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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Sep 30, 2026, cs.CVJ/K move · Enter open · S save

Invariant Shape Analysis of Surfaces with Spherical Topology

T. Shaska, M. -R. Siadat

Spherical Harmonics Statistical Shape Model Symmetry Surface Representation

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Spherical harmonic descriptors of closed 3D shapes depend on the parameterization, the pose and the scale of the surface, and the standard rotation-invariant reductions, the power spectrum and the bispectrum, discard the relative orientation of the harmonic bands and cannot distinguish a shape from its mirror image. We construct a descriptor that removes all three dependencies exactly and loses nothing else: a conformal parameterization normalized by its conformal barycenter, followed by polynomial invariants of the rotation group. Identifying each harmonic band with a binary form turns the rotation quotient into classical invariant theory and makes reflections visible as the sign of an invariant, so chirality is recorded. The descriptor is complete for the truncated expansion, stable in the orbit distance, and comes with numerical diagnostics. Benchmarks confirm the guarantees, and on bilateral anatomical structures the descriptor separates mirror-image pairs from asymmetric pairs, which parity-blind descriptors cannot.

$L$	$n$	$n-3$	$N(L)$	$r(L)$	$n-3-r(L)$	$N^{+}(L)$	$r^{+}(L)$	$n-3-r^{+}(L)$
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

Abstract

Explore similar work

Invariant Shape Analysis of Surfaces with Spherical Topology

Auditing Training-Free 3D Shape Retrieval with Diffused Geodesic Moments

Higher order PCA-like rotation-invariant features for detailed shape descriptors modulo rotation

$L$	$n$	$n-3$	$N(L)$	$r(L)$	$n-3-r(L)$	$N^{+}(L)$	$r^{+}(L)$	$n-3-r^{+}(L)$
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