Classification of stiffness and oscillations in initial value problems

The spectrum of a linear, constant coefficient operator $A$ can help us characterize the system $\dot{x}=Ax$ satisfactorily, but in the case of a nonlinear dynamical system such methods are not suitable. In this thesis we discuss the insufficiency of only studying the eigenvalues along the Jacobian of the solution trajectory and discuss possible indicators to better characterize such systems. In Söderlind et al. \cite{soderlind-stiffness} a stiffness indicator was derived, and we seek to investigate the possibility to define an oscillation indicator, i.e. an indicator that accurately captures the phenomenon known as oscillations. This is of scientific interest since a rigorous and computationally relevant characterization of oscillations is still missing. Furthermore, we discuss problems that are not due to nonlinearity, but non-normality, and derive a normality indicator. For applications one would need computationally inexpensive indicators, and we make suggestions of such, called estimators, mimicking the behavior of the stiffness indicator and the proposed oscillation indicator. In order to demonstrate the theory sixteen computational experiments serve to illustrate a variety of different phenomena.Lösningarna till linjära system med konstanta koefficienter, $\dot{x}=Ax$, har varit kända i över hundra år och deras karaktär bestäms med hjälp av egenvärdena till matrisen $A$. Det är inte konstigt att konceptet egenvärde förbryllar matematik- och ingenjörs-studenter: Vad är den korrekta tolkningen? Vad betyder det? Det är inte ett enkelt begrepp att förstå, eftersom det finns många olika perspektiv på hur det ska tolkas. Inom numerisk analys talar egenvärdena om huruvida din metod kommer att konvergera och hur snabbt detta sker, men i populationsekologi förutspår egenvärdena de långsiktiga förhållandena mellan olika arter i ett ekosystem. I kvantmekanik kan egenvärdena vara energitillstånd för en partikel i en kvantbrunn och i hållfasthetsläran talar de om för dig hur du ska designa en bro för att motstå starka vindar och jordbävningar. När man väl förstår vilket omfattande begrepp egenvärden utgör kommer ett ännu större problem -- för icke-linjära dynamiska system räcker det inte att studera egenvärden. Det är dessa problem som är av intresse ute i näringslivet. För att kunna lösa sådana problem behöver vi kunna karakterisera dem, eftersom olika problem kräver olika lösningsmetoder. Detta är inte problem som du kan skriva ner för hand, utan består av miljontals ekvationer som behöver lösas, vilket även är svårt för datorer att göra om inte rätt metoder används. I denna avhandling presenteras några förslag på hur styvhet och oscillation kan karakteriseras. Idag finns det inga vedertagna definitioner för dessa fenomen, men begreppen har existerat i forskningsvärden i över sextio år. En anledning till att det inte finns är för att det rör sig om komplexa fenomen som inte bara har en specifik egenskap. Styva ekvationer är en typ av differentialekvationer där en del numeriska metoder (explicita) är numeriskt instabila om inte steglängden är väldigt liten. Även för moderna datorer kan detta innebära långa simuleringstider och ofta överväger man att istället använda implicita metoder för att bli av med steglängdskravet. Oscillationer är inte endast lösningar till periodiska system utan även till kvasiperiodiska system och kaotiska system. Dessa kan ha olika egenskaper, t.ex. kan de vara invarianta längs med en lösningstrajektorie eller ha stabila självsvängningar

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

Bilinear Parameterization For Differentiable Rank-Regularization

Low rank approximation is a commonly occurring problem in many computer vision and machine learning applications. There are two common ways of optimizing the resulting models. Either the set of matrices with a given rank can be explicitly parametrized using a bilinear factorization, or low rank can be implicitly enforced using regularization terms penalizing non-zero singular values. While the former approach results in differentiable problems that can be efficiently optimized using local quadratic approximation, the latter is typically not differentiable (sometimes even discontinuous) and requires first order subgradient or splitting methods. It is well known that gradient based methods exhibit slow convergence for ill-conditioned problems. In this paper we show how many non-differentiable regularization methods can be reformulated into smooth objectives using bilinear parameterization. This allows us to use standard second order methods, such as Levenberg--Marquardt (LM) and Variable Projection (VarPro), to achieve accurate solutions for ill-conditioned cases. We show on several real and synthetic experiments that our second order formulation converges to substantially more accurate solutions than competing state-of-the-art methods.Comment: 17 page

Trust Your IMU: Consequences of Ignoring the IMU Drift

In this paper, we argue that modern pre-integration methods for inertial measurement units (IMUs) are accurate enough to ignore the drift for short time intervals. This allows us to consider a simplified camera model, which in turn admits further intrinsic calibration. We develop the first-ever solver to jointly solve the relative pose problem with unknown and equal focal length and radial distortion profile while utilizing the IMU data. Furthermore, we show significant speed-up compared to state-of-the-art algorithms, with small or negligible loss in accuracy for partially calibrated setups. The proposed algorithms are tested on both synthetic and real data, where the latter is focused on navigation using unmanned aerial vehicles (UAVs). We evaluate the proposed solvers on different commercially available low-cost UAVs, and demonstrate that the novel assumption on IMU drift is feasible in real-life applications. The extended intrinsic auto-calibration enables us to use distorted input images, making tedious calibration processes obsolete, compared to current state-of-the-art methods

Radially distorted planar motion compatible homographies

Fast and accurate homography estimation is essential to many computer vision applications, including scene degenerate cases and planarity detection. Such cases arise naturally in man-made environments, and failure to handle them will result in poor positioning estimates. Most modern day consumer cameras are affected by some level of radial distortion, which must be compensated for in order to get accurate estimates. This often demands calibration procedures, with specific scene requirements, and off-line processing. In this paper a novel polynomial solver for radially distorted planar motion compatible homographies is presented. The proposed algorithm is fast and numerically stable, and is proven on both synthetic and real data to work well inside a RANSAC loop

Fast Non-minimal Solvers for Planar Motion Compatible Homographies

This paper presents a novel polynomial constraint for homographies compatible with the general planar motion model. In this setting, compatible homographies have five degrees of freedom-instead of the general case of eight degrees of freedom-and, as a consequence, a minimal solver requires 2.5 point correspondences. The existing minimal solver, however, is computationally expensive, and we propose using non-minimal solvers, which significantly reduces the execution time of obtaining a compatible homography, with accuracy and robustness comparable to that of the minimal solver. The proposed solvers are compared with the minimal solver and the traditional 4-point solver on synthetic and real data, and demonstrate good performance, in terms of speed and accuracy. By decomposing the homographies obtained from the different methods, it is shown that the proposed solvers have future potential to be incorporated in a complete Simultaneous Localization and Mapping (SLAM) framework

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

A unified optimization framework for low-rank inducing penalties

In this paper we study the convex envelopes of a new class of functions. Using this approach, we are able to unify two important classes of regularizers from unbiased non-convex formulations and weighted nuclear norm penalties. This opens up for possibilities of combining the best of both worlds, and to leverage each method’s contribution to cases where simply enforcing one of the regularizers are insufficient. We show that the proposed regularizers can be incorporated in standard splitting schemes such as Alternating Direction Methods of Multipliers (ADMM), and other subgradient methods. Furthermore, we provide an efficient way of computing the proximal operator. Lastly, we show on real non-rigid structure-from-motion (NRSfM) datasets, the issues that arise from using weighted nuclear norm penalties, and how this can be remedied using our proposed method

Generalization of Parameter Recovery in Binocular Vision for a Planar Scene

In this paper, we consider a mobile platform with two cameras directed towards the floor. In earlier work, this specific problem geometry has been considered under the assumption that the cameras have been mounted at the same height. This paper extends the previous work by removing the height constraint, as it is hard to realize in real-life applications. We develop a method based on an equivalent problem geometry, and show that much of previous work can be reused with small modification to account for the height difference. A fast solver for the resulting nonconvex optimization problem is devised. Furthermore, we propose a second method for estimating the height difference by constraining the mobile platform to pure translations. This is intended to simulate a calibration sequence, which is not uncommon to impose. Experiments are conducted using synthetic data, and the results demonstrate a robust method for determining the relative parameters comparable to previous work

Relative pose estimation in binocular vision for a planar scene using inter-image homographies

In this paper we consider a mobile platform with two cameras directed towards the floor mounted the same distance from the ground, assuming planar motion and constant internal parameters. Earlier work related to this specific problem geometry has been carried out for monocular systems, and the main contribution of this paper is the generalization to a binocular system and the recovery of the relative translation and orientation between the cameras. The method is based on previous work on monocular systems, using sequences of inter-image homographies. Experiments are conducted using synthetic data, and the results demonstrate a robust method for determining the relative parameters