Comparative analysis of analytical solutions of sixth-, fourth-, and second-order PDEs for parametric surface reconstruction using Genetic Algorithms

Abstract 3D surface reconstruction from point clouds aims to generate continuous surface representations of physical objects from discrete spatial samples. This problem is of fundamental importance across diverse domains, including robotics, industrial design, and cultural heritage preservation. Among various surface representation schemes, Partial Differential Equation-based surfaces have emerged as a promising approach due to their strong mathematical foundations, compact representation, and high-order continuity properties. Despite these advantages, a key challenge lies in the analytical solution of PDEs used for surface modelling. Existing approaches often rely on numerical solvers, which are computationally expensive and prone to discretization errors. Although analytical solutions to fourth-order PDEs have been applied to parametric surface reconstruction, several limitations remain: (1) comparisons among different PDE orders (e.g., second-, fourth-, and sixth-order) have not been systematically explored; (2) hyperparameters within PDE solutions are typically tuned manually, which is inefficient and often suboptimal; and (3) multiple analytical solutions to the same PDE are rarely compared in terms of their fitting accuracy; and (4) PDEs with non-zero sculpting force are rarely investigated for surface reconstruction. 
To address these issues, this study conducts a comprehensive analysis of closed-form PDE solutions of varying orders for parametric surface reconstruction from point clouds. The Genetic Algorithm is employed to automatically determine optimal hyperparameter values, thereby improving reconstruction quality. Furthermore, we perform an extensive comparison among different PDE orders and among multiple analytical solutions derived from the same PDE. Lastly, we also investigate the impact of the sculpting force in the PDEs for surface reconstruction. Experimental results across several datasets of varying complexity demonstrate that the proposed framework achieves accurate and smooth surface reconstruction while significantly reducing manual intervention. This work provides new insights into the practical use of PDE-based parametric surfaces and establishes a foundation for future research in data-driven geometric modeling and surface reconstruction.