arXiv:2607.28021 2026 Geometry 1 ideas extracted · analyzed Aug 31, 2026
The paper supplies a concrete topological-geometric identity that turns an Euler Characteristic Transform into an exact perimeter measurement for any compact tame planar shape. The important transferable asset is that subtracting the Euler characteristic of an anchor point removes the divergent, position-dependent background, so the resulting integral is translation-independent and requires no arbitrary cutoff radius. This can become an isometry-robust geometric feature or auxiliary target for image, point-cloud, and shape networks. The most practical first use is a lightweight ECT perimeter branch concatenated to a CNN or point-cloud classifier, with a second experiment using it as a perimeter regularizer for segmentation.
Compute a translation- and rotation-robust perimeter feature from the Euler Characteristic Transform and append it to learned shape features. Unlike a finite-radius ECT comparison, the point-anchor subtraction cancels the constant Euler-characteristic tail exactly, eliminating the need to tune a spatial cutoff.