Decoy and disclosure radii of invariant shape descriptors

Tanush Shaska; Lubjana Beshaj

cs.CVOct 6, 2026

Decoy and disclosure radii of invariant shape descriptors

Authors: Tanush Shaska, Lubjana Beshaj

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 $L \geq 4$ . 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 $7 \, \mathrm{m}$ .

Figures & tables

Figure 1. The surfaces $c_{+}$ (left) and $c_{-}$ (right) of Prop. 5.3 , drawn as radial surfaces $\rho_{0}+f_{1}+f_{2}$ with $\rho_{0}=1$ , $r_{1}=0.25$ , and $r_{2}=0.35$ , where $f_{1}(p)=\langle a,p\rangle$ and $f_{2}(p)=p^{\top}Qp$ . Their band powers coincide. Their orbit distance is $\delta(c_{+},c_{-})\approx 0.487$ .

$L$	$n$	$n-3$	$N(L)$	$r(L)$	$n-3-r(L)$	$N^{+}(L)$	$r^{+}(L)$	$n-3-r^{+}(L)$
2	8	5	4	4	1	4	4	1
3	15	12	10	10	2	10	10	2
4	24	21	16	16	5	17	17	4
5	35	32	22	22	10	24	24	8
6	48	45	32	32	13	37	37	8
7	63	60	42	42	18	50	50	10

Table 1. Sizes, generic ranks, and generic fiber dimensions of the pools

\Phi_{L}

and

\Phi^{+}_{L}

Figure 2. The enrolled surface $c$ (left) and the decoy $c^{\prime}$ (right), drawn with $f_{0}=24$ ; $c^{\prime}$ is rotated by the rotation $R_{0}$ of Ex. 7.4 (2). All $32$ components of $\Phi_{6}$ agree within relative tolerance $2\cdot 10^{-18}$ ; see Rem. 7.5 .

$L$	components	$V_{\mathrm{pool}}$ ( $\mathrm{km}^{3}$ )	$\lvert V_{\mathrm{pool}}-V_{L}\rvert/V_{\mathrm{mesh}}$	$(V_{\mathrm{pool}}-V_{\mathrm{mesh}})/V_{\mathrm{mesh}}$	$\gamma_{L}$ ( $\mathrm{m}$ )
2	4	0.061969	$1.1\cdot 10^{-15}$	$-0.477\,\%$	9.71
3	7	0.062009	$8.4\cdot 10^{-16}$	$-0.412\,\%$	9.01
4	13	0.062186	$3.6\cdot 10^{-16}$	$-0.127\,\%$	4.97
5	19	0.062204	$2.4\cdot 10^{-16}$	$-0.098\,\%$	4.32
6	29	0.062227	$<10^{-16}$	$-0.063\,\%$	3.45
7	39	0.062232	$3.6\cdot 10^{-16}$	$-0.054\,\%$	3.18

Table 2. Volume of Bennu from the even pool.

V_{\mathrm{pool}}

is the right-hand side of Eq. 4 evaluated on the powers and even bispectra;

V_{L}

is the volume of the truncated radial model;

\gamma_{L}=\lVert\rho-\rho_{0}-f_{1}-\dots-f_{L}\rVert_{L^{2}(d\mu)}

is the truncation error of the radial function. The column “components” counts the powers and even bispectra.

Figure 3. Bennu: the radar shape model of [ 13 ] , its $L=6$ model $c$ , the mirror image $\tau c$ , and the decoy $d_{+}$ , labeled in each panel. The last two are rotated to their numerically best alignment with $c$ . All four are drawn at the same scale. The color shows the distance from the origin in meters; see the color bar.

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$L$	$n$	$n-3$	$N(L)$	$r(L)$	$n-3-r(L)$	$N^{+}(L)$	$r^{+}(L)$	$n-3-r^{+}(L)$
2	8	5	4	4	1	4	4	1
3	15	12	10	10	2	10	10	2
4	24	21	16	16	5	17	17	4
5	35	32	22	22	10	24	24	8
6	48	45	32	32	13	37	37	8
7	63	60	42	42	18	50	50	10

$L$	$n$	$n-3$	$N(L)$	$r(L)$	$n-3-r(L)$	$N^{+}(L)$	$r^{+}(L)$	$n-3-r^{+}(L)$
2	8	5	4	4	1	4	4	1
3	15	12	10	10	2	10	10	2
4	24	21	16	16	5	17	17	4
5	35	32	22	22	10	24	24	8
6	48	45	32	32	13	37	37	8
7	63	60	42	42	18	50	50	10

$L$	$n$	$n-3$	$N(L)$	$r(L)$	$n-3-r(L)$	$N^{+}(L)$	$r^{+}(L)$	$n-3-r^{+}(L)$
2	8	5	4	4	1	4	4	1
3	15	12	10	10	2	10	10	2
4	24	21	16	16	5	17	17	4
5	35	32	22	22	10	24	24	8
6	48	45	32	32	13	37	37	8
7	63	60	42	42	18	50	50	10