For decades, it has been widely accepted that the gold standard for two-view triangulation is to minimize the cost based on reprojection errors. In this work, we challenge this idea. We propose a novel alternative to the classic midpoint method that leads to significantly lower 2D errors and parallax errors. It provides a numerically stable closed-form solution based solely on a pair of backprojected rays. Since our solution is rotationally invariant, it can also be applied for fisheye and omnidirectional cameras. We show that for small parallax angles, our method outperforms the state-of-the-art in terms of combined 2D, 3D and parallax accuracy, while achieving comparable speed.
refer to those triangulation methods that minimize the cost based on reprojection errors. Assuming that the image measurements are independently perturbed by the noise in the same distribution of certain types, the optimal methods find the maximum likelihood (ML) solution. For Gaussian and Laplacian distribution, the ML solution is to minimize the @math norm or @math norm of the reprojection errors, respectively @cite_9 . This can be found in closed form by solving a polynomial of degree six or eight @cite_23 . Alternatively, the @math solution can be obtained using iterative correction methods @cite_27 @cite_35 . While these iterative methods do not guarantee global optimality, they were shown to be faster and more stable. For a uniform distribution, minimizing the @math norm leads to the ML estimate for the lower bound of the noise @cite_34 , and the solution is obtained by solving a quartic polynomial @cite_1 . Unlike the @math and @math cost, the @math cost has a simple shape with a single minimum, but it is relatively more sensitive to noise and outliers @cite_16 .
@cite_35 @cite_9 @cite_1 @cite_16 @cite_27 @cite_23 @cite_34
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The problem of estimating the phases (angles) of a complex unit-modulus vector @math from their noisy pairwise relative measurements @math , where @math is a complex-valued Gaussian random matrix, is known as phase synchronization. The maximum likelihood estimator (MLE) is a solution to a unit--modulus-constrained quadratic programming problem, which is nonconvex. Existing works have proposed polynomial-time algorithms such as a semidefinite programming (SDP) relaxation or the generalized power method (GPM). Numerical experiments suggest that both of these methods succeed with high probability for @math up to @math , yet existing analyses only confirm this observation for @math up to @math . In this paper, we bridge the gap by proving that the SDP relaxation is tight for @math , and GPM converges to the global optimum under the same regime. Moreover, we establish a linear convergence rate for GPM, and derive a tight... We address the problem of camera motion and 3D structure reconstruction from line correspondences across multiple views, from initialization to final bundle adjustment. One of the main difficulties when dealing with line features is their algebraic representation. First, we consider the triangulation problem. Based on Plucker coordinates to represent the 3D lines, we propose a maximum likelihood algorithm, relying on linearizing the Plucker constraint and on a Plucker correction procedure, computing the closest Plucker coordinates to a given 6-vector. Second, we consider the bundle adjustment problem, which is essentially a nonlinear optimization process on camera motion and 3D line parameters. Previous overparameterizations of 3D lines induce gauge freedoms and or internal consistency constraints. We propose the orthonormal representation, which allows handy nonlinear optimization of 3D lines using the minimum four parameters with an unconstrained optimization engine. We compare our algorithms to existing ones on simulated and real data. Results show that our triangulation algorithm outperforms standard linear and bias-corrected quasi-linear algorithms, and that bundle adjustment using our orthonormal representation yields results similar to the standard maximum likelihood trifocal tensor algorithm, while being usable for any number of views. We propose and analyze a discontinuous Galerkin approximation for the Stokes problem. The finite element triangulation employed is not required to be conforming and we use discontinuous pressures and velocities. No additional unknown fields need to be introduced, but only suitable bilinear forms defined on the interfaces between the elements, involving the jumps of the velocity and the average of the pressure. We consider hp approximations using ℚk′–ℚk velocity-pressure pairs with k′ = k + 2, k + 1, k. Our methods show better stability properties than the corresponding conforming ones. We prove that our first two choices of velocity spaces ensure uniform divergence stability with respect to the mesh size h. Numerical results show that they are uniformly stable with respect to the local polynomial degree k, a property that has no analog in the conforming case. An explicit bound in k which is not sharp is also proven. Numerical results show that if equal order approximation is chosen for the velocity and pressure, no spurious pressure modes are present but the method is not uniformly stable either with respect to h or k. We derive a priori error estimates generalizing the abstract theory of mixed methods. Optimal error estimates in h are proven. As for discontinuous Galerkin methods for scalar diffusive problems, half of the power of k is lost for p and hp pproximations independently of the divergence stability. In this work, we present a Direct Least-Squares (DLS) method for computing all solutions of the perspective-n-point camera pose determination (PnP) problem in the general case (n ≥ 3). Specifically, based on the camera measurement equations, we formulate a nonlinear least-squares cost function whose optimality conditions constitute a system of three third-order polynomials. Subsequently, we employ the multiplication matrix to determine all the roots of the system analytically, and hence all minima of the LS, without requiring iterations or an initial guess of the parameters. A key advantage of our method is scalability, since the order of the polynomial system that we solve is independent of the number of points. We compare the performance of our algorithm with the leading PnP approaches, both in simulation and experimentally, and demonstrate that DLS consistently achieves accuracy close to the Maximum-Likelihood Estimator (MLE).