HCC: Small: Variational surfacing of unorganized spatial curves

Three-dimensional (3D) surface models are widely used in many domains, such as engineering, design, manufacturing, medicine, and entertainment. In many use cases, the surface models are obtained from curve inputs. For example, engineers traditionally construct an object by first defining its wireframe; designers routinely conceptualize new shapes by sketching curves; and doctors often create 3D models of organs by delineating the organ boundaries on 2D slices of an image volume. Crucial to these applications is the ability to reconstruct a complete surface from a set of sparse, un-organized, and possibly noise-ridden curve segments. This research will develop a suite of algorithms that offer such ability with improved efficiency, surface quality, and generality over existing methods. Ultimately, the project will make it easier for everyone - novices and professionals alike - to create 3D models by drawing curves. Curve inputs are particularly challenging for surface reconstruction due to their sparsity, imprecision, and lack of connectivity. This project builds on the success of recent variational methods for surface construction from sparse and noisy point clouds, and introduces new problem formulations, optimization strategies, and surface representations for generating surfaces from 3D curves. The research will extend existing variational methods in three significant ways: improved efficiency for large inputs and interactive applications; better surface shapes that leverage higher-order geometric information of curves; and a greater variety of outputs including piecewise-smooth surfaces, surfaces with open boundaries, and non-manifold surface networks. The project will also produce prototypes of interactive, curve-driven modeling software that complements existing modeling tools with the unique capability of direct surface generation from unorganized and error-laden curves. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.