This paper presents a solution to the problem of pose estimation in the presence of heavy radial distortion and a potentially large number of outliers. The main contribution is an algorithm that solves for radial distortion, focal length and camera pose using a minimal set of four point correspondences between 3D world points and image points. We use a RANSAC loop to find a set of inliers and an initial estimate for bundle adjustment. Unlike previous approaches where one starts out by assuming a linear projection model, our minimal solver allows us to handle large radial distortions already at the RANSAC stage. We demonstrate that with the inclusion of radial distortion in an early stage of the process, a broader variety of cameras can be handled than was previously possible. In the experiments, no calibration whatsoever is applied to the camera. Instead we assume square pixels, zero skew and centered principal point. Although these assumptions are not strictly true, we show that good results are still obtained and by that conclude that the proposed method is applicable to uncalibrated photographs.

Correspondence problems are of great importance in computer vision. They appear as subtasks in many applications such as object recognition, merging partial 3D reconstructions and image alignment. Automatically matching features from appearance only is difficult and errors are frequent. Thus, it is necessary to use geometric consistency to remove incorrect correspondences. Typically heuristic methods like RANSAC or EM-like algorithms are used, but they risk getting trapped in local optima and are in no way guaranteed to find the best solution. This paper illustrates how pairwise constraints in combination with graph methods can be used to efficiently find optimal correspondences. These ideas are implemented on two basic geometric problems, 3D-3D registration and 2D-3D registration. The developed scheme can handle large rates of outliers and cope with multiple hypotheses. Despite the combinatorial explosion, the resulting algorithm which has been extensively evaluated on real data, yields competitive running times compared to state of the art.

A number of minimal problems of structure from motion for cameras with radial distortion have recently been studied and solved in some cases. These problems are known to be numerically very challenging and in several cases there exist no known practical algorithm yielding solutions in floating point arithmetic. We make some crucial observations concerning the floating point implementation of Gröbner basis computations and use these new insights to formulate fast and stable algorithms for two minimal problems with radial distortion previously solved in exact rational arithmetic only: (i) simultaneous estimation of essential matrix and a common radial distortion parameter for two partially calibrated views and six image point correspondences and (ii) estimation of fundamental matrix and two different radial distortion parameters for two uncalibrated views and nine image point correspondences. We demonstrate on simulated and real experiments that these two problems can be efficiently solved in floating point arithmetic.

Where am I and what am I seeing? This is a classical vision problem and this paper presents a solution based on efficient use of a combination of 2D and 3D features. Given a model of a scene, the objective is to find the relative camera location of a new input image. Unlike traditional hypothesize-and-test methods that try to estimate the unknown camera position based on 3D model features only, or alternatively, based on 2D model features only, we show that using a mixture of such features, that is, a hybrid correspondence set, may improve performance. We use minimal cases of structure-from-motion for hypothesis generation in a RANSAC engine. For this purpose, several new and useful minimal cases are derived for calibrated, semi-calibrated and uncalibrated settings. Based on algebraic geometry methods, we show how these minimal hybrid cases can be solved efficiently. The whole approach has been validated on both synthetic and real data, and we demonstrate improvements compared to previous work.

This paper presents techniques for improving the numerical stability of Grobner basis solvers for polynomial equations. Recently Grobner basis methods have been used successfully to solve polynomial equations arising in global optimization e.g. three view triangulation and in many important minimal cases of structure from motion. Such methods work extremely well for problems of reasonably low degree, involving a few variables. Currently, the limiting factor in using these methods for larger and more demanding problems is numerical difficulties. In the paper we (i) show how to change basis in the quotient space R[x]/I and propose a strategy for selecting a basis which improves the conditioning of a crucial elimination step, (ii) use this technique to devise a Grobner basis with improved precision and (iii) show how solving for the eigenvalues instead of eigenvectors can be used to improve precision further while retaining the same speed. We study these methods on some of the latest reported uses of Grobner basis methods and demonstrate dramatically improved numerical precision using these new techniques making it possible to solve a larger class of problems than previously.