done so in the context of reverse engineering. We review their di#erent approaches below. However, we first note that a generally useful framework for geometric constraints is provided by Triggs [21]; although he uses it for the solution of problems in computer vision, he notes that it is of much wider applicability. 2.1 Constraints in surface fitting A wide range of work considers constraints in the context of fitting a single (possibly piecewise) free form surface. Typical constraints ....

B. Triggs, "Optimal estimation of matching constraints", In: 3D Structure from Multiple Images of Large-scale Environments (SMILE 98), Eds: R. Koch and L. Van Gool, Lecture Notes in Computer Science, Springer-Verlag, 1998 30

....According to the previous section, quasi linear estimators perform well under certain constraints and from various points of view. The question is now, how can we devise a good plane homography estimator based on the quasi linear principle. Such an estimator follows the reduced approach [17] (i.e. minimizes a geometric residual in both images) and incorporates the external global constraint of compatibility with a given epipolar geometry, reducing the number of unknowns from 8 to 3 for each homography. Our constraints to build a good estimator are then: minimize a symmetric ....

....normalization: the algorithm does not yield gross errors, it is slightly less accurate than a non linear estimation and so is appropriate to choose a model for the fundamental matrix. Once the parameterization has been chosen, one can use two kinds of criteria: the reduced and the direct approach [17]. Let us investigate first the reduced approach criterion. We are only looking for y, the parameters of the epipolar geometry which optimize: F argmin (d2(x , Fx) d2(x, FTx ) 19) where the previously chosen parameterization is enforced via the constraint F f(v) and where d(x, 1) is the ....

B. Triggs. Optimal estimation of matching constraints. In 3D Structure from Multiple Images of Large-scale Environments SMILE'98. Springer Verlag, 1998.

....in the expression for the number of essential degrees of freedom of the scene, we have to take into account 15 degrees of gauge freedom left by the arbitrary choice for the projective basis of the reconstruction. Gauge freedom is defined as the internal freedom of choice for a coordinate system [43]. It can be fixed using a particular formulation for the structure [17] or for the camera matrices [6] Due to the complexity of structure parameterization, we have chosen to absorb the gauge freedom into the parameterization of motion. In the next two sections, we describe respectively our ....

B. Triggs. Optimal estimation of matching constraints. In 3D Structure from Multiple Images of Large-scale Environments' SMILE'98. Springer Verlag, June 1998.

....are then given from plane homographies using equation 1. However, an estimated homography does not correspond to a world plane in general. We show how to constrain the estimation so that the estimated homography really corresponds to a world plane. We have chosen to follow the reduced approach [13] which consists in minimizing a geometric residual in both images. This criterion is known to give stable results and involves only the parameters of the model (here a plane homography) into the optimization whereas the direct approach [13] also involves the data. The drawback of such a criterion ....

....plane. We have chosen to follow the reduced approach [13] which consists in minimizing a geometric residual in both images. This criterion is known to give stable results and involves only the parameters of the model (here a plane homography) into the optimization whereas the direct approach [13] also involves the data. The drawback of such a criterion is that it is non linear due to the use of the Eu clidean distance and the need to compute the inverse homography. We show that it can be quasi linearly optimized. Let us start from the inconsistent criterion corresponding to any 2D ....

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B. Triggs. Optimal estimation of matching constraints. InSMILE'98, June 1998.

.... in terms of a set of actual image data points that are thought to be well matched [43] For 4 or more views, the exponentially many constraints on multilinear tensors have been derived algebraically, but they are redundant and nonlinear, and methods for enforcing them are not yet well established [47]. 2.6 Projective Structure and Motion The geometry of multiple views has been modeled as a collection of epipolar relations by several others. Laveau, et al. use collections of fundamental matrices to implicitly represent 3D surfaces via novel view prediction [24] Rothwell, et al. demonstrate ....

B. Triggs. Optimal Estimation of Matching Constraints. In R. Koch and L. Van Gool, editors, European Workshop on 3D Structure from Multiple Images of Largescale Environments, SMILE '98. Springer Verlag LNCS, 1998.

....problems in computer vision may be couched in terms of parameter estimation. Accordingly, much effort has gone into the development of sophisticated techniques for generating estimates of parameters. Some of these techniques utilise covariance information characterising uncertainty in the data [1, 2]. This paper is concerned with the application of a recently introduced covariance based method [3] to the problem of estimating the fundamental matrix (see also [1, 4 10] However, we assume here that, as is often the case, covariance information is unavailable. A 3D point in a scene ....

B. Triggs, "Optimal estimation of matching constraints," in 3D Structure from Multiple Images of Large-Scale Environments, European Workshop, Freiburg, Germany. 1998, vol. 1506 of Lecture Notes in Computer Science, pp. 63--77, Springer-Verlag, Berlin.

....coefficients [3,17] and in many other areas. Because such problems are typically very sensitive to noise, there has recently been considerable interest in assessing how parameter estimation might be improved if additional information is available characterising the uncertainty of the data [1, 4, 10, 11, 16]. This uncertainty information is usually expressed in terms of covariance matrices. This paper investigates conditions under which the use of covariance matrices enables parameter estimates of improved quality to be obtained. Several novel experiments are carried out under carefully controlled ....

B. Triggs. Optimal estimation of matching constraints. In 3D Structure from Multiple Images of Large-Scale Environments, European Workshop, Freiburg, Germany, volume 1506 of Lecture Notes in Computer Science, pages 63--77, 1998.

....such as non holonomic links on a mobile robot [10] involve implicit constraints on parameter positions or velocities. On the other hand, even if an explicit parameterization of a geometrical object (curve, surface, etc. is available, it is always simpler to dene it from implicit equations [16, 14, 8, 5]. It is another deliberate choice to have these manifolds embedded in a nite dimensional Euclidean space. In fact, dioeerential manifolds can be intrinsically built from their parameterization, without any reference to an arbitrary Euclidean space. However as defended for instance by Demazure ....

B. Triggs. Optimal estimation of matching constraints. In R. Koch and L. Van Gool, editors, Workshop on 3D Structure from Multiple Images of Large-scale Environments SMILE'98, Lecture Notes in Computer Science, 1998.

....expresses in a differential form the fact that b ML is a minimiser of a constrained problem. This system can be solved numerically by employing, say, the NewtonRaphson method. Finding b ML (and hence b ML ) in this way constitutes the so called direct approach to maximum likelihood fitting [15 17]. In contrast, finding b ML (and hence b ML ) by resorting to J # ML embodies the so called reduced approach [17] C. Approximated Maximum Likelihood Estimator To obtain an approximation of J # ML , first note that the definition of involves minimisation subject to the constraints ....

....numerically by employing, say, the NewtonRaphson method. Finding b ML (and hence b ML ) in this way constitutes the so called direct approach to maximum likelihood fitting [15 17] In contrast, finding b ML (and hence b ML ) by resorting to J # ML embodies the so called reduced approach [17]. C. Approximated Maximum Likelihood Estimator To obtain an approximation of J # ML , first note that the definition of involves minimisation subject to the constraints T u(x i ) 0 (i = 1; n) This minimisation can effectively be handled with the use of Lagrange multipliers. ....

B. Triggs, "Optimal estimation of matching constraints," in 3D Structure from Multiple Images of Large-Scale Environments (SMILE), ECCV'98 workshop organised by the CUMULI, PANORAMA and VANGUARD projects, R. Koch and L. Van Gool, Eds., Freiburg, Germany, June 6--7, 1998, 1998, Available from http://www.inrialpes.fr/movi/people/Triggs.

....and discard. In 3D reconstruction the 3D scene coordinate frame can be identified with the 3D camera coordinate frame at the first image, which specifies the first camera position as the canonical frame for reconstruction [27] 2 However this procedure sometimes introduces singularities [52], and there is also the issue of independence of the algorithm to different choices of coordinate frame and data ordering, which are usually violated by direct elimination. We argue in [30] that in 3D vision the best strategy is usually to leave the gauge freedoms in the representation, and deal ....

....because the vectors and matrices required (e.g. the Jacobian matrices) are smaller. Coordinate Frame Independence Where there are arbitrary choices of coordinate frame, ideally the particular choice made should not affect the result of subsequent computations. This point has been made in [5, 52]. Given that we are employing the statistical tool of affine least squares in the VSDF, the best that can be hoped for is independence to global affine transformations of coordinates. We have shown in other work [30] that the Euclidean structure from motion method described in this paper possesses ....

B. Triggs. Optimal estimation of matching constraints. In Proc. SMILE Workshop (associated with ECCV'98), 1998.

....Q C 1 according to Theorem 3. However, this enforcement is not necessary at each step of the iterations. We can alternatively repeat the above update until the solution converges and then transform the solution so that it satisfies the gauge C. We call this approach, first discussed by Triggs [9], the free gauge approach. 15. NORMAL FORM So far, the covariance matrix of an estimators of is defined for a particular gauge. However, it can be defined independently of gauges. Let 2 T be an arbitrary true value, and its estimator under an arbitrary gauge. Since and may have ....

B. Triggs, Optimal estimation of matching constraint, in R. Koch and L. Van Gool (eds.), 3D Structure from Multiple Images of Large-Scale Environments, Lecture Notes in Computer Science, 1506, Springer, 1998, pp. 63--77.

....For instance, one could apply a factorization algorithm to a subset of features with complete data, and use the motion parameters thus computed to compute the rest of the structure. In this way it would be easy to develop a coordinate system independent starting point for optimization. Triggs [20] discusses solutions to the problem of gauge freedoms in the optimization, which occur frequently in vision. He describes the main two approaches, gauge fixing conditions, which break the degeneracy by the introduction of artificial constraints, such as imposing a canonical frame (see below) and ....

....reduction of the error) exit, else return to step 1. 8 Results Details of the projective bundle adjustment algorithms implemented in other work is sketchy. We have designed two alternative approaches with features similar to [4] and [3] and which fit into the classification scheme of Triggs [20] as follows: Elimination: in this method we enforce a canonical frame P (j) set to (I 3 Theta3 j0) and fix other elements of the projection matrices and structure vectors so as to reduce the parameters to a minimal representation. The canonical frame is enforced in [4] Free Gauge: here we ....

B. Triggs. Optimal estimation of matching constraints. In Proc. SMILE Workshop (associated with ECCV'98), 1998.

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B. Triggs, Optimal Estimation of Matching Constraints, Proceedings of European Workshop on 3D Structure from Multiple Images of Large-scale Environments, June 1998. 6

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B. Triggs. Optimal estimation of matching constraints. In Proc. SMILE Workshop (associated with ECCV'98), 1998.

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B. Triggs. Optimal estimation of matching constraints. In 3D Structure from Multiple Images of Large-scale Environments SMILE'98. Springer Verlag, June 1998.

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