BESTAnP: Bi-Step Efficient and Statistically Optimal Estimator for Acoustic-n-Point Problem

Chao Yang; Guangyang Zeng; Hongxu Zhao; Junfeng Wu; Lingpeng Chen; Wenliang Sheng; Yunling Shao; Yuze Hong; Ziyang Hong

REVIEW 3 major objections 5 minor 28 references

BESTAnP: Bi-Step Efficient and Statistically Optimal Estimator for Acoustic-n-Point Problem

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The acoustic-n-point problem is solved in closed form by separating ranges and azimuths, with one Gauss-Newton step reaching the Cramer-Rao bound.

desk verdict Real new AnP solver with strong experiments, but the optimality proof is outsourced to an unreviewed preprint and the coplanar degeneracy is unaddressed. read the letter →

arxiv 2411.17521 v1 pith:RYLCBSO7 submitted 2024-11-26 cs.RO

Wenliang Sheng , Hongxu Zhao , Lingpeng Chen , Guangyang Zeng , Yunling Shao , Yuze Hong , Chao Yang , Ziyang Hong

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Junfeng Wu

This is my paper · ORCID

classification cs.RO

keywords acoustic-n-point2Dforward-lookingsonarposeestimationrange-basedlocalizationbiaseliminationconsistentestimatorCramer-Raolowerboundodometry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that estimating a sonar's full six-degree pose from distance and azimuth measurements of known 3D points does not require iterative search. It argues that distance measurements alone determine the sonar's position through range-based localization, and that azimuth measurements then determine its rotation through an eigendecomposition, after correcting for measurement noise bias. A single Gauss-Newton step from this closed-form starting point is claimed to produce an estimator whose covariance approaches the Cramer-Rao lower bound as the number of points grows. If correct, this makes real-time acoustic pose estimation practical on low-power onboard computers and enables sonar-only odometry in unknown environments.

What carries the argument

The carrying object is the $6 \times 6$ matrix $\mathbf{Q}_{BE} = (1/n)\mathbf{B}^\top\mathbf{B} - \hat{\sigma}_\theta^2 \mathbf{S}$, where the rows of $\mathbf{B}$ are $[\tan\theta_i(\mathbf{W}p_i - \hat{\mathbf{t}}_{BE})^\top,\ (\hat{\mathbf{t}}_{BE} - \mathbf{W}p_i)^\top]$ and $\mathbf{S}$ contains the sample scatter of the translated 3D points. Its noise-free counterpart $\mathbf{Q}^*$ has a nullspace containing $\mathbf{r}^* = [\mathbf{r}_1^{*\top},\ \mathbf{r}_2^{*\top}]^\top$, so the unit eigenvector of the smallest eigenvalue of $\mathbf{Q}_{BE}$ is the consistent estimate of the rotation rows; subtracting $\hat{\sigma}_\theta^2 \mathbf{S}$ removes the bias caused by angle noise. Translation comes separately from the bias-eliminated linear least-squares problem $\min_{\mathbf{x}\in\mathbb{R}^4}\|\mathbf{A}\mathbf{x}-\mathbf{b}\|^2$ with $\mathbf{A} = [-2\mathbf{W}p_i^\top,\ 1]$ and $b_i = d_i^2 - \|\mathbf{W}p_i\|^2$.

What would settle it

Run the method on synthetic scenes where all 3D points lie in a single plane while the sonar looks at them with realistic pose and noise levels. If BESTAnP's translation RMSE no longer decreases at the predicted rate with more points, or the rotation estimate stays biased while the plane geometry is held fixed, the non-coplanarity premise is doing the work; if the RMSE still tracks the Cramer-Rao bound on coplanar configurations, the premise can be relaxed.

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Extended reading notes

Core claim

The paper's central claim is that the measured partial spherical coordinates split the acoustic-n-point problem cleanly: the true distance $d_i^* = \|\mathbf{W}p_i - \mathbf{t}^*\|$ makes translation a range-only localization problem, while the azimuth relation $\tan\theta_i^* = \mathbf{r}_2^*(\mathbf{W}p_i - \mathbf{t}^*) / \mathbf{r}_1^*(\mathbf{W}p_i - \mathbf{t}^*)$ turns rotation into a linear null-space problem once translation is fixed. The paper constructs a bias-eliminated matrix $\mathbf{Q}_{BE} = \mathbf{Q} - \hat{\sigma}_\theta^2 \mathbf{S}$ whose smallest eigenvector yields the first two rows of the rotation matrix consistently, then projects the result onto $SO(3)$. It proves $\sqrt{n}$-consistency of both the translation and rotation estimators and shows that a single Gauss-Newton iteration from this start is asymptotically equivalent to the maximum-likelihood estimate, so the covariance reaches the Cramer-Rao lower bound as $n$ grows.

Load-bearing premise

The central claim rests on the 3D points being spatially non-coplanar and in generic positions, because the translation step needs a unique sphere intersection and the rotation step needs the azimuth matrix's nullspace to be exactly one-dimensional; near-coplanar point sets, which occur naturally with walls and floors underwater, are not analyzed.

Editorial extensions

If this is right

Full six-degree sonar pose estimation becomes practical on embedded platforms, since the closed-form core uses only linear least squares and an eigendecomposition, with a single Gauss-Newton refinement step.
The accuracy of acoustic pose estimation approaches the Cramer-Rao lower bound in the large-point regime, giving users a statistically grounded baseline rather than an ad hoc iterative solution.
Sonar-only odometry becomes feasible: the paper demonstrates trajectories estimated by alternately running AnP pose tracking and 3D triangulation without external sensors.
Existing AnP methods that iterate on part of the translation or rely on planar or weak-perspective approximations are no longer needed for full-pose accuracy in generic scenes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension the paper leaves implicit is a minimal-point version of the AnP solver: the nullspace structure of $\mathbf{Q}_{BE}$ suggests that a small set of correspondences could seed RANSAC-style robust estimation, analogous to minimal solvers in the PnP literature.
The non-coplanarity premise means planar underwater structures such as walls, floors, and flat ship hulls are a degeneracy risk; the paper does not analyze this regime, so a practical system would need to detect and fall back to constrained models for such scenes.
The decoupling logic may transfer to any range-and-single-angle sensor, such as radar or ultrawideband localization, where distance and one angle are measured but elevation is unavailable; the same two-step bias-eliminated estimator could apply after adapting the measurement model.
The paper's comparison treats angle noise as added to $\tan\theta$; for sonars with wide azimuth apertures or high noise levels, the equivalence to direct angle noise weakens, so a useful extension would be a bias correction that works directly on $\theta$ residuals.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, the paper does deliver a genuinely new algorithmic result: it decouples the AnP problem into a range-only translation estimate and an azimuth-only rotation estimate, producing a closed-form six-degree pose. That is the first such solver in the AnP literature, and the experimental section backs it up — Monte Carlo scaling, pool tests, and an odometry integration, all with open-source code. The claimed speedup (over 10x, about 1 ms on a Pi 5) is plausible from the algorithm's structure.

Second, the headline "statistically optimal" claim is not actually proved here. Theorem 3, the single-Gauss-Newton optimality result, is deferred to the authors' own unpublished preprint [27], and the CRLB curves appear without derivation. The paper is honest about this, but it means you cannot take the optimality label on faith. The consistency part (Theorem 2) is proved in the appendices, and the translation step is directly from [26], which is a published, peer-reviewed paper.

The real soft spot is degeneracy. The translation step inverts A^T A, which requires the 3D points to be non-coplanar. The paper states this in Section IV-A, but the abstract and problem statement do not. For planar or near-planar scenes — walls, floors, which are common in sonar — the translation estimate is singular or ill-conditioned, and the whole pipeline breaks. The stress-test note is right that the failure mechanism is rank deficiency in A, not an enlarged nullspace of Q*. This is a genuine limitation and should be analyzed or at least discussed. The experiments carefully use non-coplanar structures, so the reported results are not misleading, but the scope is narrower than the abstract implies.

Minor issues: the noise model adds noise to tan(theta) rather than theta; the paper justifies this by simulation, which is fine. The single-GN refinement means "closed-form" is slightly generous, but the initial estimator is closed-form, so that's more a wording quibble.

Overall, this is a useful, well-executed paper. It deserves peer review. I would send it to a robotics or estimation venue, with a request that the authors either prove Theorem 3 in the paper or clearly mark it as a dependency, and add a degeneracy discussion.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes BESTAnP, a two-step estimator for the acoustic-n-point (AnP) problem. The first step estimates the sonar translation from distance-only measurements via a bias-eliminated linear least-squares formulation, and the second step estimates the rotation by eigendecomposition of a bias-corrected Gram matrix built from azimuth observations. A single Gauss-Newton iteration is then applied to refine the pose. The paper claims that the two-step estimator is √n-consistent, that one GN iteration yields an estimator with the same asymptotic distribution as the maximum-likelihood estimator, that the covariance approaches the Cramér-Rao lower bound, and that the method is the first closed-form six-degree-of-freedom AnP solver. The claims are supported by Monte Carlo simulations, a Raspberry Pi timing study, real pool experiments, and an embedded sonar-based odometry demonstration.

Significance. If the technical claims hold, BESTAnP would be a practically valuable AnP solver: it is non-iterative up to a single GN refinement, has an explicit consistency analysis, and is demonstrated to be over ten times faster than state-of-the-art methods while matching their accuracy. The manuscript is commendable for shipping open-source code and for validating the algorithm in both synthetic and real underwater scenarios, including a sonar-only odometry pipeline. The main caveats are that the headline statistical-optimality result is not proved in this paper, the CRLB is used without derivation, and the non-coplanarity/generic-position assumption behind the closed-form solution is stated only locally and is not analyzed as a degeneracy limitation.

major comments (3)

[Section IV-C, Theorem 3] The closed-form translation step inverts A^T A in Eq. (6), and the rotation step requires rank(B*)=5 in Eq. (9). The manuscript states, immediately before Eq. (6), that the 3D points must be spatially non-coplanar, and it assumes 'generic positions' after Eq. (9), but the abstract and Section III present the six-degree closed-form solution and the asymptotic-optimality claim without these preconditions. For coplanar point sets, such as a sonar observing a wall or floor, A is rank-deficient and \hat t_BE in Eq. (7) is not uniquely defined; for near-coplanar sets the estimator is ill-conditioned. This is a load-bearing restriction on the central claim. Please state the assumption in the problem formulation and abstract, and add a degeneracy/conditioning analysis or a fallback procedure for planar scenes.
[Section V-A, Figs. 2–6] Theorem 3 is the basis for the 'statistically optimal' and 'covariance approaching the CRLB' claims, but its proof is not contained in this manuscript. The text says 'For the proof of Theorem 3, please refer to the proof of Theorem 7 in [27]', where [27] is the authors' own arXiv preprint on camera-robot pose estimation from points and lines. Because the observation model, parameterization, and estimation problem differ from the AnP likelihood (5), the reduction is not immediate. Please either provide a self-contained proof of Theorem 3 or state precisely the mapping from the AnP problem to Theorem 7 of [27], including all regularity conditions that must be verified.
[Section IV-A, Eq. (7)] The CRLB curves in Figs. 2–6 are displayed without a definition or derivation. The paper does not give the CRLB expression for the observation model (4), nor does it state the conditions under which the single-GN estimator attains it. Since 'statistically optimal' is a headline contribution, please include the CRLB formula (or a precise citation with the exact expression) and explain how it is computed for the plotted experiments.

minor comments (5)

[Algorithm 1] The first sentence reads 'We has introduced the noise-free sonar measurement model'; it should be 'We have introduced'.
[Appendix A] The input list includes only 'azimuth angle measurements tan θi', but step 4 determines the sign by checking cos θ_i and the reprojected cos θ̂_i. Please specify whether the signed azimuth angle (or its cosine) is available as input, since tan alone does not determine the sign of cos.
[Fig. 8 caption] The proof of Lemma 1 refers to 'Lemma 3' before Lemma 3 has been stated (it appears in Appendix B); please reorder or renumber the lemmas for readability.
[Section V-A.3] The caption contains the typo 'Dual-palne'; it should be 'Dual-plane'.
The sentence 'the App method performs even better than the CRLB' should be clarified as an effect of bias in the App estimator, since the CRLB is a bound for unbiased estimators; otherwise the statement may confuse readers.

Circularity Check

1 steps flagged · score 4.0 of 10

The AnP-specific decoupling is independent, but the paper's headline statistical-optimality claim is delegated to a theorem from the authors' own prior work, making the support partially self-citational.

self citation load bearing [Section IV-C, Theorem 3]
"For the proof of Theorem 3, please refer to the proof of Theorem 7 in [27]. Theorem 3 implies that ˆRGN and ˆtGN have the same asymptotic property that ˆRML and ˆtML possess. In other words, ˆRGN and ˆtGN are statistically optimal and can asymptotically reach the CRLB as n increases."

The paper's second headline contribution—statistical optimality with covariance approaching the CRLB—is not proved in this manuscript. The proof is entirely delegated to Theorem 7 of [27], which is prior work by the same research group. The assumptions and regularity conditions of that theorem are not restated or verified for the specific AnP maximum-likelihood problem (5), including its noise model on tan θ and its Gauss-Newton Jacobian in Appendix C. The central claim therefore rests on a self-citation chain rather than on a self-contained derivation. This is load-bearing because the abstract and introduction promote the CRLB-approaching property as a main result, even though the AnP-specific decoupling and the closed-form estimators themselves are independent contributions.

full rationale

BESTAnP's core AnP-specific derivation is not circular: the translation estimator (6)-(7) is constructed from range-only least squares, the rotation estimator (9)-(11) is constructed from azimuth-only least squares plus eigendecomposition, and the bias-eliminated matrix QBE = Q − σ̂θ² S is derived from the measurement statistics rather than defined to yield the target pose. The resulting closed-form estimates are benchmarked against external methods and against the CRLB, so the main algorithmic content has independent substance. No fitted parameter is renamed as a prediction, and no known result is merely relabeled. The most serious concern is Theorem 3, where the optimality result is imported by reference to Theorem 7 of the authors' own prior paper [27] without proof or condition checking in this manuscript; Theorem 1 and the supporting lemmas from [26] are also self-citations, though they concern standard range-localization consistency and are less central to the novelty. These self-citations raise the circularity score to 4 but do not make the whole derivation equivalent to its inputs. The degeneracy of coplanar point sets is a correctness and robustness limitation, not a circularity, so it is not counted here.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The algorithm itself has no hand-tuned parameters, but the odometry pipeline uses an unspecified outlier-rejection threshold. The main theoretical load is carried by prior results from the authors' own group and by the approximate noise model, both listed above.

free parameters (1)

Outlier rejection threshold (reprojection error) in sonar-based odometry = not stated
Section V-B says outliers are removed using reprojection error but the threshold is not specified; this affects the reported trajectory accuracy in Tables II and IV.

assumptions (5)

domain assumption The 3D points are spatially non-coplanar and in generic position, so A has rank 4 and B* has rank 5.
Section IV-A needs non-coplanar points for translation identifiability in (6)-(7); Section IV-B needs rank(B*)=5 so the rotation nullspace is one-dimensional. Near-planar sonar scenes are not analyzed.
domain assumption Gaussian noise is added to tan theta_i rather than theta_i, with constant variance and independent of distance noise.
Eq. (4) defines this as the measurement model; the ML problem (5), the bias correction (10), and the CRLB claims all depend on it. The paper only approximates real sensor behavior and verifies it by simulation (Fig. 2) and one Gaussianity check (Fig. 9).
domain assumption Theorem 1 of [26], Lemmas 2 and 3 of [26]/[27], and Theorem 7 of [27] are valid as stated.
The paper imports the sqrt(n)-consistency of the translation estimator, the eigenvalue identity, the LLN-type bound, and the single-GN optimality theorem without proof. None are machine-checked; [27] is an unreviewed preprint.
standard math Eigendecomposition and SVD projection are continuous, so sqrt(n)-consistency is preserved.
Used in Theorem 2 and the projection onto SO(3) in Section IV-B; standard perturbation theory.
domain assumption The Cramer-Rao lower bound for model (4) exists and is the relevant lower bound.
CRLB curves are displayed in Figures 4-6 but no formula is derived; the asymptotic optimality statement relies on standard ML regularity conditions and on correct model specification.

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Cite this review

Pith. "Pith review of BESTAnP: Bi-Step Efficient and Statistically Optimal Estimator for Acoustic-n-Point Problem." pith.science (2026). https://pith.science/paper/RYLCBSO7

@misc{pith2026241117521,
  author       = {Pith},
  title        = {Pith review of: BESTAnP: Bi-Step Efficient and Statistically Optimal Estimator for Acoustic-n-Point Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RYLCBSO7}},
  note         = {Machine review of arXiv:2411.17521}
}

read the original abstract

We consider the acoustic-n-point (AnP) problem, which estimates the pose of a 2D forward-looking sonar (FLS) according to n 3D-2D point correspondences. We explore the nature of the measured partial spherical coordinates and reveal their inherent relationships to translation and orientation. Based on this, we propose a bi-step efficient and statistically optimal AnP (BESTAnP) algorithm that decouples the estimation of translation and orientation. Specifically, in the first step, the translation estimation is formulated as the range-based localization problem based on distance-only measurements. In the second step, the rotation is estimated via eigendecomposition based on azimuth-only measurements and the estimated translation. BESTAnP is the first AnP algorithm that gives a closed-form solution for the full six-degree pose. In addition, we conduct bias elimination for BESTAnP such that it owns the statistical property of consistency. Through simulation and real-world experiments, we demonstrate that compared with the state-of-the-art (SOTA) methods, BESTAnP is over ten times faster and features real-time capacity in resource-constrained platforms while exhibiting comparable accuracy. Moreover, for the first time, we embed BESTAnP into a sonar-based odometry which shows its effectiveness for trajectory estimation.

Figures

Figures reproduced from arXiv: 2411.17521 by the authors.

**Figure 1.** Illustration of the AnP problem. Given n 3D points in the world frame and their corresponding 2D measurements in the sonar image, the pose of the FLS with respect to the world frame needs to be estimated. mapping (SLAM) system for autonomous navigation [4], [5]. However, there are difficulties in developing an AnP algorithm, such as the non-convexity of the problem and the lack of efficient feature extraction and as… view at source ↗

**Figure 2.** Effect of different noise-adding schemes. 101 102 103 Point Number 10-4 10-3 10-2 10-1 RMSE CRLBt 0 GN RMSEt 1 GN RMSEt 2 GN RMSEt 5 GN RMSEt CRLBR 0 GN RMSER 1 GN RMSER 2 GN RMSER 5 GN RMSER RMSEt RMSER [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** RMSE comparison under varied noise intensities. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (6 more)

**Figure 5.** Figure 5: RMSE comparison under varied point numbers. [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: When 3D points are tightly clustered, the App algo [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 7.** Figure 7: Comparison of trajectory estimation results from [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 8.** Figure 8: (a) consists of two non-coplanar acrylic plates and 14 steel balls scattered on them; the cube structure shown in [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]

**Figure 9.** Figure 9: Empirical distributions of sonar measurement devia [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]

**Figure 10.** Figure 10: Trajectory estimation in real-world experiments. [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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