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The version of record of this article, first published in Discrete & Computational Geometry, is available online at Publisher’s website: https://doi.org/10.1007/s00454-022-00395-8, Let S be a set of n points in general position in the plane, and let Xk,(S) be the number of convex k-gons with vertices in S that have exactly points of S in their interior. We prove several equalities for the numbers Xk,(S). This problem is related to the Erd¿os–Szekeres theorem. Some of the obtained equations also extend known equations for the numbers of empty convex polygons to polygons with interior points. Analogous results for higher dimension are shown as well., Research of C.H. was partially supported by project MTM2015-63791-R (MINECO/ FEDER), PID-2019-104129GB-I00/MCIN/AEI/10.13039/501100011033, and by project Gen. Cat. DGR 2017SGR1336. D.O. was partially supported by project PAPIIT IG100721 and CONACyT 282280. P. P-L. was partially supported by project DICYT 041933PL Vicerrectoría de Investigación, Desarrollo e Innovación USACH (Chile), and Programa Regional STICAMSUD 19-STIC-02. Research of B.V. was partially supported by the Austrian Science Fund (FWF) within the collaborative DACH project Arrangements and Drawings as FWF Project I 3340-N35. We thank an anonymous referee for helpful comments. This project has been supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 734922, Postprint (author's final draft)

In recent decades, with the prevalence of affordable RGB-D cameras and LiDAR (Light Detection and Ranging) scanners, point cloud representation has become increasingly practical and popular in many computer vision applications, such as structure from motion (SfM), simultaneous localization and mapping (SLAM). Since an acquisition of point clouds is usually limited to only one perspective, several acquisitions are required to cover the whole area of interest. Point cloud registration is then applied as a fundamental step for finding an optimal transformation between the partially overlapped point cloud fragments and recovering a complete underlying geometry. Subsequently, with the reconstructed 3D model represented as point clouds, performing semantic scene understanding is necessary for many applications, such as autonomous driving and augmented reality. In this thesis, we present our contributions in solving these two problems, namely, point cloud registration and point cloud semantic understanding. First, we present two methods for efficient and robust point cloud registration based on the key idea of decomposing the point cloud registration pipeline into three learnable sub-modules. In the first method, we design a keypoint detector sub-module and a keypoint descriptor sub-module for efficient local feature extraction, emphasizing the importance of a reliable keypoint detector and demonstrating the superiority of joint learning of both detection and description tasks. In the second method, we propose a correspondence filtering sub-module to improve the robustness towards large outlier ratio. We explicitly incorporate the spatial consistency constrained by rigid transformations for pruning outlier correspondences. Second, we study the problem of semantic understanding on registered point clouds. Specifically, we propose a LiDAR-camera fusion solution for 3D perception. Our studies investigate the inherent difficulties of LiDAR-camera fusion and reveal a crucial as

Obtaining 3D realistic models of urban scenes from accurate range data is nowadays an important research topic, with applications in a variety of fields ranging from Cultural Heritage and digital 3D archiving to monitoring of public works. Processing massive point clouds acquired from laser scanners involves a number of challenges, from data management to noise removal, model compression and interactive visualization and inspection. In this paper, we present a new methodology for the reconstruction of 3D scenes from massive point clouds coming from range lidar sensors. Our proposal includes a panorama-based compact reconstruction where colors and normals are estimated robustly through an error-aware algorithm that takes into account the variance of expected errors in depth measurements. Our representation supports efficient, GPU-based visualization with advanced lighting effects. We discuss the proposed algorithms in a practical application on urban and historical preservation, described by a massive point cloud of 3.5 billion points. We show that we can achieve compression rates higher than 97% with good visual quality during interactive inspections., Peer Reviewed, Postprint (author's final draft)