Surface approximation, discrete varifolds, and regularized first variation

Shape visualization and processing are fundamental tasks in many fields from mechanical engineering to physics, biology, chemical engineering, medicine, astronomy, etc., and are the subject of very active research in image processing and computer graphics. For there is a huge variety of applications and a large variety of capture systems, there are many discrete models for representing a shape: point clouds, meshes, pixel/voxel representations, splines, level sets representations, etc. All these models carry very different informations, and it is hopeless to look after equivalence relations between them. It makes sense however to try to look at all these models in a common framework, possibly weaker, which would provide a common formalism for studying a large variety of both discrete and continuous shapes. Such a common framework has been proposed in [Bue14, Bue15]: the class of varifolds. Varifolds have been introduced by Almgren in 1965 [Alm65] to study the existence of critical points of the area functional. They have several nice properties in a variational context: compactness, mass continuity, criteria of rectifiability, a notion of multiplicity, and a weak notion of curvature called the first variation [All72, Sim83]. In addition, the varifold structure is flexible enough to describe not only classical continuous objects as curves, surfaces, rectifiable sets, etc., but also ”discrete” objects like meshes, point clouds, volumetric representations. etc. Previous contributions in the literature also involve tools from geometric measure theory to define a convenient notion of curvature or to study more generally discrete surface approximation. For instance, a unified notion of curvature measure valid both for surfaces and their discrete approximations, and based on normal cycles, is introduced in [CSM06]. First defined for surfaces and triangulations [Mor08], it has been recently extended in [CCLT09] to more general discretizations like point clouds. The accuracy of the approximation of the surface is measured in terms of Hausdorff distance while the error between the curvature measure of the surface and the curvature measure of the approximation is controlled in terms of the Bounded Lipschitz distance which is similar to the Wasserstein distance. Shape comparison is another application of geometric measure theory: Charon and Trouvé[CT13] endow triangulated surfaces with a varifold structure, and define a distance between varifolds, both computable from a numerical point of view and adapted to shape matching. In the first section of this paper, we follow [Bue15] and introduce discrete volumetric varifolds and point cloud varifolds to represent both volumetric surface models and point clouds. We study convergence issues, and in particular, in Theorem 2.6 , we estimate in terms of the Bounded Lipschitz distance the quality of the approximation of rectifiable varifolds by either discrete volumetric varifolds or point cloud varifolds.