Differential entropy-based analysis of point clouds: novel applications and methods

Point clouds are three-dimensional (3D) representations generated from data acquired through various technologies, including 3D scanners, advanced cameras, and depth sensors. These representations have brought about significant advancements in numerous sectors, such as autonomous driving, augmented reality, and Industry 4.0. In the modern industrial landscape, point clouds play an indispensable role. They are instrumental in real-time monitoring and optimization of production processes, enhancing the analysis of machine and robot performance. These 3D representations also facilitate the manipulation and training of industrial robots, enabling them to recognise objects and collaborate safely with humans. Additionally, point clouds are of paramount significance in industrial quality control, enabling the verification of product specifications against predefined criteria. The versatility of point clouds extends to representing systems of objects that can be simplified as points on a plane or in space. A wide array of analyses and applications can be conducted using point clouds. The focus of this study was on point cloud analysis based on differential entropy, a concept that quantifies the information contained in the data. Differential entropy of a point cloud provides insights into the spatial distribution of points. In this work, a thorough examination of this concept led to the development of three primary innovations: a novel formulation of differential entropy for point clouds, the creation of a new method called Differential Entropy for Deviation Analysis (DEDA), which is geared towards the analysis of geometric deviation for quality control in industrial production, and the introduction of the Differential Entropy-based Compactness Index (DECI), a compactness index designed specifically for analysing systems that can be schematised as point clouds, with a particular focus on the transportation system, proposed as a risk index. The modification of the formulation of differential entropy for univariate and multivariate normal distributions has eliminated problems associated with the vanishing determinant of the covariance matrix, as present in the formula, and ensured that the entropic contribution of each point in the point cloud assumes a zero or positive value. This new formulation has paved the way for the development and demonstrated the effectiveness of the DEDA method and the DECI index. In the field of quality control, the DEDA method has proven to be reliable and robust. Geometric deviation analysis on samples produced through Additive Manufacturing (AM), conducted using both traditional methods based on Euclidean distance calculations between points in compared point clouds and the new DEDA method, have underscored the potential of the newly proposed method. The DEDA method, notably, provides a synthetic quality index that determines the product's quality level, as opposed to traditional methods that rely on the interpretation of mean and standard deviation, which can often lead to challenging or incorrect quality level assessments. Furthermore, the robustness assessment of the DEDA method has demonstrated its ability to overcome several limitations inherent in classical methods, including the presence of holes, background noise, density variations between compared point clouds, and the non-commutativity of Euclidean distance. The DEDA method also lays the foundation for the development of a new point clouds registration method. The latest innovation, the DECI index, allows for the derivation of a descriptive index of the compactness of a point cloud. While this index is suitable for various applications, it has been applied to monitor maritime traffic, serving as a quantifiable measure of risk in a specific sea area. Leveraging information on maritime traffic, DECI values were evaluated on an hourly, monthly, and annual basis. This enabled a comprehensive examination of maritime traffic, shedding light on the most and least congested time periods. DECI-based analysis could be instrumental in implementing measures to alleviate congestion in the busiest sea areas and facilitating qualitative comparisons.