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Off-Grid Point-Scatterer Localization from Sparse Limited-Aperture Data via Hankel Completion

Jingzhi Li, Xiliang Lu, Juntao You

arXiv:2609.08665v1math.NAcs.NA

TL;DR

Sparse, limited-aperture measurements make off-grid localization of planar point scatterers a structured recovery problem. The paper uses aperture-admissible Hankel completion followed by MUSIC localization, and proves exact and stable recovery with a localization bound under stated conditions.

  • Problem

    The paper addresses recovering off-grid locations of sparse planar scatterers from measurements available only over a restricted receiver geometry.

    Source: §1 · p. 1

  • Method

    The framework completes missing coherent data with an aperture-admissible low-rank Hankel pencil, then applies MUSIC for localization.

    Source: §1 · p. 2

  • Results

    Under verifiable geometric and minimum-separation conditions, exact recovery, bounded-noise stability, and an explicit localization bound are established with O(s log^4 |S|) samples.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The theory provides quantitative criteria for selecting sample-efficient aperture-admissible pencils based on pencil geometry and source conditioning.

    Source: §1 · p. 2

    The analyzed forward model is the Born point-scatterer model; noise, finite-radius effects, and multiple scattering are treated as perturbations.

    Source: §2.1 · p. 3

Abstract

from arXiv · show

We study the localization of $s$ distinct planar point scatterers from sparsely sampled single-incidence, single-frequency acoustic far-field data over a limited receiver aperture. Under the first Born approximation, the coherent data form a two-dimensional off-grid exponential sum and admit a low-rank Hankel representation. A limited aperture generally admits multiple aperture-admissible Hankel pencils with different lifted dimensions, sampling multiplicities, and Vandermonde conditioning. Hence, the choice of the Hankel pencil affects both data completion and subsequent localization. In this work, we introduce a unified framework for structured completion and off-grid localization based on general aperture admissible Hankel pencils. We establish exact recovery in the noiseless case and stable recovery under bounded noise from a number of random aperture samples that is linear in $s$, up to factors determined by the pencil geometry, source conditioning, and logarithmic terms, together with an explicit localization bound in terms of the completion error. The analysis separates the contribution of the pencil geometry from the conditioning induced by the source configuration and provides verifiable sufficient conditions for stable completion and MUSIC resolution. These results yield quantitative criteria for selecting sample efficient Hankel pencils under a prescribed physical aperture. Numerical experiments on connected and disconnected apertures demonstrate the effects of pencil geometry, sampling, and noise.

1 Introduction

The paper targets off-grid localization of sparse planar scatterers from sparse measurements over a restricted receiver aperture. It develops aperture-aware Hankel completion and localization, with recovery guarantees and criteria for choosing the pencil.

1 Introduction

  • Aperture geometry and sparse acquisition constrain the available data and the off-grid locations that must be recovered.The setting is single-incidence, fixed-frequency scattering from planar point scatterers; accessible regions may be nonrectangular or disconnected.Source: §1 · p. 1
  • The framework combines structured low-rank completion of missing coherent data with two-sided MUSIC localization and least-squares amplitude estimation.The Hankel pencil is chosen to match the physically accessible aperture.Source: §1 · p. 2
  • Exact recovery, bounded-noise stability, and an explicit localization bound are established, with sample complexity depending on pencil geometry and source conditioning.Under verifiable geometric and minimum-separation conditions, the stated sample bound reduces to O(s log^4 |S|).Source: §1 · p. 2
  • Rectangular decompositions control ΓA,B, while Cartesian cores and minimum separation control Vandermonde conditioning and MUSIC resolution.These bounds provide quantitative criteria for selecting sample-efficient aperture-admissible pencils.Source: §1 · p. 2
  • Numerical experiments on connected and disconnected apertures validate the method and pencil-selection criteria under sparse sampling and noise.The supplied passages do not report specific numerical comparisons.Source: §1 · p. 2

2 Born data model and aperture discretization

The data model represents planar point-scatterer measurements under the first Born approximation. The aperture defines accessible frequencies, while the observed subset supplies the sparse measurements used to recover locations and strengths.

2.1 Planar point-scatterer model

  • Under the first Born approximation, the far-field model is the forward model analyzed, with measurement noise and departures from the Born point-scatterer model included in a perturbation term.The passage identifies finite-radius effects and multiple scattering among those departures.Source: §2.1 · p. 3
  • Two calibrated reference fields can convert intensity measurements into two real equations that determine the complex coherent datum when Im(r1r2) ≠ 0.The subsequent analysis starts from coherent samples, either measured directly or obtained by this preprocessing.Source: §2.1 · p. 3

2.2 Limited aperture and the recovery problem

  • The recovery objective is to estimate the scatterer locations and strengths from the partially observed aperture data.The frequency set is determined by accessible receiver directions and the selected grid spacings.Source: §2.2 · pp. 3–4
  • The receiver aperture induces a finite accessible frequency set, and observations consist of noisy data on a subset of that set.A conical aperture of half-angle θmax gives the disk ∥q∥2 ≤ ω sin θmax.Source: §2.2 · p. 4

3 Partial-aperture data recovery via low-rank Hankel completion

The reconstruction completes data on a Hankel pencil whose index sets respect the physical aperture, then estimates continuous scatterer parameters from the completed matrix. Pencil geometry affects completion guarantees, while the localization stage uses both signal subspaces.

3 Partial-aperture data recovery via low-rank Hankel completion

  • The reconstruction first completes coefficients on an aperture-admissible sum set and then estimates continuous scatterer parameters from the completed Hankel matrix.The section develops the two stages, with recovery and localization guarantees deferred to Section 4.Source: §3 · p. 4

3.1 Aperture-admissible Hankel pencils

  • An aperture-admissible pencil has finite index sets A and B with A + B ⊂ Sap, so every lifted entry corresponds to an accessible frequency.Nonrectangular and disconnected frequency sets can therefore yield nonstandard admissible pencils.Source: §3.1 · p. 4
  • When all scatterer coefficients are nonzero and both Vandermonde factors have full column rank, the aperture-admissible Hankel matrix has rank s.Different pencils can change lifted dimensions, fiber multiplicities, and source-dependent Vandermonde conditioning, which the completion theory retains.Source: §3.1 · p. 4

3.2 Completion and parameter estimation

  • Completion minimizes the nuclear norm of the Hankel lift subject to observed-data constraints, with a noise-tolerant constraint when observation error is bounded.The stated programs are convex; the experiments use a scaled ADMM implementation.Source: §3.2 · p. 5
  • After completion, two-sided MUSIC estimates locations from the rank-s signal subspaces, and least squares estimates amplitudes.The continuous location parameters are subject to periods induced by the frequency grid.Source: §3.2 · pp. 5–6
  • The perturbation result covers ideal continuous MUSIC selection, not search-grid initialization or nonlinear-solver error used in computation.In practice, minimization is initialized on a search grid and followed by local refinement.Source: §3.2 · p. 6

4 Aperture-pencil recovery and localization theory

The section develops recovery and localization guarantees for aperture-admissible Hankel pencils, separating source conditioning from pencil geometry. It gives conditions for completion and MUSIC resolution, then derives geometric criteria for pencil design.

4.1 Recovery parameters

  • The source-dependent parameter is the normalized inverse lower frame bound of the two Vandermonde factors, while Γ_A,B measures energy accumulation along low-multiplicity fibers.For size-balanced pencils decomposable into a fixed number of uniformly scale-comparable rectangles, c_A,B = O(1) and Γ_A,B = O(log^2 N).Source: §4.1 · pp. 6–7

4.2 Completion and localization guarantees

  • A sufficient sample count is O(Γ_A,B μ_1 c_A,B s log^2 N) for exact completion with high probability; the same conditions also yield stability under bounded noise.For decomposable pencils, Γ_A,B = O(log^2 N).Source: §4.2 · p. 7
  • If the Cartesian-core, separation, and translate conditions hold, the sufficient MUSIC condition (4.8) holds uniformly over all separated source sets.The admissible constant depends on A, B, F, s, Δ, Δq1, and Δq2.Source: §4.2 · p. 8

4.3 Verifiable geometry and pencil design

  • A fixed number of uniformly scale-comparable rectangular pieces gives Γ_A,B = O(log^2 N), supporting sample-efficient completion guarantees.The geometric bound separates pencil-dependent factors from source conditioning.Source: §4.3 · p. 8
  • For fixed uniformly scale-comparable pieces, O(μ_1 c_A,B s log^4 N) samples suffice for exact completion with high probability.The stated guarantee assumes Assumption 4.1, nonzero amplitudes, defined μ_1, and the decomposition in Proposition 4.9.Source: §4.3 · p. 8
  • For two cases with A+B⊂Sap, c_A,B=251/66 and 41/12 (both <4), with Γ_A,B≤19C1 log^2(eN) in each; for A+B=Sap, c_A,B=6 and Γ_A,B≤16C1 log^2(eN).Source: §4.3 · p. 9
  • For the two factorizations generating the same aperture, the balanced factorization has the smaller sufficient geometric factor than the unbalanced one.The bounds are Γ_bal ≤ 441/121 C1 log^2(eN) and Γ_un ≤ 441/36 C1 log^2(eN).Source: §4.3 · p. 9

5 Numerical experiments

The experiments evaluate the full completion-and-localization procedure on connected and disconnected apertures, then examine pencil balance, noise sensitivity, and computational time. Results illustrate accurate localization from sparse observations and better recovery with a balanced pencil in the tested comparison.

5 Numerical experiments

  • The study tests the full reconstruction procedure on connected and disconnected apertures, then examines pencil geometry, noise sensitivity, and computational time.Synthetic data use randomly selected admissible indices; success is defined by relative localization error at most 5%.Source: §5 · p. 9

5.1 Localization on connected apertures

  • With 13 of 123 coefficients at 40 dB, two sector-aperture scatterers are accurately localized; with 46 observations at 40 dB, five elliptical-aperture scatterers, including a close pair, are accurately localized.The corresponding reconstructions appear in Figures 3 and 4.Source: §5.1 · p. 10, Figure 3 · p. 10, Figure 4 · p. 10

5.2 Disconnected acquisition windows

  • The construction accommodates disconnected apertures: four separated 3 × 4 windows use 32 observations at 30 dB, and nine separated 3 × 3 windows use 33 observations at 30 dB.Figures 5 and 6 show the acquisition geometry, observations, pencil factors, and localization results.Source: §5.2 · pp. 10–11, Figure 5 · p. 11, Figure 6 · p. 11

5.3 Pencil balance, noise stability, and computational time

  • With the same sum set of N = 441, the balanced pencil reaches reliable recovery with fewer observations and gives smaller localization errors through the transition region.The balance factors are 441/121 for the balanced pencil and 441/36 for the unbalanced pencil; the comparison uses 30 paired trials at 40 dB.Source: §5.3 · p. 11, Figure 7 · p. 12
  • In the four-window experiment, localization error decreases as SNR rises, while computational time increases with sum-set size under the same sampling rule.Figure 8 reports interquartile-range error bars over 30 trials.Source: §5.3 · p. 12, Figure 8 · p. 12

6 Concluding remarks

The paper develops a geometry-adapted Hankel framework for localizing finite planar point scatterers from sparse limited-aperture Born data. It provides recovery and localization guarantees, criteria for pencil selection, and numerical demonstrations; deterministic sampling and more general scattering models remain future work.

6 Concluding remarks

  • The framework incorporates acquisition geometry into Hankel pencil design and quantifies recovery through Vandermonde conditioning, pencil balance, and fiber accumulation.It provides a criterion for selecting aperture-admissible pencils.Source: §6 · p. 12
  • The framework provides exact and stable completion guarantees and a subsequent MUSIC localization bound for finite planar point scatterers.The conclusion describes it as a geometry-adapted framework for sparse limited-aperture Born data.Source: §6 · p. 12
  • Numerical experiments illustrate the proposed design on nonrectangular and disconnected apertures.Source: §6 · p. 12
  • Extensions to deterministic sampling and more general scattering models are left for future work.Source: §6 · p. 13

Declaration on the use of artificial intelligence

The authors disclose using generative AI tools during revision for language editing, structural reorganization, bibliography checks, and notation, formula, and proof consistency checks; they assume responsibility for the content.

Declaration on the use of artificial intelligence

  • During revision, generative AI tools assisted with language editing, structural reorganization, bibliographic cross-checking, and consistency checks, while the authors retained responsibility for all content.Source: Declaration on the use of artificial… · p. 13

A Proofs

The appendix defines matrix inner-product and rank-one operator notation, describes the Hankel subspace basis and associated projections, and gives a formula for projection onto the rank-s tangent space.

A Proofs

  • The Frobenius inner product is defined as tr(X*Y), and X ⊗ Y maps M to ⟨Y, M⟩F X.Source: A Proofs · p. 13
  • Each Pk is a 0-1 partial permutation matrix, and the Gk atoms have disjoint supports that form an orthonormal basis of HA,B.The stated norms are ∥Pk∥2 = 1, ∥Gk∥F = 1, and ∥Gk∥2 = w(k)^−1/2.Source: A Proofs · p. 13
  • The appendix introduces the orthogonal projector onto HA,B and defines the rank-s tangent space using a compact SVD.Source: A Proofs · p. 13
  • The tangent-space projection is given by PT(M) = UU*M + MVV* − UU*MVV*.Source: A Proofs · p. 13

A.1 Technical lemmas

The appendix lists technical lemmas and a sampling condition; it states that estimates are proved in Section A.8.

A.1 Technical lemmas

  • Lemma A.1 states a singular-subspace incoherence result using μ1 from Definition 4.2.Source: §A.1 · p. 13
  • Lemma A.2 assumes Cartesian rectangle pairs, with the G-seminorm formed from the rectangle pair's fibers.Source: §A.1 · p. 13
  • Lemma A.3 states an estimate for every a, b in S using μ1, while Lemma A.5 gives a Bernoulli sampling condition and probability bound.The listed estimates are proved in Section A.8; C_W is identified as a positive numerical constant.Source: §A.1 · pp. 13–14

A.2 Proof of Proposition 4.3

Under the stated source-separation condition, the proof establishes full column rank and a singular-value bound for the Vandermonde factors used in the proposition.

A.2 Proof of Proposition 4.3

  • For each aperture index set, a translated square subgrid yields full column rank and σ_s(V_KC) > 0.9n_C under the assumed separation.The proof takes n_C = 2⌊L_C/2⌋ ≥ L_C/2 and uses that translation changes the Vandermonde matrix only by a diagonal unitary factor.Source: §A.2 · p. 14
  • Both Vandermonde factors therefore have full column rank, which gives the proposition’s claimed bound after substitution into (4.1).Source: §A.2 · p. 14

A.3 Proof of Proposition 4.6

The proof establishes that the MUSIC residual vanishes at the source locations, stays positive elsewhere, and has a uniform lower bound tied to distance from the source set.

A.3 Proof of Proposition 4.6

  • Uniform full column rank of V_A and V_B holds over the Δ-separated configurations when A contains a translate of {0,…,s}², with the B case following by conjugation.Translation leaves the normalized one-sided MUSIC residual unchanged.Source: §A.3 · p. 14
  • For every t outside the source set T, the MUSIC residual R(t) is strictly positive.Degree-at-most-s interpolation polynomials supported on {0,…,s}² establish rank s+1 after adjoining the test steering vector.Source: §A.3 · p. 14
  • The residual has a uniform lower bound c_M min{d_per(t,T), Δ} over the field of view.Local rank conditions yield a distance-based bound near each source; compactness and positivity away from the sources make the constants uniform.Source: §A.3 · pp. 14–15

A.4 Proof of Proposition 4.9

These appendix proofs establish the completion and localization guarantees through weighted estimates, certificate arguments, and perturbation bounds, while deriving supporting geometric and sampling estimates.

A.4 Proof of Proposition 4.9

  • The Proposition 4.9 proof partitions matrix entries by (u,v) blocks and applies weighted Cauchy–Schwarz before summing over k.The resulting estimate is related to (4.5).Source: §A.4 · p. 15

A.5 Proof of Theorem 4.4

  • The certificate argument proves uniqueness of the nuclear-norm completion minimizer with probability at least 1−N^-10.It shows every nonzero feasible perturbation strictly increases the objective, then uses injectivity to conclude the coefficient perturbation is zero.Source: §A.5 · p. 15

A.6 Proof of Theorem 4.5

  • The noisy completion estimate combines certificate inequalities and bounds w_max by min{|A|,|B|}, yielding dependence on s, aperture dimensions, log N, and ε.Source: §A.6 · p. 16

A.7 Proof of Theorem 4.7

  • Under the completion-error condition, the s smallest local minima remain uniquely associated with disjoint source neighborhoods.The proof preserves strong convexity within each neighborhood and keeps its minimum below minima outside; Algorithm 3.2’s selected minima can be matched to those neighborhoods.Source: §A.7 · p. 17

A.8.1 Proof of Lemma A.1

  • Lemma A.1 bounds an inverse-Gram quantity by μ₁s/|A| and gives the corresponding bound for B.Source: §A.8.1 · p. 17

A.8.2 Proof of Lemma A.2

  • For interval index sets, the convolution multiplicity is w_i(t)=min{t+1,a_i,b_i,a_i+b_i−1−t}; the proof groups dyadic levels across at most nine monotonicity pairs.It sums at most 4(1+log |A₀+B₀|)² nonempty dyadic level pairs.Source: §A.8.2 · pp. 17–18

A.8.3–A.8.4 Proofs of Lemmas A.3 and A.4

  • The A.4 estimates expand (A.1) for the kernel estimate and bound its projector expression by three terms; for the first two, nonzero entries overlap at most once per corresponding row or column of Ga.Source: §A.8.3 · p. 18

A.8.5 Proof of Lemma A.5

  • A golfing construction using independent Bernoulli batches produces a certificate while its tangent component decays geometrically.The batch estimates hold with total failure probability at most δ, and the construction yields the stated tangent and normal component bounds.Source: §A.8.5 · pp. 19–21
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