Homotopy Continuation Made Easy: Regression-based Online Simulation of Starting Problem-Solution Pairs

While automatically generated polynomial elimination templates have sparked great progress in the field of 3D computer vision, there remain many problems for which the degree of the constraints or the number of unknowns leads to intractability. In recent years, homotopy continuation has been introduced as a plausible alternative. However, the method currently depends on expensive parallel tracking of all possible solutions in the complex domain, or a classification network for starting problem-solution pairs trained over a limited set of real-world examples. Our innovation consists of employing a regression network trained in simulation to directly predict a solution from input correspondences, followed by an online simulator that invents a consistent problem-solution pair. Subsequently, homotopy continuation is applied to track that single solution back to the original problem. We apply this elegant combination to generalized camera resectioning, and also introduce a new solution to the challenging generalized relative pose and scale problem. As demonstrated, the proposed method successfully compensates the raw error committed by the regressor alone, and leads to state-of-the-art efficiency and success rates while running on CPU resources, only.

fCOP: Focal Length Estimation from Category-level Object Priors

In the realm of computer vision, the perception and reconstruction of the 3D world through vision signals heavily rely on camera intrinsic parameters, which have long been a subject of intense research within the community. In practical applications, without a strong scene geometry prior like the Manhattan World assumption or special artificial calibration patterns, monocular focal length estimation becomes a challenging task. In this paper, we propose a method for monocular focal length estimation using category-level object priors. Based on two well-studied existing tasks: monocular depth estimation and category-level object canonical representation learning, our focal solver takes depth priors and object shape priors from images containing objects and estimates the focal length from triplets of correspondences in closed form. Our experiments on simulated and real world data demonstrate that the proposed method outperforms the current state-of-the-art, offering a promising solution to the long-standing monocular focal length estimation problem.

GS-EVT: Cross-Modal Event Camera Tracking based on Gaussian Splatting

Reliable self-localization is a foundational skill for many intelligent mobile platforms. This paper explores the use of event cameras for motion tracking thereby providing a solution with inherent robustness under difficult dynamics and illumination. In order to circumvent the challenge of event camera-based mapping, the solution is framed in a cross-modal way. It tracks a map representation that comes directly from frame-based cameras. Specifically, the proposed method operates on top of gaussian splatting, a state-of-the-art representation that permits highly efficient and realistic novel view synthesis. The key of our approach consists of a novel pose parametrization that uses a reference pose plus first order dynamics for local differential image rendering. The latter is then compared against images of integrated events in a staggered coarse-to-fine optimization scheme. As demonstrated by our results, the realistic view rendering ability of gaussian splatting leads to stable and accurate tracking across a variety of both publicly available and newly recorded data sequences.

Line-based 6-DoF Object Pose Estimation and Tracking With an Event Camera

Pose estimation and tracking of objects is a fundamental application in 3D vision. Event cameras possess remarkable attributes such as high dynamic range, low latency, and resilience against motion blur, which enables them to address challenging high dynamic range scenes or high-speed motion. These features make event cameras an ideal complement over standard cameras for object pose estimation. In this work, we propose a line-based robust pose estimation and tracking method for planar or non-planar objects using an event camera. Firstly, we extract object lines directly from events, then provide an initial pose using a globally-optimal Branch-and-Bound approach, where 2D-3D line correspondences are not known in advance. Subsequently, we utilize event-line matching to establish correspondences between 2D events and 3D models. Furthermore, object poses are refined and continuously tracked by minimizing event-line distances. Events are assigned different weights based on these distances, employing robust estimation algorithms. To evaluate the precision of the proposed methods in object pose estimation and tracking, we have devised and established an event-based moving object dataset. Compared against state-of-the-art methods, the robustness and accuracy of our methods have been validated both on synthetic experiments and the proposed dataset. The source code is available at https://github.com/Zibin6/LOPET.

EVIT: Event-based Visual-Inertial Tracking in Semi-Dense Maps Using Windowed Nonlinear Optimization

Event cameras are an interesting visual exteroceptive sensor that reacts to brightness changes rather than integrating absolute image intensities. Owing to this design, the sensor exhibits strong performance in situations of challenging dynamics and illumination conditions. While event-based simultaneous tracking and mapping remains a challenging problem, a number of recent works have pointed out the sensor's suitability for prior map-based tracking. By making use of cross-modal registration paradigms, the camera's ego-motion can be tracked across a large spectrum of illumination and dynamics conditions on top of accurate maps that have been created a priori by more traditional sensors. The present paper follows up on a recently introduced event-based geometric semi-dense tracking paradigm, and proposes the addition of inertial signals in order to robustify the estimation. More specifically, the added signals provide strong cues for pose initialization as well as regularization during windowed, multi-frame tracking. As a result, the proposed framework achieves increased performance under challenging illumination conditions as well as a reduction of the rate at which intermediate event representations need to be registered in order to maintain stable tracking across highly dynamic sequences. Our evaluation focuses on a diverse set of real world sequences and comprises a comparison of our proposed method against a purely event-based alternative running at different rates.

Motion and Structure from Event-based Normal Flow

Recovering the camera motion and scene geometry from visual data is a fundamental problem in the field of computer vision. Its success in standard vision is attributed to the maturity of feature extraction, data association and multi-view geometry. The recent emergence of neuromorphic event-based cameras places great demands on approaches that use raw event data as input to solve this fundamental problem.Existing state-of-the-art solutions typically infer implicitly data association by iteratively reversing the event data generation process. However, the nonlinear nature of these methods limits their applicability in real-time tasks, and the constant-motion assumption leads to unstable results under agile motion. To this end, we rethink the problem formulation in a way that aligns better with the differential working principle of event cameras.We show that the event-based normal flow can be used, via the proposed geometric error term, as an alternative to the full flow in solving a family of geometric problems that involve instantaneous first-order kinematics and scene geometry. Furthermore, we develop a fast linear solver and a continuous-time nonlinear solver on top of the proposed geometric error term.Experiments on both synthetic and real data show the superiority of our linear solver in terms of accuracy and efficiency, and indicate its complementary feature as an initialization method for existing nonlinear solvers. Besides, our continuous-time non-linear solver exhibits exceptional capability in accommodating sudden variations in motion since it does not rely on the constant-motion assumption.

An N-Point Linear Solver for Line and Motion Estimation with Event Cameras

Event cameras respond primarily to edges--formed by strong gradients--and are thus particularly well-suited for line-based motion estimation. Recent work has shown that events generated by a single line each satisfy a polynomial constraint which describes a manifold in the space-time volume. Multiple such constraints can be solved simultaneously to recover the partial linear velocity and line parameters. In this work, we show that, with a suitable line parametrization, this system of constraints is actually linear in the unknowns, which allows us to design a novel linear solver. Unlike existing solvers, our linear solver (i) is fast and numerically stable since it does not rely on expensive root finding, (ii) can solve both minimal and overdetermined systems with more than 5 events, and (iii) admits the characterization of all degenerate cases and multiple solutions. The found line parameters are singularity-free and have a fixed scale, which eliminates the need for auxiliary constraints typically encountered in previous work. To recover the full linear camera velocity we fuse observations from multiple lines with a novel velocity averaging scheme that relies on a geometrically-motivated residual, and thus solves the problem more efficiently than previous schemes which minimize an algebraic residual. Extensive experiments in synthetic and real-world settings demonstrate that our method surpasses the previous work in numerical stability, and operates over 600 times faster.

Six-Point Method for Multi-Camera Systems with Reduced Solution Space

Relative pose estimation using point correspondences (PC) is a widely used technique. A minimal configuration of six PCs is required for generalized cameras. In this paper, we present several minimal solvers that use six PCs to compute the 6DOF relative pose of a multi-camera system, including a minimal solver for the generalized camera and two minimal solvers for the practical configuration of two-camera rigs. The equation construction is based on the decoupling of rotation and translation. Rotation is represented by Cayley or quaternion parametrization, and translation can be eliminated by using the hidden variable technique. Ray bundle constraints are found and proven when a subset of PCs relate the same cameras across two views. This is the key to reducing the number of solutions and generating numerically stable solvers. Moreover, all configurations of six-point problems for multi-camera systems are enumerated. Extensive experiments demonstrate that our solvers are more accurate than the state-of-the-art six-point methods, while achieving better performance in efficiency.

Tight Fusion of Events and Inertial Measurements for Direct Velocity Estimation

Traditional visual-inertial state estimation targets absolute camera poses and spatial landmark locations while first-order kinematics are typically resolved as an implicitly estimated sub-state. However, this poses a risk in velocity-based control scenarios, as the quality of the estimation of kinematics depends on the stability of absolute camera and landmark coordinates estimation. To address this issue, we propose a novel solution to tight visual-inertial fusion directly at the level of first-order kinematics by employing a dynamic vision sensor instead of a normal camera. More specifically, we leverage trifocal tensor geometry to establish an incidence relation that directly depends on events and camera velocity, and demonstrate how velocity estimates in highly dynamic situations can be obtained over short time intervals. Noise and outliers are dealt with using a nested two-layer RANSAC scheme. Additionally, smooth velocity signals are obtained from a tight fusion with pre-integrated inertial signals using a sliding window optimizer. Experiments on both simulated and real data demonstrate that the proposed tight event-inertial fusion leads to continuous and reliable velocity estimation in highly dynamic scenarios independently of absolute coordinates. Furthermore, in extreme cases, it achieves more stable and more accurate estimation of kinematics than traditional, point-position-based visual-inertial odometry.

Online Stability Improvement of Groebner Basis Solvers using Deep Learning

Over the past decade, the Gr\"obner basis theory and automatic solver generation have lead to a large number of solutions to geometric vision problems. In practically all cases, the derived solvers apply a fixed elimination template to calculate the Gr\"obner basis and thereby identify the zero-dimensional variety of the original polynomial constraints. However, it is clear that different variable or monomial orderings lead to different elimination templates, and we show that they may present a large variability in accuracy for a certain instance of a problem. The present paper has two contributions. We first show that for a common class of problems in geometric vision, variable reordering simply translates into a permutation of the columns of the initial coefficient matrix, and that -- as a result -- one and the same elimination template can be reused in different ways, each one leading to potentially different accuracy. We then prove that the original set of coefficients may contain sufficient information to train a classifier for online selection of a good solver, most notably at the cost of only a small computational overhead. We demonstrate wide applicability at the hand of generic dense polynomial problem solvers, as well as a concrete solver from geometric vision.