Decoy and disclosure radii of invariant shape descriptors
Abstract
A recognizer that compares rotation-invariant descriptors sees a surface only up to the fiber of the descriptor. We measure this fiber by its radius in the orbit distance from the enrolled surface. A large radius admits decoys, that is, distant shapes that pass the matcher. A small radius discloses the enrolled shape to anyone who captures the stored value. For star-shaped surfaces truncated to spherical harmonics of degree at most L, with n coefficients, a descriptor of generic rank r has generic fibers of dimension n−3−r modulo rotations. The standard pool of band powers, even bispectra, and three invariants of the degree-three band therefore admits decoy families of dimension $5$, $13$, $20$ at L=4,6,8. Its rank first reaches n−3 at L=16, and a mirror decoy remains at every L. The odd bispectra remove the mirror decoy generically for L ≥ 4. Yet at fixed mean radius the same pool determines the enclosed volume exactly, and it does not determine whether a surface meets a clearance requirement. We certify two cases by exact and interval arithmetic. At L=6 a decoy matches all $32$ invariants to relative precision 2 · 10−18 at orbit distance at least $0.87$ times the norm of the enrolled tuple. For the radar shape model of asteroid (101955) Bennu, the pool recovers the modeled volume, misses the handedness, and leaves the keep-out radius uncertain by more than 7 m.
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BibTeX
@misc{shaska2026decoy,
title = {{Decoy and disclosure radii of invariant shape descriptors}},
author = {Tanush Shaska and Lubjana Beshaj},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08410},
eprint = {2610.08410},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08410},
}
RIS
TY - UNPB
TI - Decoy and disclosure radii of invariant shape descriptors
AU - Tanush Shaska
AU - Lubjana Beshaj
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08410
UR - https://arxiv.org/abs/2610.08410
AB - A recognizer that compares rotation-invariant descriptors sees a surface only up to the fiber of the descriptor. We measure this fiber by its radius in the orbit distance from the enrolled surface. A large radius admits decoys, that is, distant shapes that pass the matcher. A small radius discloses the enrolled shape to anyone who captures the stored value. For star-shaped surfaces truncated to spherical harmonics of degree at most L, with n coefficients, a descriptor of generic rank r has generic fibers of dimension n−3−r modulo rotations. The standard pool of band powers, even bispectra, and three invariants of the degree-three band therefore admits decoy families of dimension $5$, $13$, $20$ at L=4,6,8. Its rank first reaches n−3 at L=16, and a mirror decoy remains at every L. The odd bispectra remove the mirror decoy generically for L ≥ 4. Yet at fixed mean radius the same pool determines the enclosed volume exactly, and it does not determine whether a surface meets a clearance requirement. We certify two cases by exact and interval arithmetic. At L=6 a decoy matches all $32$ invariants to relative precision 2 · 10^−18 at orbit distance at least $0.87$ times the norm of the enrolled tuple. For the radar shape model of asteroid (101955) Bennu, the pool recovers the modeled volume, misses the handedness, and leaves the keep-out radius uncertain by more than 7 m.
ER -