We present Gradient-SDF, a novel representation for 3D geometry that combines the advantages of implict and explicit representations. By storing at every voxel both the signed distance field as well as its gradient vector field, we enhance the capability of implicit representations with approaches originally formulated for explicit surfaces. As concrete examples, we show that (1) the Gradient-SDF allows us to perform direct SDF tracking from depth images, using efficient storage schemes like hash maps, and that (2) the Gradient-SDF representation enables us to perform photometric bundle adjustment directly in a voxel representation (without transforming into a point cloud or mesh), naturally a fully implicit optimization of geometry and camera poses and easy geometry upsampling. Experimental results confirm that this leads to significantly sharper reconstructions. Since the overall SDF voxel structure is still respected, the proposed Gradient-SDF is equally suited for (GPU) parallelization as related approaches.

We propose a new formulation for the bundle adjustment problem which relies on nullspace marginalization of landmark variables by QR decomposition. Our approach, which we call square root bundle adjustment, is algebraically equivalent to the commonly used Schur complement trick, improves the numeric stability of computations, and allows for solving large-scale bundle adjustment problems with single-precision floating-point numbers. We show in real-world experiments with the BAL datasets that even in single precision the proposed solver achieves on average equally accurate solutions compared to Schur complement solvers using double precision. It runs significantly faster, but can require larger amounts of memory on dense problems. The proposed formulation relies on simple linear algebra operations and opens the way for efficient implementations of bundle adjustment on hardware platforms optimized for single-precision linear algebra processing.

In this paper we propose a novel square root sliding-window bundle adjustment suitable for real-time odometry applications. The square root formulation pervades three major aspects of our optimization-based sliding-window estimator: for bundle adjustment we eliminate landmark variables with nullspace projection; to store the marginalization prior we employ a matrix square root of the Hessian; and when marginalizing old poses we avoid forming normal equations and update the square root prior directly with a specialized QR decomposition. We show that the proposed square root marginalization is algebraically equivalent to the conventional use of Schur complement (SC) on the Hessian. Moreover, it elegantly deals with rank-deficient Jacobians producing a prior equivalent to SC with Moore–Penrose inverse. Our evaluation of visual and visual-inertial odometry on real-world datasets demonstrates that the proposed estimator is 36% faster than the baseline. It furthermore shows that in single precision, conventional Hessian-based marginalization leads to numeric failures and reduced accuracy. We analyse numeric properties of the marginalization prior to explain why our square root form does not suffer from the same effect and therefore entails superior performance.

Continuous-time trajectory representation has recently gained popularity for tasks where the fusion of high-frame-rate sensors and multiple unsynchronized devices is required. Lie group cumulative B-splines are a popular way of representing continuous trajectories without singularities. They have been used in near real-time SLAM and odometry systems with IMU, LiDAR, regular, RGB-D and event cameras, as well as for offline calibration. These applications require efficient computation of time derivatives (velocity, acceleration), but all prior works rely on a computationally suboptimal formulation. In this work we present an alternative derivation of time derivatives based on recurrence relations that needs O(k) instead of O(k^2) matrix operations (for a spline of order k) and results in simple and elegant expressions. While producing the same result, the proposed approach significantly speeds up the trajectory optimization and allows for computing simple analytic derivatives with respect to spline knots. The results presented in this paper pave the way for incorporating continuous-time trajectory representations into more applications where real-time performance is required.

This paper tackles the problem of data abstraction in the context of 3D point sets. Our method classifies points into different geometric primitives, such as planes and cones, leading to a compact representation of the data. Being based on a semi-global Hough voting scheme, the method does not need initialization and is robust, accurate, and efficient. We use a local, low-dimensional parameterization of primitives to determine type, shape and pose of the object that a point belongs to. This makes our algorithm suitable to run on devices with low computational power, as often required in robotics applications. The evaluation shows that our method outperforms state-of-the-art methods both in terms of accuracy and robustness.

The use of structural information in 3D scanning is becoming more and more popular. However, most approaches exploit this structural information either in the form of geometric primitives (mostly planes) or known rigid bodies, but not both. We overcome this limitation and propose an object representation that combines primitive and non-primitive objects using one unified formulation that is based on signed distance fields. Object pose manifolds are introduced to represent the rigid movement of primitives and non-primitives in a natural way. We show that different components of volumetric scanning, such as global trajectory optimization or geometry completion and denoising, benefit from our formulation.