Corrected curvature measures

We propose a new mathematical and computational tool for infering the geometry of shapes known only through approximations, such as triangulated or digital surfaces. The main idea is to decouple the position of the shape boundary from its normal vector field. To do so, we extend a classical tool of geometric measure theory, the normal cycle, so that it takes as input not only a surface but also a normal vector field. We formalize it as a current in the oriented Grassmann bundle. By choosing adequate differential forms, we define geometric measures like area, mean and Gaussian curvatures. We then show the stability of these measures when both position and normal input data are approximations of the underlying continuous shape. As a byproduct, our tool is able to correctly estimate curvatures over polyhedral approximations of shapes with explicit bounds, even when their natural normal are not correct, as long as an external convergent normal vector field is provided. We show that this framework induces state-of-the-art curvature estimations on polyhedral surfaces, digital surfaces and even point clouds with normal data. This talks gather joint works with Pascal Romon, Boris Thibert, David Coeurjolly and Céline Labart.

May 20, 2025 4:00 PM — 5:00 PM