The paper introduces an efficient algebraic solver for the 4-sender/6-receiver (4s/6r) Time-of-Arrival (ToA) self-localization problem, which involves determining the relative positions of all receivers and senders given their pairwise distance measurements. The problem is addressed through a new parametrization combining Cayley--Menger determinants with an implicitization technique. The proposed algorithm proceeds in three steps. First, a 148 x 211 Macaulay matrix is constructed from the coefficients of the original polynomial system. Second, PLU decomposition of this matrix yields a 63 x 63 matrix pair. Finally, up to 38 real solutions are obtained via generalized eigendecomposition followed by a validation step. Experiments on synthetic noise-free data demonstrate that the proposed solver outperforms existing methods by approximately three orders of magnitude in numerical accuracy while achieving an average runtime of 1.3x faster than the fastest alternative. Experiments on a real-world acoustic dataset confirm that, when integrated within a RANSAC framework, the solver provides a reliable initial guess for bundle adjustment refinement.
Estimating camera geometry typically involves solving minimal problems formulated as systems of multivariate polynomial equations, which often pose computational challenges when using existing Gröbner-basis or resultant-based methods due to matrix inversion needed in the online solver. Here we propose a sampling-based, matrix inversion-free method that constructs the solvers using sparse hidden-variable resultants. The determinant polynomial in the hidden variable is efficiently reconstructed via inverse fast Fourier transform interpolation from sampled evaluations, avoiding symbolic expansion. Solving this polynomial yields the hidden variable, and the remaining unknowns are recovered by identifying rank-1 deficient submatrices and applying Cramer's rule. A greatest common divisor-based criterion ensures robust submatrix identification under noise. Experiments on diverse minimal problems demonstrate that the proposed solver achieves strong numerical stability and competitive runtime, particularly for small-scale problems, providing a practical alternative to traditional Gröbner-basis and resultant-based solvers.
We consider the problems of computing the optimal rank-1 Hankel and Toeplitz-structured approximation of arbitrary matrices under L2 and L1-norm error. Such problems arise naturally in engineered systems, including the basic few-shot signal Direction-of-Arrival (DoA) estimation problem that is of importance to modern autonomous systems applications. We develop accurate and computationally efficient structured matrix decomposition algorithms for both formulations and then derive analytically grounded small-sample-support DoA estimators for practical sensing system deployments. The resulting estimators under the L2 and L1 norms are formally shown to be maximum-likelihood optimal under white Gaussian and Laplace noise, respectively. The estimators are further validated through extensive simulation studies and real-world data experiments in few-shot DoA inference.
Solving non-linear least-squares problem for pose estimation (rotation and translation) is often a time consuming yet fundamental problem in several real-time computer vision applications. With an adequate rotation parametrization, the optimization problem can be reduced to the solution of a~system of polynomial equations and solved in closed form. Recent advances in efficient closed form solvers utilizing resultant matrices have shown a promising research direction to decrease the computation time while preserving the estimation accuracy. In this paper, we propose a new class of resultant-based solvers that exploit Sylvester forms to further reduce the complexity of the resolution. We demonstrate that our proposed methods are numerically as accurate as the state-of-the-art solvers, and outperform them in terms of computational time. We show that this approach can be applied for pose estimation in two different types of problems: estimating a pose from 3D to 3D correspondences, and estimating a pose from 3D points to 2D points correspondences.