Generalized extrinsic distortion and applications

• Examine existing notions of distortion, a discrete analog of curvature.
• Classify them into intrinsic and extrinsic based on angles used.
• Generalize extrinsic distortion to nD and derive a mean curvature weighting.
• Empirically examine the behavior of the operator on 3-manifolds in 4D.

Curvature is a key feature in shape analysis and its estimation on discrete simplicial complexes benefits many geometry processing applications. However, its study has mostly remained focused on 2D manifolds and computationally practical extensions to higher dimensions remain an active area of computer science research. We examine the existing notions of distortion, an analog of curvature in the discrete setting, and classify them into two categories: intrinsic and extrinsic, depending on whether they use the interior or the dihedral angles of the tessellation. We then propose a generalization of extrinsic distortion to ce:italic> D /ce:italic> D and derive a weighting that can be used to compute mean curvature on tessellated hypersurfaces. We analyze the behavior of the operator on 3-manifolds in 4D and compare it to the well-known Laplace–Beltrami operator using ground truth hypersurfaces defined by functions of three variables, and a segmentation application, showing it to behave as well or better while being intuitively simple and easy to implement.

Average normalized RMS error of weighted distortion (blue) vs the Laplace–Beltrami operator (red) as a function of increasing resolution on (a) uniform grid, (b) irregular, (c) diamond meshes, and also (d) as a function of the number of tetrahedra incident per vertex on irregular tessellations.Figure optionsDownload high-quality image (106 K)Download as PowerPoint slide