不变形状描述符的诱饵半径与披露半径

中文摘要 AI 辅助

一种比较旋转不变描述符的识别器，只能看到曲面直至描述符的纤维。我们通过该纤维在轨道距离上相对于注册曲面的半径来度量它。半径大则允许诱饵存在，即能通过匹配器的远处形状。半径小则向任何获取存储值的人披露注册形状。对于截断至球谐函数次数至多$L$（含$n$个系数）的星形曲面，一个一般秩为$r$的描述符具有模旋转的维数为$n-3-r$的一般纤维。标准的带功率、偶双谱以及三次带三个不变量的集合，因此在$L=4,6,8$时允许维数分别为$5$、$13$、$20$的诱饵族。其秩首次达到$n-3$是在$L=16$，且在每个$L$下都存在镜像诱饵。奇双谱在$L\geq 4$时一般性地移除镜像诱饵。然而，在固定平均半径下，同一集合能精确确定所包围的体积，但无法确定曲面是否满足净空要求。我们通过精确算术和区间算术验证了两个案例。在$L=6$时，一个诱饵匹配所有$32$个不变量，相对精度为$2 \cdot 10^{-18}$，轨道距离至少为注册元组范数的$0.87$倍。对于小行星(101955)贝努的雷达形状模型，该集合恢复了建模体积，但遗漏了手性，并使禁入半径的不确定性超过$7 \\, \mathrm{m}$。

英文摘要

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 $L \geq 4$. 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 $2 \cdot 10^{-18}$ 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 $7 \, \mathrm{m}$.

发表机构

• Oakland University（奥克兰大学）
• United States Military Academy（美国军事学院）

机构由 AI 辅助整理，请以论文原文为准。