A clever elimination strategy for efficient minimal solvers

We present a new insight into the systematic generation of minimal solvers in computer vision, which leads to smaller and faster solvers. Many minimal problem formulations are coupled sets of linear and polynomial equations where image measurements enter the linear equations only. We show that it is useful to solve such systems by first eliminating all the unknowns that do not appear in the linear equations and then extending solutions to the rest of unknowns. This can be generalized to fully non-linear systems by linearization via lifting. We demonstrate that this approach leads to more efficient solvers in three problems of partially calibrated relative camera pose computation with unknown focal length and/or radial distortion. Our approach also generates new interesting constraints on the fundamental matrices of partially calibrated cameras, which were not known before.Comment: 13 pages, 7 figure

Radially-Distorted Conjugate Translations

This paper introduces the first minimal solvers that jointly solve for affine-rectification and radial lens distortion from coplanar repeated patterns. Even with imagery from moderately distorted lenses, plane rectification using the pinhole camera model is inaccurate or invalid. The proposed solvers incorporate lens distortion into the camera model and extend accurate rectification to wide-angle imagery, which is now common from consumer cameras. The solvers are derived from constraints induced by the conjugate translations of an imaged scene plane, which are integrated with the division model for radial lens distortion. The hidden-variable trick with ideal saturation is used to reformulate the constraints so that the solvers generated by the Grobner-basis method are stable, small and fast. Rectification and lens distortion are recovered from either one conjugately translated affine-covariant feature or two independently translated similarity-covariant features. The proposed solvers are used in a \RANSAC-based estimator, which gives accurate rectifications after few iterations. The proposed solvers are evaluated against the state-of-the-art and demonstrate significantly better rectifications on noisy measurements. Qualitative results on diverse imagery demonstrate high-accuracy undistortions and rectifications. The source code is publicly available at https://github.com/prittjam/repeats

A sparse resultant based method for efficient minimal solvers

Many computer vision applications require robust and efficient estimation of camera geometry. The robust estimation is usually based on solving camera geometry problems from a minimal number of input data measurements, i.e. solving minimal problems in a RANSAC framework. Minimal problems often result in complex systems of polynomial equations. Many state-of-the-art efficient polynomial solvers to these problems are based on Gr\"obner bases and the action-matrix method that has been automatized and highly optimized in recent years. In this paper we study an alternative algebraic method for solving systems of polynomial equations, i.e., the sparse resultant-based method and propose a novel approach to convert the resultant constraint to an eigenvalue problem. This technique can significantly improve the efficiency and stability of existing resultant-based solvers. We applied our new resultant-based method to a large variety of computer vision problems and show that for most of the considered problems, the new method leads to solvers that are the same size as the the best available Gr\"obner basis solvers and of similar accuracy. For some problems the new sparse-resultant based method leads to even smaller and more stable solvers than the state-of-the-art Gr\"obner basis solvers. Our new method can be fully automatized and incorporated into existing tools for automatic generation of efficient polynomial solvers and as such it represents a competitive alternative to popular Gr\"obner basis methods for minimal problems in computer vision

Sparse resultant based minimal solvers in computer vision and their connection with the action matrix

Many computer vision applications require robust and efficient estimation of camera geometry from a minimal number of input data measurements, i.e., solving minimal problems in a RANSAC framework. Minimal problems are usually formulated as complex systems of sparse polynomials. The systems usually are overdetermined and consist of polynomials with algebraically constrained coefficients. Most state-of-the-art efficient polynomial solvers are based on the action matrix method that has been automated and highly optimized in recent years. On the other hand, the alternative theory of sparse resultants and Newton polytopes has been less successful for generating efficient solvers, primarily because the polytopes do not respect the constraints on the coefficients. Therefore, in this paper, we propose a simple iterative scheme to test various subsets of the Newton polytopes and search for the most efficient solver. Moreover, we propose to use an extra polynomial with a special form to further improve the solver efficiency via a Schur complement computation. We show that for some camera geometry problems our extra polynomial-based method leads to smaller and more stable solvers than the state-of-the-art Grobner basis-based solvers. The proposed method can be fully automated and incorporated into existing tools for automatic generation of efficient polynomial solvers. It provides a competitive alternative to popular Grobner basis-based methods for minimal problems in computer vision. We also study the conditions under which the minimal solvers generated by the state-of-the-art action matrix-based methods and the proposed extra polynomial resultant-based method, are equivalent. Specifically we consider a step-by-step comparison between the approaches based on the action matrix and the sparse resultant, followed by a set of substitutions, which would lead to equivalent minimal solvers.Comment: arXiv admin note: text overlap with arXiv:1912.1026

MeshLoc: Mesh-Based Visual Localization

Visual localization, i.e., the problem of camera pose estimation, is a central component of applications such as autonomous robots and augmented reality systems. A dominant approach in the literature, shown to scale to large scenes and to handle complex illumination and seasonal changes, is based on local features extracted from images. The scene representation is a sparse Structure-from-Motion point cloud that is tied to a specific local feature. Switching to another feature type requires an expensive feature matching step between the database images used to construct the point cloud. In this work, we thus explore a more flexible alternative based on dense 3D meshes that does not require features matching between database images to build the scene representation. We show that this approach can achieve state-of-the-art results. We further show that surprisingly competitive results can be obtained when extracting features on renderings of these meshes, without any neural rendering stage, and even when rendering raw scene geometry without color or texture. Our results show that dense 3D model-based representations are a promising alternative to existing representations and point to interesting and challenging directions for future research.Comment: to be published in the proceedings of ECCV 2022, code repository: https://github.com/tsattler/meshloc_releas

Efficient solutions to the relative pose of three calibrated cameras from four points using virtual correspondences

We study the challenging problem of estimating the relative pose of three calibrated cameras. We propose two novel solutions to the notoriously difficult configuration of four points in three views, known as the 4p3v problem. Our solutions are based on the simple idea of generating one additional virtual point correspondence in two views by using the information from the locations of the four input correspondences in the three views. For the first solver, we train a network to predict this point correspondence. The second solver uses a much simpler and more efficient strategy based on the mean points of three corresponding input points. The new solvers are efficient and easy to implement since they are based on the existing efficient minimal solvers, i.e., the well-known 5-point relative pose and the P3P solvers. The solvers achieve state-of-the-art results on real data. The idea of solving minimal problems using virtual correspondences is general and can be applied to other problems, e.g., the 5-point relative pose problem. In this way, minimal problems can be solved using simpler non-minimal solvers or even using sub-minimal samples inside RANSAC. In addition, we compare different variants of 4p3v solvers with the baseline solver for the minimal configuration consisting of three triplets of points and two points visible in two views. We discuss which configuration of points is potentially the most practical in real applications

A minimal solution to the autocalibration of radial distortion

Epipolar geometry and relative camera pose computation are examples of tasks which can be formulated as minimal problems and solved from a minimal number of image points. Finding the solution leads to solving systems of algebraic equations. Often, these systems are not trivial and therefore special algorithms have to be designed to achieve numerical robustness and computational efficiency. In this paper we provide a solution to the problem of estimating radial distortion and epipolar geometry from eight correspondences in two images. Unlike previous algorithms, which were able to solve the problem from nine correspondences only, we enforce the determinant of the fundamental matrix be zero. This leads to a system of eight quadratic and one cubic equation in nine variables. We simplify the system by eliminating six of these variables. Then, we solve the system by finding eigenvectors of an action matrix of a suitably chosen polynomial. We show how to construct the action matrix without computing complete Gröbner basis, which provides an efficient and robust solver. The quality of the solver is demonstrated on synthetic and real data. 1