Computation of Circular Area and Spherical Volume Invariants via Boundary Integrals
This provides a method for feature detection in applications like anthropology, specifically for analyzing broken bone fragments, but it is incremental as it builds on existing integral techniques.
The authors tackled the problem of computing circular area and spherical volume invariants for curves and surfaces by expressing them as boundary integrals using the Divergence Theorem, resulting in a simple computational algorithm that avoids discretizing the ambient space and is analytically computable on triangulated meshes.
We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the Divergence Theorem to express the area and volume integrals as line and surface integrals, respectively, against particular kernels; our results also extend to higher dimensional hypersurfaces. The resulting surface integrals are computable analytically on a triangulated mesh. This gives a simple computational algorithm for computing the spherical volume invariant for triangulated surfaces that does not involve discretizing the ambient space. We discuss potential applications to feature detection on broken bone fragments of interest in anthropology.