I‑AIMS Institute for Artificial Intelligence
and Mathematical Sciences
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Decoy and disclosure radii of invariant shape descriptors

Lubjana Beshaj, Tanush Shaska

I-AIMS preprint sh-183 · 2026

A rotation-invariant shape descriptor has two complementary limitations: different shapes can pass the same matcher, while a released value can disclose the enrolled geometry. Both limitations are governed by the descriptor’s fiber modulo rotations. We measure geometric ambiguity by the fiber radius, and task ambiguity by the range of a specified invariant quantity on the fiber. A descriptor of generic rank r on coefficients of dimension n has regular generic fibers of dimension n − 3 − r modulo rotations. For the spectral pool of band powers, even bispectra, and band-three invariants, the computed dimensions at L = 4, 6, 8 are 5, 13, 20; rank saturation at L = 16 leaves at least a mirror pair. Adding odd bispectra removes the generic mirror ambiguity for L ≥ 4 and gives rank saturation at L = 10, in agreement with known degree-three orbit-recovery results. We obtain finite-tolerance bounds and an exact noisy-quadrupole radius, with linear sensitivity at distinct spectra and square-root sensitivity at repeated eigenvalues. We also prove that, at fixed mean radius, the even pool determines enclosed volume exactly despite its shape ambiguity, and give explicit descriptor- and radial-error bounds. Two positive radial surfaces provide an exact counterexample to a clearance decision while sharing the descriptor and volume. On the radar shape model of asteroid (101955) Bennu, the even pool at L = 6 recovers the modeled volume but not the handedness, and certified decoys that match the descriptor and the modeled volume within stated finite tolerances have keep-out radii differing by more than 7 m. Observer-centered reconstruction bounds, combined-release analysis, and a certified high-band benchmark complete a framework for auditing what a geometric representation can certify and what it discloses.

MSC classes: Primary 13A50, 68M25; Secondary 14P10, 68T45, 68U05

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