In this paper, we introduce GS-LIVM, a real-time photo-realistic LiDAR-Inertial-Visual mapping framework with Gaussian Splatting tailored for outdoor scenes. Compared to existing methods based on Neural Radiance Fields (NeRF) and 3D Gaussian Splatting (3DGS), our approach enables real-time photo-realistic mapping while ensuring high-quality image rendering in large-scale unbounded outdoor environments. In this work, Gaussian Process Regression (GPR) is employed to mitigate the issues resulting from sparse and unevenly distributed LiDAR observations. The voxel-based 3D Gaussians map representation facilitates real-time dense mapping in large outdoor environments with acceleration governed by custom CUDA kernels. Moreover, the overall framework is designed in a covariance-centered manner, where the estimated covariance is used to initialize the scale and rotation of 3D Gaussians, as well as update the parameters of the GPR. W e evaluate our algorithm on several outdoor datasets, and the results demonstrate that our method achieves state-of-the-art performance in terms of mapping efficiency and rendering quality. The source code is available on GitHub .

Learning-based visual relocalizers exhibit leading pose accuracy, but require hours or days of training. Since training needs to happen on each new scene again, long training times make learning-based relocalization impractical for most applications, despite its promise of high accuracy. In this paper we show how such a system can actually achieve the same accuracy in less than 5 minutes. We start from the obvious: a relocalization network can be split in a scene-agnostic feature backbone, and a scene-specific prediction head. Less obvious: using an MLP prediction head allows us to optimize across thousands of view points simultaneously in each single training iteration. This leads to stable and extremely fast convergence. Furthermore, we substitute effective but slow end-to-end training using a robust pose solver with a curriculum over a reprojection loss. Our approach does not require privileged knowledge, such a depth maps or a 3D model, for speedy training. Overall, our approach is up to 300x faster in mapping than state-of-the-art scene coordinate regression, while keeping accuracy on par.

We present an approach to construct appropriate and efficient emulators for Hamiltonian flow maps. Intended future applications are long-term tracing of fast charged particles in accelerators and magnetic plasma confinement configurations. The method is based on multi-output Gaussian process regression on scattered training data. To obtain long-term stability the symplectic property is enforced via the choice of the matrix-valued covariance function. Based on earlier work on spline interpolation we observe derivatives of the generating function of a canonical transformation. A product kernel produces an accurate implicit method, whereas a sum kernel results in a fast explicit method from this approach. Both correspond to a symplectic Euler method in terms of numerical integration. These methods are applied to the pendulum and the H\'enon-Heiles system and results compared to an symmetric regression with orthogonal polynomials. In the limit of small mapping times, the Hamiltonian function can be identified with a part of the generating function and thereby learned from observed time-series data of the system's evolution. Besides comparable performance of implicit kernel and spectral regression for symplectic maps, we demonstrate a substantial increase in performance for learning the Hamiltonian function compared to existing approaches.