Parametric curve and Partial Differential Equation-based parametric surface reconstruction from point clouds.

Abstract

There are two primary types of representations used to model curves and 3D surfaces in the digital world: explicit and implicit. Explicit representations include parametric representations, poly- gons, and similar methods, while implicit representations encom- pass level sets, distance functions, and Constructive Solid Ge- ometry (CSG), among others. Parametric models are especially popular in 3D modelling, design, and manufacturing because they are mathematically defined, making them easy to edit, transmit, and construct. However, real-world data often exists in a discrete form, typically as point sets, rather than in a parametric form. Therefore, it is crucial to reconstruct a parametric representation that closely approximates real-world data. In the context of parametric curve reconstruction from point sets, two key objectives are to approximate the underlying structure of the point sets as accurately as possible while minimising the num- ber of curves used. Existing methods often struggle to balance these two factors effectively. To address this gap, we have developed a method that mimics the process of human vectorization of image boundaries. The boundary points are first segmented into multiple segments us- ing corner points. Within each segment, the bisection method is employed to identify the largest subset of points that a single curve can fit. Additional curves are introduced only when the fitting error exceeds a predefined threshold. This process contin- ues until all points in the segment are fitted, thereby minimising the number of B´ezier curves required. Additionally, symmetric shape boundaries within the point sets are detected, further re- ducing the number of curves needed. My method also allows for the selection of the optimal parameterization method on a case- by-case basis to minimise fitting error, which is a critical step in parametric curve reconstruction. Comparisons with both con- temporary and classical methods demonstrate that my approach outperforms existing techniques. For parametric 3D surface reconstruction from point sets, B´ezier, B-spline, and NURBS surfaces have been extensively studied. However, these methods share common drawbacks, such as the need for large data storage and complex geometry processing. Moreover, maintaining good continuity between reconstructed parametric patches is often challenging. In contrast, PDE-based methods for surface reconstruction are advantageous due to their low storage requirement, strong fitting capabilities and the rel- ative ease of achieving good continuity, as most are boundary- based approaches. The primary challenge with PDE-based sur- face reconstruction methods lies in solving partial differential equations, which is why most studies have focused on implicit PDE-based shape reconstruction, despite its computational ex- pense. To address these challenges, we propose a novel method that uses an accurate closed-form solution to a fourth-order PDE for recon- structing 3D parametric surfaces from point clouds. This method offers powerful fitting capabilities and is computationally efficient. However, postprocessing is necessary to ensure continuity, as this is not inherently guaranteed in the model. Additionally, the pa- rameterization of point sets in the initial method is not sufficiently effective. To overcome these limitations, my subsequent work in- tegrates linearly blended Coons patches with an analytical solu- tion of a specific fourth-order PDE. This combined approach not only ensures good positional continuity between reconstructed parametric patches but also uses the Coons patch as an effec- tive tool for accurate parameterization of the point sets. Lastly, tangential continuity is achieved by combining the bicubic Coons patch and a special deformation surface.