Variational Mesh Offsetting by Smoothed Winding Number

Surface mesh offsetting is a fundamental operation in various applications (e.g., shape modeling). Implicit methods that contour a volumetric distance field are robust at handling intersection defects, but it is challenging to apply shape control (e.g., preserving sharp features in the input shape) and to avoid undesired topology changes. Explicit methods, which move vertices towards the offset surface (with possible adaptivity), can address the above issues, but it is hard to avoid intersection issues. To combine the advantages of both, we propose a variational framework that takes mesh vertex locations as variables while simultaneously involving a smooth winding-number field associated with the mesh. Under various shape regularizations (e.g., sharp feature preservation) formulated on the mesh, the objective function mainly requires that the input mesh lie on the offset contour of the field induced by the resulting mesh. Such a combination inherits the ability to apply flexible shape regularizations from explicit methods and significantly alleviates intersection issues because of the field. Moreover, the optimization problem is numerically friendly by virtue of the differentiability of the field w.r.t. the mesh vertices. Results show that we can offset a mesh while preserving sharp features of the original surface, restricting selected parts to quadric surfaces and penalizing intersections.