Metric 3D-reconstruction from Unordered and Uncalibrated Image Collections

In this thesis the problem of Structure from Motion (SfM) for uncalibrated and unordered image collections is considered. The proposed framework is an adaptation of the framework for calibrated SfM proposed by Olsson-Enqvist (2011) to the uncalibrated case. Olsson-Enqvist's framework consists of three main steps; pairwise relative rotation estimation, rotation averaging, and geometry estimation with known rotations. For this to work with uncalibrated images we also perform auto-calibration during the first step. There is a well-known degeneracy for pairwise auto-calibration which occurs when the two principal axes meet in a point. This is unfortunately common for real images. To mitigate this the rotation estimation is instead performed by estimating image triplets. For image triplets the degenerate congurations are less likely to occur in practice. This is followed by estimation of the pairs which did not get a successful relative rotation from the previous step. The framework is successfully applied to an uncalibrated and unordered collection of images of the cathedral in Lund. It is also applied to the well-known Oxford dinosaur sequence which consists of turntable motion. Image pairs from the turntable motion are in a degenerate conguration for auto-calibration since they both view the same point on the rotation axis

Computational Methods for Computer Vision : Minimal Solvers and Convex Relaxations

Robust fitting of geometric models is a core problem in computer vision. The most common approach is to use a hypothesize-and-test framework, such as RANSAC. In these frameworks the model is estimated from as few measurements as possible, which minimizes the risk of selecting corrupted measurements. These estimation problems are called minimal problems, and they can often be formulated as systems of polynomial equations. In this thesis we present new methods for building so-called minimal solvers or polynomial solvers, which are specialized code for solving such systems. On several minimal problems we improve on the state-of-the-art both with respect to numerical stability and execution time.In many computer vision problems low rank matrices naturally occur. The rank can serve as a measure of model complexity and typically a low rank is desired. Optimization problems containing rank penalties or constraints are in general difficult. Recently convex relaxations, such as the nuclear norm, have been used to make these problems tractable. In this thesis we present new convex relaxations for rank-based optimization which avoid drawbacks of previous approaches and provide tighter relaxations. We evaluate our methods on a number of real and synthetic datasets and show state-of-the-art results

Non-Convex Rank/Sparsity Regularization and Local Minima

This paper considers the problem of recovering either a low rank matrix or a sparse vector from observations of linear combinations of the vector or matrix elements. Recent methods replace the non-convex regularization with $\ell_1$ or nuclear norm relaxations. It is well known that this approach can be guaranteed to recover a near optimal solutions if a so called restricted isometry property (RIP) holds. On the other hand it is also known to perform soft thresholding which results in a shrinking bias which can degrade the solution. In this paper we study an alternative non-convex regularization term. This formulation does not penalize elements that are larger than a certain threshold making it much less prone to small solutions. Our main theoretical results show that if a RIP holds then the stationary points are often well separated, in the sense that their differences must be of high cardinality/rank. Thus, with a suitable initial solution the approach is unlikely to fall into a bad local minima. Our numerical tests show that the approach is likely to converge to a better solution than standard $\ell_1$/nuclear-norm relaxation even when starting from trivial initializations. In many cases our results can also be used to verify global optimality of our method

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

Beyond Gr\"obner Bases: Basis Selection for Minimal Solvers

Many computer vision applications require robust estimation of the underlying geometry, in terms of camera motion and 3D structure of the scene. These robust methods often rely on running minimal solvers in a RANSAC framework. In this paper we show how we can make polynomial solvers based on the action matrix method faster, by careful selection of the monomial bases. These monomial bases have traditionally been based on a Gr\"obner basis for the polynomial ideal. Here we describe how we can enumerate all such bases in an efficient way. We also show that going beyond Gr\"obner bases leads to more efficient solvers in many cases. We present a novel basis sampling scheme that we evaluate on a number of problems

Analytic Solutions to the Dynamic Programming sub-problem in Hybrid Vehicle Energy Management

The computationally demanding Dynamic Programming (DP) algorithm is frequently used in academic research to solve the energy management problem of an Hybrid Electric Vehicle (HEV). This paper is focused exclusively on how the computational demand of such a computation can be reduced. The main idea is to use a local approximation of the gridded cost-to-go and derive an analytic solution for the optimal torque split decision at each point in the time and state grid. Thereby it is not necessary to quantize the torque split and identify the optimal decision by interpolating in the cost-to-go. Two different approximations of the cost-to-go are considered in the paper: i) a local linear approximation, and ii) a quadratic spline approximation. The results indicate that computation time can be reduced by orders of magnitude with only a slight degradation in simulated fuel economy. Furthermore, with a spline approximated cost-to-go it is also possible to significantly reduce the memory storage requirements. A parallel Plug-in HEV is considered in the paper but the method is also applicable to an HEV

Influence of State of Charge Estimation Uncertainty on Energy Management Strategies for Hybrid Electric Vehicles

This paper studies how the optimal energy management of a hybrid electric vehicle and a plug-in hybrid electric vehicle is affected by uncertain estimates of the battery state of charge. A simple model for the battery dynamics and the state of charge estimation is postulated, inspired by the known characteristics of previously proposed estimation schemes. Based on the assumption that the drive cycle is perfectly known, the effects of state of charge estimation uncertainty is studied by including the estimation uncertainty in the optimization of the energy management strategy. The simulations indicate lower battery usage and higher fuel consumption as the estimation uncertainty increases

Practical Robust Two-View Translation Estimation

Outliers pose a problem in all real structure from motion systems. Due to the use of automatic matching methods one has to expect that a (sometimes very large) portion of the detected correspondences can be incorrect. In this paper we propose a method that estimates the relative translation between two cameras and simultaneously maximizes the number of inlier correspondences. Traditionally, outlier removal tasks have been addressed using RANSAC approaches. However, these are random in nature and offer no guarantees of finding a good solution. If the amount of mismatches is large, the approach becomes costly because of the need to evaluate a large number of random samples. In contrast, our approach is based on the branch and bound methodology which guarantees that an optimal solution will be found. While most optimal methods trade speed for optimality, the proposed algorithm has competitive running times on problem sizes well beyond what is common in practice. Experiments on both real and synthetic data show that the method outperforms state-of-the-art alternatives, including RANSAC, in terms of solution quality. In addition, the approach is shown to be faster than RANSAC in settings with a large amount of outliers