Robust Estimation of Motion Parameters and Scene Geometry : Minimal Solvers and Convexification of Regularisers for Low-Rank Approximation

In the dawning age of autonomous driving, accurate and robust tracking of vehicles is a quintessential part. This is inextricably linked with the problem of Simultaneous Localisation and Mapping (SLAM), in which one tries to determine the position of a vehicle relative to its surroundings without prior knowledge of them. The more you know about the object you wish to track—through sensors or mechanical construction—the more likely you are to get good positioning estimates. In the first part of this thesis, we explore new ways of improving positioning for vehicles travelling on a planar surface. This is done in several different ways: first, we generalise the work done for monocular vision to include two cameras, we propose ways of speeding up the estimation time with polynomial solvers, and we develop an auto-calibration method to cope with radially distorted images, without enforcing pre-calibration procedures.We continue to investigate the case of constrained motion—this time using auxiliary data from inertial measurement units (IMUs) to improve positioning of unmanned aerial vehicles (UAVs). The proposed methods improve the state-of-the-art for partially calibrated cases (with unknown focal length) for indoor navigation. Furthermore, we propose the first-ever real-time compatible minimal solver for simultaneous estimation of radial distortion profile, focal length, and motion parameters while utilising the IMU data.In the third and final part of this thesis, we develop a bilinear framework for low-rank regularisation, with global optimality guarantees under certain conditions. We also show equivalence between the linear and the bilinear framework, in the sense that the objectives are equal. This enables users of alternating direction method of multipliers (ADMM)—or other subgradient or splitting methods—to transition to the new framework, while being able to enjoy the benefits of second order methods. Furthermore, we propose a novel regulariser fusing two popular methods. This way we are able to combine the best of two worlds by encouraging bias reduction while enforcing low-rank solutions

Homography-Based Positioning and Planar Motion Recovery

Planar motion is an important and frequently occurring situation in mobile robotics applications. This thesis concerns estimation of ego-motion and pose of a single downwards oriented camera under the assumptions of planar motion and known internal camera parameters. The so called essential matrix (or its uncalibrated counterpart, the fundamental matrix) is frequently used in computer vision applications to compute a reconstruction in 3D of the camera locations and the observed scene. However, if the observed points are expected to lie on a plane - e.g. the ground plane - this makes the determination of these matrices an ill-posed problem. Instead, methods based on homographies are better suited to this situation.One section of this thesis is concerned with the extraction of the camera pose and ego-motion from such homographies. We present both a direct SVD-based method and an iterative method, which both solve this problem. The iterative method is extended to allow simultaneous determination of the camera tilt from several homographies obeying the same planar motion model. This extension improves the robustness of the original method, and it provides consistent tilt estimates for the frames that are used for the estimation. The methods are evaluated using experiments on both real and synthetic data.Another part of the thesis deals with the problem of computing the homographies from point correspondences. By using conventional homography estimation methods for this, the resulting homography is of a too general class and is not guaranteed to be compatible with the planar motion assumption. For this reason, we enforce the planar motion model at the homography estimation stage with the help of a new homography solver using a number of polynomial constraints on the entries of the homography matrix. In addition to giving a homography of the right type, this method uses only \num{2.5} point correspondences instead of the conventional four, which is good \eg{} when used in a RANSAC framework for outlier removal

Planar motion bundle adjustment

In this paper we consider trajectory recovery for two cameras directed towards the floor, and which are mounted rigidly on a mobile platform. Previous work for this specific problem geometry has focused on locally minimising an algebraic error between inter-image homographies to estimate the relative pose. In order to accurately track the platform globally it is necessary to refine the estimation of the camera poses and 3D locations of the feature points, which is commonly done by utilising bundle adjustment; however, existing software packages providing such methods do not take the specific problem geometry into account, and the result is a physically inconsistent solution. We develop a bundle adjustment algorithm which incorporates the planar motion constraint, and devise a scheme that utilises the sparse structure of the problem. Experiments are carried out on real data and the proposed algorithm shows an improvement compared to established generic methods

Planar motion bundle adjustment

In this paper we consider trajectory recovery for two cameras directed towards the floor, and which are mounted rigidly on a mobile platform. Previous work for this specific problem geometry has focused on locally minimising an algebraic error between inter-image homographies to estimate the relative pose. In order to accurately track the platform globally it is necessary to refine the estimation of the camera poses and 3D locations of the feature points, which is commonly done by utilising bundle adjustment; however, existing software packages providing such methods do not take the specific problem geometry into account, and the result is a physically inconsistent solution. We develop a bundle adjustment algorithm which incorporates the planar motion constraint, and devise a scheme that utilises the sparse structure of the problem. Experiments are carried out on real data and the proposed algorithm shows an improvement compared to established generic methods