  Citation: Jung, W.; Hyeon, J.; Doh, N. Robust Cuboid Modeling from Noisy and Incomplete 3D Point Clouds Using Gaussian Mixture Model. Remote Sens. 2022, 14, 5035. https:// doi.org/10.3390/rs14195035 Academic Editors: Kourosh Khoshelham, Martin Weinmann, Johannes Otepka and Di Wang Received: 29 August 2022 Accepted: 4 October 2022 Published: 9 October 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). remote sensing Article Robust Cuboid Modeling from Noisy and Incomplete 3D Point Clouds Using Gaussian Mixture Model Woonhyung Jung 1 , Janghun Hyeon 1 and Nakju Doh 2,3, * 1 School of Electrical Engineering, Korea University, Seoul 02841, Korea 2 Institute of Convergence Science, Korea University, Seoul 02841, Korea 3 CTO of TeeLabs, Seoul 02857, Korea * Correspondence: nakju@korea.ac.kr Abstract: A cuboid is a geometric primitive characterized by six planes with spatial constraints, such as orthogonality and parallelism. These characteristics uniquely define a cuboid. Therefore, previous modeling schemes have used these characteristics as hard constraints, which narrowed the solution space for estimating the parameters of a cuboid. However, under high noise and occlusion conditions, a narrowed solution space may contain only false or no solutions, which is called an over-constraint. In this paper, we propose a robust cuboid modeling method for point clouds under high noise and occlusion conditions. The proposed method estimates the parameters of a cuboid using soft constraints, which, unlike hard constraints, do not limit the solution space. For this purpose, a cuboid is represented as a Gaussian mixture model (GMM). The point distribution of each cuboid surface owing to noise is assumed to be a Gaussian model. Because each Gaussian model is a face of a cuboid, the GMM shares the cuboid parameters and satisfies the spatial constraints, regardless of the occlusion. To avoid an over-constraint in the optimization, only soft constraints are employed, which is the expectation of the GMM. Subsequently, the soft constraints are maximized using ana- lytic partial derivatives. The proposed method was evaluated using both synthetic and real data. The synthetic data were hierarchically designed to test the performance under various noise and oc- clusion conditions. Subsequently, we used real data, which are more dynamic than synthetic data and may not follow the Gaussian assumption. The real data are acquired by light detection and ranging- based simultaneous localization and mapping with actual boxes arbitrarily located in an indoor space. The experimental results indicated that the proposed method outperforms a previous cuboid model- ing method in terms of robustness. Keywords: cuboid modeling; geometric primitive; point cloud; 3D modeling; object mesh; LiDAR 1. Introduction Geometric primitive-based modeling is a widely used method for abstracting point clouds for applications such as scene reconstruction [1–6], rendering [7–11], and shape processing [12,13], as geometric primitives are lighter and easier to manipulate than raw point clouds. Moreover, these characteristics can reduce the labor needed for inspection, assessment, and management. Hence, Geometric primitive-based modeling is also useful in the fields of extraction of bridge components [14–17] for inspection purposes. Among geometric primitives, the cuboid model is the most practical for substituting point clouds acquired by simultaneous localization and mapping (SLAM). A cuboid can substitute a large portion of point clouds as it is frequently observable in man-made environments. Moreover, a cuboid model is light and easy to manipulate because it renders six spatially related planar patches simultaneously with nine parameters. However, most point clouds generated by SLAM contain many defects, such as noise and occlusion. Therefore, a cuboid modeling method must be robust to remain useful. A traditional cuboid modeling method [4,10,18–20] consists of two processes: plane detection and spatial constraint validation, which impose hard constraints to narrow the Remote Sens. 2022, 14, 5035. https://doi.org/10.3390/rs14195035 https://www.mdpi.com/journal/remotesensing