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Decoy and disclosure radii of invariant shape descriptors

Tanush Shaska, Lubjana Beshaj

arXiv:2610.08410v1cs.CVastro-ph.EPcs.CRmath.AG

TL;DR

Rotation-invariant descriptors leave a fiber of shapes consistent with the enrolled value, creating decoy or disclosure ambiguity that depends on the fiber’s radius. The paper analyzes this radius for spherical-harmonic shape models and certifies both geometric ambiguity and task-specific outcomes, including a close descriptor match at L=6.

  • Problem

    A rotation-invariant recognizer sees only a descriptor, so the fiber can contain distinct shapes that remain indistinguishable from that representation.

    Source: §1 · p. 1

  • Method

    The paper measures descriptor fibers by their radius in orbit distance and uses generic rank to assess the dimension of decoy families modulo rotations.

    Source: §1 · pp. 1–2

  • Results

    At L=6, an integer and rational tuple match all 32 invariants to relative precision 2 · 10^-18 while their orbit distance is at least 0.87 times the enrolled tuple norm.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The spectral pool can fix enclosed volume while leaving handedness and a clearance decision unresolved, so whole-shape ambiguity and task-specific uncertainty are distinct.

    Source: §12 · p. 28

    The disclosure result is an information bound, not a reconstruction algorithm or probabilistic privacy guarantee, and it gives no bound on the effort needed to find a consistent point.

    Source: §3 · p. 8

Abstract

from arXiv · show

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 $7 \, \mathrm{m}$.

1. Introduction

The paper measures a rotation-invariant descriptor’s fiber by orbit distance, treating large radii as decoy opportunities and small radii as disclosure. It analyzes the standard spectral pool’s ambiguities and task-specific limits, including certified examples.

1. Introduction

  • For compact semialgebraic admissible sets, a separating descriptor’s stability inequality bounds orbit distance by a Hölder power of descriptor discrepancy.The cited prior result is also read here as a disclosure bound for a released descriptor known to accuracy ε.Source: §1 · p. 2
  • For the standard spectral pool, generic decoy-family dimensions are 5, 13, and 20 at L=4, 6, and 8; its rank first reaches n−3 at L=16.At L=16 the continuous ambiguity disappears on the free maximal-rank locus.Source: §1 · p. 2
  • The scalar-valued pool leaves a mirror-image decoy at every L, while adding odd bispectra removes it generically for L≥4.The augmented pool first reaches rank n−3 at L=10.Source: §1 · p. 2
  • At L=6, an integer and a rational tuple match all 32 invariants to relative precision 2 · 10^-18 while remaining at least 0.87 times the enrolled tuple norm apart in orbit distance.The rational comparison is exact, and interval arithmetic certifies the distance bound.Source: §1 · p. 2
  • On the quadrupole band, the power fiber is a sphere, and adjoining det Q makes the exact decoy radius zero.With exact power known, the radius at determinant tolerance ε grows linearly for distinct eigenvalues and like ε^1/2 at a repeated eigenvalue.Source: §1 · p. 2
  • At fixed constant radial term, the even pool determines enclosed volume exactly, yet an explicit descriptor-matched pair has opposite outcomes for a clearance requirement.The pair consists of positive radial surfaces and can be checked by hand.Source: §1 · p. 2
  • The paper applies the analysis to Bennu’s radar shape model, with certified descriptor, distance, positivity, and task bounds for the rounded L=6 model.The supplied passage describes the application but does not give its outcome values.Source: §1 · p. 3
  • The coefficient model assumes a fully observed closed surface that is star-shaped about a specified origin; higher-genus surfaces and partial views are outside its scope.A centroid convention is an additional constraint, not a consequence of positivity or omitting the constant harmonic.Source: §1 · p. 3

2. Descriptors, fibers, and radii

The section defines the coefficient model for star-shaped surfaces, rotation-invariant descriptors, fibers, and orbit distance. It then defines a tolerance-dependent radius that measures ambiguity around an enrolled tuple and distinguishes that centered measure from uncertainty based only on a released value.

2. Descriptors, fibers, and radii

  • A star-shaped surface is represented by a positive radial function on the sphere, and its degree-L truncation supplies the coefficient tuple with the constant band treated separately when physical radial solids are compared.The constant term is fixed or retained separately in the volume and clearance statements.Source: §2 · p. 4
  • Rotations act diagonally on the coefficient bands, and orbit distance is the minimum coefficient-tuple distance over all rotations.The resulting distance is a metric on the orbit space.Source: §2 · pp. 4–5
  • A descriptor is an SO(3)-invariant polynomial map, and its fiber contains all coefficient tuples with the enrolled tuple’s descriptor value.It separates orbits when each fiber is a single orbit.Source: §2 · p. 5
  • For a compact admissible set and tolerance ε, the radius is the largest orbit distance from the enrolled tuple among admissible tuples whose descriptor discrepancy is within ε.The maximum exists, and the radius is nondecreasing in ε.Source: §2 · p. 5
  • The centered radius measures both the farthest accepted decoy and the farthest fiber-consistent shape from the enrolled tuple, so large values favor the decoy adversary and small values favor disclosure.These interpretations use the same fixed admissible set and tolerance.Source: §2 · p. 5
  • A nonsingular weighting of descriptor discrepancy preserves exact fibers and Jacobian ranks but can change positive-tolerance radii, while observation-consistent sets remain conditional on the specified admissible set.Componentwise relative error requires a positive absolute floor when descriptor components can vanish.Source: §2 · pp. 5–6

3. Separating descriptors disclose

Separating descriptors bound both accepted decoys and disclosure under finite descriptor error on compact semialgebraic sets. The section also gives a constructive orbit reconstruction on the dipole and quadrupole bands and states what the information bounds do not provide.

3. Separating descriptors disclose

  • For a compact semialgebraic set, a separating descriptor bounds the radius at tolerance ε by C_K ε^α_K and gives radius zero at exact tolerance.A released value known within ε places every consistent point within orbit distance C_K(2ε)^α_K of the enrolled tuple.Source: §3 · p. 7
  • On U2, six invariant values determine (a,Q) up to rotation on the distinct-eigenvalue locus through a cubic equation and a linear system, with separation also holding for repeated eigenvalues.The reconstruction uses the eigenvalues of Q, a Vandermonde system for squared coordinates, and an identity fixing the sign of a1a2a3.Source: §3 · p. 7
  • These results are information bounds, not reconstruction algorithms or probabilistic privacy guarantees, and they do not bound the computation needed to find a consistent point.They assume no distribution on the admissible set, and a large shape diameter need not hide a particular attribute.Source: §3 · p. 8

4. Fibers of descriptors

The section relates descriptor rank to the size of its fibers modulo rotations, then considers what happens when descriptors are released together.

4. Fibers of descriptors

  • The argument uses the free maximal-rank locus, where the rotation action is free and the descriptor differential has maximal rank.There, the regular fiber is an invariant embedded manifold before quotienting by rotations.Source: §4 · p. 8
  • When r < n −3, the regular fiber modulo rotations has dimension n −3 −r and contains decoys near generic enrolled surfaces.When r = n −3, the regular fiber is a finite union of rotation orbits.Source: §4 · p. 8
  • A descriptor with fewer than n −3 components admits decoys near every generic enrolled surface, regardless of the components’ values.The result applies on an open dense set and to every compact neighborhood of such a surface.Source: §4 · p. 9
  • Combining releases gives a fiber whose dimension is determined by the rank of the concatenated descriptor, not by adding component ranks.For separately budgeted errors, the jointly consistent set is an intersection and its diameter cannot exceed any individual diameter.Source: §4 · p. 9

5. Band powers

Band powers leave families of shapes indistinguishable, while adding the quadrupole determinant can separate them. The section also quantifies orbit-radius behavior under exact and tolerant observations.

5. Band powers

  • For generic coefficients, band powers have rank L and fibers of dimension L^2 + L −3 modulo rotations.The fiber is a product of spheres, allowing independent rotations of the bands.Source: §5 · p. 9
  • For the fixed-power sphere, the lower radius bound is attained when det Q = 0, and the upper bound when Q has a repeated eigenvalue.The compact set K is assumed to contain the sphere of matrices with the same norm as Q.Source: §5 · p. 10
  • On H2, adjoining det Q to P2 gives a separating descriptor with zero exact radius.The power fixes tr Q^2 and the determinant fixes the spectrum, which determines Q up to rotation.Source: §5 · p. 10
  • With exact power known, the fixed-spectrum radius is linear to first order, while the uniform tolerance exponent 1/2 is sharp.These statements do not describe joint noise in both invariants.Source: §5 · p. 11
  • The c+ and c− surfaces have identical band powers but orbit distance δ(c+, c−) ≈0.487.The figure uses ρ0 = 1, r1 = 0.25, and r2 = 0.35.Source: Figure 1 · p. 12

6. The spectral pool

The section defines the even spectral pool and augmented pool, then determines their generic ranks, fiber dimensions, and mirror ambiguity. It also describes how the rank claims are certified.

The spectral pool

  • The spectral pool ΦL combines band powers, even bispectra, and three invariants of the degree-three sextic; Φ+L adds odd bispectra.The spectral pool is also called the even pool, and the augmented pool includes the odd bispectra divided by i.Source: §6 · pp. 12–13
  • The augmented pool reaches generic rank n−3 at L=10, with fiber dimensions 4, 8, and 6 at L=4, 6, and 8.For L≤9 its rank is below n−3.Source: §6 · pp. 13–14
  • Exact modular Jacobian calculations certify the generic ranks, using an integer evaluation point and a nonzero minor modulo 2^31−1.The accompanying program reports a maximal independent row set and a nonzero determinant for each tested rank.Source: §6 · p. 14
  • The even pool has generic fiber dimensions 5, 13, and 20 at L=4, 6, and 8, respectively, and its rank first reaches n−3 at L=16.For L≤15 its generic rank is below n−3.Source: §6 · pp. 13–14, Table 1 · p. 13
  • The augmented pool’s rank saturation at L=10 is not a new generic-recovery threshold; the section instead examines the even pool’s radii, mirror ambiguity, and task information.Prior recovery results cited in the section establish generic recovery for the augmented invariant set from L≥10.Source: §6 · p. 14
  • The even pool retains a mirror decoy at every L, whereas odd bispectrum B234 distinguishes the mirror generically for L≥4; neither result proves global orbit separation.At L=16 the even pool has only finitely many rotation classes on the free maximal-rank locus, including at least the mirror pair.Source: §6 · p. 15

7. An explicit finite-tolerance benchmark

An explicit L=6 pair demonstrates that a finite-tolerance match can be far from the enrolled tuple. The authors certify the match and distance using exact arithmetic and interval bounds.

Certification

  • The component comparison is exact rational arithmetic, while the orbit-distance lower bound is certified by interval arithmetic over rotations.The covering procedure subdivides rotation-parameter boxes until every box has a validated lower bound.Source: §7 · pp. 17–18

Certification status

  • The rational finite-tolerance match does not establish that c′ belongs to the exact fiber or to a particular connected component.The distance bounds do establish that c′ is distinct from both c and its mirror τc.Source: §7 · p. 18

Construction

  • Numerical continuation found the candidate decoy, but the certificate verifies the final rational pair independently of that search.The continuation is explicitly not part of the certificate.Source: §7 · p. 18

Certified L = 6 benchmark

  • The componentwise relative match is accepted at tolerance 10^-14, and the two surfaces’ band powers agree to the same precision.No component of Φ6(c) vanishes, and the band-power agreement implies their energies agree in every band to that precision.Source: §7 · p. 18
  • With f0=24, both radial functions exceed 4 on the sphere.Figure 2 renders both as radial surfaces with the same constant band.Source: §7 · p. 18, Figure 2 · p. 19
  • All 32 Φ6 components match within relative tolerance 2 · 10^-18, while δ(c,c′) is at least 0.87 times ∥c∥.Figure 2 depicts the enrolled surface and the decoy after rotation R0.Source: §7 · p. 17, Figure 2 · p. 19

8. Task-specific certification and disclosure

Task intervals measure what a descriptor reveals about a rotation-invariant task, not just about shape. Exact collisions or local fiber variation can make task values ambiguous, while a first-order test does not settle global ambiguity.

Task information

  • At exact tolerance, the smallest worst-case scalar-estimation error is half the task interval’s width, attained by its midpoint.At exact tolerance, every task interval is a singleton exactly when the task factors continuously through the descriptor.Source: §8 · pp. 19–20
  • An exact descriptor collision makes any descriptor-based predictor give both shapes the same value, so differing task values prevent it from being correct for both.At positive tolerance, the impossibility applies when both shapes are consistent with a common observation, not merely when their descriptors are close.Source: §8 · p. 20

Differential test for task ambiguity

  • If the task derivative is nonzero along a descriptor-fiber direction, every neighborhood of a regular point contains same-fiber shapes with task values on opposite sides of its value.The conclusion also holds within an admissible set containing a neighborhood of that point.Source: §8 · p. 20
  • The rank test detects first-order task variation only; it does not measure its size or rule out jumps between disconnected fiber components.At L = 16, full local rank does not prevent the mirror pair from having opposite values of χ.Source: §8 · p. 20

9. Volume disclosed by the even spectral pool

At fixed constant radial term, the even spectral pool determines modeled enclosed volume exactly, regardless of shape-fiber size. The section also gives error qualifications and an exact example where identical descriptors and volume fail to settle clearance.

9. Volume disclosed by the even spectral pool

  • At fixed ρ0, an explicitly computable vector of even-pool invariants determines enclosed volume exactly for every positive modeled radial function.Every exact fiber among positive models has zero volume-task width, irrespective of its shape dimension or diameter.Source: §9 · p. 21

9.1. Normalization and exact coefficient computation

  • Rational calibration computes the normalization coefficients, and the accompanying program propagates componentwise absolute error bounds but does not test descriptor realizability or validate supplied error bounds.The calibration uses rational arithmetic only.Source: §9.1 · p. 22

9.2. Descriptor and radial-model error

  • Modeled-volume error must account for descriptor and radial-model error; physical volume additionally depends on truncation and acquisition errors.An uncertain constant term must be included in the radial error or bounded separately.Source: §9.2 · p. 22

9.3. An exact clearance test with identical volume

  • Two positive radial surfaces with ρ0 = 3 have the same even pool and volume, 216π/5, yet opposite outcomes for containment in the origin-centered ball of radius 9/2.The example fixes the reference origin and imposes no centroid normalization.Source: §9.3 · p. 23
  • For that exact descriptor value, the clearance-task interval has width at least 5/12, forcing every scalar estimator’s worst-case outer-radius error to be at least 5/24, while volume’s exact task width is zero.The two objects are noncongruent even under O(3).Source: §9.3 · p. 23

10. Admissible shapes and a representation audit

The audit framework specifies an admissible shape model, geometric bounds, and an error budget before drawing task conclusions. It is intended for evaluators of documented representations, with unresolved cases kept distinct from certificates.

10.1. Origin, scale, and the feasible domain

  • The admissible domain can impose coefficient-norm and radial-bound constraints, but positivity is an assumption because truncating a positive function need not preserve positivity.A declared origin and a centroid convention differ; recentering generally changes the radial function and its harmonic coefficients.Source: §10.1 · pp. 23–24

10.2. A computable geometric bound

  • A computable bound on the radial perturbation gives a conservative outer-radius task certificate from coefficient norm; the stated positivity condition is sufficient, not necessary.Optimizing the task itself can yield a substantially tighter bound.Source: §10.2 · p. 24

10.3. An error budget for the actual representation

  • Descriptor error can be budgeted from a derivative bound, tuple-fit error, and stored-value error, while truncation error is propagated separately.If acquisition bounds hold only with stated probability, the resulting certificate has that probability rather than being unconditional; failure to find a conflicting shape does not certify its absence.Source: §10.2 · p. 24, §10.3 · p. 24, §10.4 · p. 25

10.4. Use in independent evaluation

  • The framework targets evaluators using one documented representation, admissible model, calibrated error budget, and task requirement; it does not treat an unspecified field observation as its input.A comparison against complete geometry does not automatically inherit a descriptor collision.Source: §10.4 · p. 25
  • A focused validation compares band powers, the even pool, and a task-augmented representation under matched acquisition and preprocessing, reporting task bounds, conflicting decisions, unresolved cases, runtime, and storage.For shared records, the audit also considers multiple records jointly and whether one task measurement is preferable to a larger invariant pool.Source: §10.4 · p. 25

11. A case study: asteroid (101955) Bennu

The Bennu case study applies the even spectral pool to a degree-six model and finds that matching descriptors preserve modeled volume but leave handedness and keep-out clearance unresolved. The clearance interval spans at least 7.416 m for the specified tolerance.

11. A case study: asteroid (101955) Bennu

  • The audit uses the radar shape model of Bennu, a closed triangulated surface with 1,348 vertices and 2,692 facets, whose origin is the center of mass.The origin lies inside the polyhedron's kernel, so the surface is star-shaped about that origin; the degree-six tuple is a rounded model approximation.Source: §11 · pp. 25–26
  • For L = 6, Corollary 9.2 bounds volume error by 5.8% of Vmesh, while the actual error is 0.063%.The table's final two columns combine truncation and quadrature errors without a validated bound.Source: Table 2 · p. 26, §11 · p. 27
  • The decoys d+ and d− were found by numerical continuation along the level set of Φ6 through c, followed by high-precision Newton iteration and rounding.Their descriptor components differ from those of c by at most 4 · 10^-19 in relative terms.Source: §11 · pp. 26–27
  • At componentwise relative descriptor tolerance 10^-14, the keep-out-radius interval is at least 7.416 m wide, while the even pool fixes modeled volume but misses handedness.The mirror matches all 32 components exactly and lies at orbit distance at least 5 m; odd bispectra determine handedness. The clearance requirements at 277 m and 273 m distinguish c, d+, and d−, respectively.Source: §11 · pp. 26–27
  • The reported Bennu results concern a degree-six radar model with vertex spacing of about 25 m, not the asteroid's finer-scale surface.The continuation imposes no centroid constraint; the audit holds the radar model's center-of-mass origin fixed rather than recentering each candidate.Source: §11 · p. 27

12. Conclusion

The conclusion distinguishes geometric ambiguity from task utility and attribute confidentiality. In the Bennu audit, the even pool fixes modeled volume but misses handedness and leaves keep-out radius uncertain by more than 7 m.

12. Conclusion

  • The fiber radius measures geometric ambiguity, but does not by itself establish task utility or attribute confidentiality.Continuous, discrete mirror, finite-precision, and task ambiguity are distinct; the spectral pool can leave shape undetermined while fixing volume.Source: §12 · p. 28
  • An explicit clearance pair shows that a second geometric decision can remain indeterminate from the same descriptor, beyond the volume attribute it fixes.The conclusion frames task intervals, observer-centered reconstruction bounds, and combined-release analysis as tools for evaluating what a retained representation certifies and discloses.Source: §12 · p. 28
  • In the Bennu audit, the even pool fixes modeled volume, misses handedness, and leaves keep-out radius uncertain by more than 7 m at tolerance 10^-14.The conclusion presents these together as an example of distinct ambiguity and disclosure phenomena.Source: §12 · p. 28
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