Affine Grassmannian has been favored for expressing proximity between lines and planes due to its theoretical exactness in measuring distances among features. Despite this advantage, the existing method can only measure the proximity without yielding the distance as an explicit function of rigid body transformation. Thus, an optimizable distance function on the manifold has remained underdeveloped, stifling its application in registration problems. This paper is the first to explicitly derive an optimizable cost function between two Grassmannian features with respect to rigid body transformation ($\mathbf{R}$ and $\mathbf{t}$). Specifically, we present a rigorous mathematical proof demonstrating that the bases of high-dimensional linear subspaces can serve as an explicit representation of the cost. Finally, we propose an optimizable cost function based on the transformed bases that can be applied to the registration problem of any affine subspace. Compared to vector parameter-based approaches, our method is able to find a globally optimal solution by directly minimizing the geodesic distance which is agnostic to representation ambiguity. The resulting cost function and its extension to the inlier-set maximizing Branch-and-Bound (BnB) solver have been demonstrated to improve the convergence of existing solutions or outperform them in various computer vision tasks. The code is available on https://github.com/joomeok/GrassmannRegistration.

Lifelong mapping is crucial for the long-term deployment of robots in dynamic environments. In this paper, we present ELite, an ephemerality-aided LiDAR-based lifelong mapping framework which can seamlessly align multiple session data, remove dynamic objects, and update maps in an end-toend fashion. Map elements are typically classified as static or dynamic, but cases like parked cars indicate the need for more detailed categories than binary. Central to our approach is the probabilistic modeling of the world into two-stage ephemerality, which represent the transiency of points in the map within two different time scales. By leveraging the spatiotemporal context encoded in ephemeralities, ELite can accurately infer transient map elements, maintain a reliable up-to-date static map, and improve robustness in aligning the new data in a more finegrained manner. Extensive real-world experiments on long-term datasets demonstrate the robustness and effectiveness of our system. The source code is publicly available for the robotics community: https://github.com/dongjae0107/ELite.

LiDAR place recognition is a crucial module in localization that matches the current location with previously observed environments. Most existing approaches in LiDAR place recognition dominantly focus on the spinning type LiDAR to exploit its large FOV for matching. However, with the recent emergence of various LiDAR types, the importance of matching data across different LiDAR types has grown significantly-a challenge that has been largely overlooked for many years. To address these challenges, we introduce HeLiOS, a deep network tailored for heterogeneous LiDAR place recognition, which utilizes small local windows with spherical transformers and optimal transport-based cluster assignment for robust global descriptors. Our overlap-based data mining and guided-triplet loss overcome the limitations of traditional distance-based mining and discrete class constraints. HeLiOS is validated on public datasets, demonstrating performance in heterogeneous LiDAR place recognition while including an evaluation for longterm recognition, showcasing its ability to handle unseen LiDAR types. We release the HeLiOS code as an open source for the robotics community at https://github.com/minwoo0611/HeLiOS.

In this study, a deep-learning-based multi-stage network architecture called Multi-Stage Depth Prediction Network (MSDPN) is proposed to predict a dense depth map using a 2D LiDAR and a monocular camera. Our proposed network consists of a multi-stage encoder-decoder architecture and Cross Stage Feature Aggregation (CSFA). The proposed multi-stage encoder-decoder architecture alleviates the partial observation problem caused by the characteristics of a 2D LiDAR, and CSFA prevents the multi-stage network from diluting the features and allows the network to learn the interspatial relationship between features better. Previous works use sub-sampled data from the ground truth as an input rather than actual 2D LiDAR data. In contrast, our approach trains the model and conducts experiments with a physically-collected 2D LiDAR dataset. To this end, we acquired our own dataset called KAIST RGBD-scan dataset and validated the effectiveness and the robustness of MSDPN under realistic conditions. As verified experimentally, our network yields promising performance against state-of-the-art methods. Additionally, we analyzed the performance of different input methods and confirmed that the reference depth map is robust in untrained scenarios.