The research objective of this award is to develop mathematical foundations, algorithmic infrastructure, and prototype software that can process massive point-cloud data directly, accurately and efficiently into suitable geometric form for product development use. The new approach is based on a moving least-squares (MLS) formulation that defines a continuous surface directly from a set of points. The MLS surface has many unique properties, such as projection procedure, simple implicit form, Cn continuity, and local computing. This research would lead to a paradigm shift in geometric processing techniques, from laborious intermediate surface reconstruction with human intervention to enabling D3M from massive point-cloud data. These new techniques are based on the MLS surface and include: a) analytical formulae for differential geometric analysis; b) Morse theory based methods for uncovering its topological structures; and c) fundamental algorithms enabling point based geometric computing with guaranteed geometric accuracy and topological robustness.
If successful, this research will result in computational tools enabling D3M from massive point-cloud data. Specifically, direct, accurate and adaptive processing would lead to dramatic time reduction in shape modeling in product design, improved product dimensional accuracy, and shortened product development cycle. Through industrial collaboration with both sensor vendors and point data users, this research can unleash the full potential of 3D scanning for a host of manufacturing industries such as aerospace, automobile, die and mold, mass customization and biomedical applications. Through its integrated research, education and outreach activities, this project will provide advanced knowledge in geometric processing and D3M for students from high schools to graduate schools and will increase domestic students? interest in science and engineering and therefore strengthen our competitiveness in the global workforce.
This project has led to significance advances in mathematical foundations and computing techniques for processing massive point-cloud data directly, accurately and efficiently into suitable geometric form for product development use. These techniques have resulted in end-to-end systems that can bypass laborious surface reconstruction and enable direct digital design and manufacturing from scan data. Mathematically, we have derived formulae and methods for differential geometric and topological analysis of the moving least-squares (MLS) surface. We used the MLS surface’s implicit form for differential geometric analysis and adopted Morse theory to uncover its topological structures. Computationally, we have utilized the above mathematical results to develop fundamental algorithms for MLS surface based surface intersection and deformation that are critical for direct digital design and manufacturing. We have focused on addressing basic properties of these operations, including geometric adaptivity, bounded error and topological robustness. Research contributions in this project includes the explicit formulas for computing curvatures on an MLS surface, its applications in slicing for rapid prototyping, algorithms for direct Boolean intersection of non-uniform rational B-spline surfaces and an MLS surface, and an algorithm for automatically generate B-spline based numerical control paths from scanned data. We have also conducted theoretical study on topological guarantee during the slicing of the scan data for 3D printing. The algorithm has also been extended to tele-fabrication where a physical part is scanned in site and re-produced by additive manufacturing in another site.
