Identification of generic polygonal domains by integral-geometric invariants

What the math gives to ML

The paper provides a global geometric signature for planar shapes: the full pairwise-distance distribution, equivalently its Mellin transform/Riesz energy function, generically identifies polygonal domains up to Euclidean isometry. The transferable asset is not merely another distance metric, but a theoretically motivated, transformation-invariant multiscale signature whose singular or rapidly varying behavior encodes geometric events such as side lengths and angles. A practical neural adaptation is to add a sampled Riesz-energy or distance-distribution matching loss to polygon or shape reconstruction, yielding an alignment-free global constraint that complements local pixel, Chamfer, or vertex losses.