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Self-Supervised Multibeam Sonar Restoration via Coherent Forward Modeling

Zixin Wang, James Hopgood, Mike Davies

arXiv:2610.31490v1eess.IVcs.CVeess.SP

TL;DR

Clean reflectivity targets are rarely available, while sonar restoration must address correlated coherent speckle and system blur together. The paper uses a coherent conditional-mean forward model with registered-frame self-supervision and REI regularization; experiments report a favorable balance across forward prediction, speckle suppression, and edge preservation. Real-sonar deblurring remains indirectly evaluated because clean latent reflectivity ground truth is unavailable.

  • Problem

    Blind-spot self-supervision is difficult for sonar because spatially correlated speckle makes target-noise independence hard to satisfy, while larger masks remove useful local context.

    Source: §I · p. 2

  • Method

    The framework trains a latent-reflectivity reconstruction using registered multi-frame observations, a coherent conditional-mean forward model, and REI regularization based on stochastic coherent simulation.

    Source: §I · p. 2

  • Results

    Experiments on Boat and Underwater Structure show improved paired forward prediction and a favorable balance between speckle suppression and edge preservation relative to representative baselines.

    Source: §V · pp. 7–8

  • Takeaways & Limitations

    The framework produces reconstructions more consistent with the physical forward model and less prone to residual speckle or excessive structural smoothing than representative baselines.

    Source: §V · p. 8

    Real-sonar deblurring is evaluated indirectly because clean latent reflectivity ground truth is unavailable; the framework also relies on registered pairs and an approximate local PSF.

    Source: §V · p. 8

Abstract

from arXiv · show

Multibeam imaging sonar produces intensity images degraded by coherent speckle and system-induced blur, while clean reflectivity targets are rarely available. Existing despeckling methods often rely on simplified intensity-domain noise models, synthetic clean targets, or blind-spot self-supervision, and typically do not jointly model deblurring and speckle suppression. We propose a physics-consistent self-supervised framework for joint despeckling and deblurring of multibeam imaging sonar. A stochastic coherent image-formation model yields a conditional-mean forward operator that relates latent scene reflectivity to expected intensity through the squared magnitude of the system point spread function. We then use registered multi-frame sonar sequences to train without clean ground truth images, showing that, under an idealized inter-frame conditional-mean condition, the expected paired loss equals the corresponding forward-prediction loss up to a network-independent constant. To stabilize the ill-conditioned inverse problem, we further adapt robust equivariant imaging using stochastic coherent simulation and introduce a paired forward prediction error metric for ground-truth-free evaluation. Experiments on two real multibeam sonar datasets demonstrate a favourable balance between forward-prediction agreement, homogeneous-region speckle suppression, and preservation of visible structural boundaries compared with representative baselines.

I. INTRODUCTION

Sonar restoration must address coherent speckle and PSF blur together despite scarce clean reflectivity targets. The paper develops a coherent forward model and self-supervised reconstruction framework, with registered-frame learning and PFPE for ground-truth-free evaluation.

I. INTRODUCTION

  • Coherent propagation couples spatially correlated speckle with PSF blur, making intensity-only restoration a joint despeckling and deblurring problem without clean reflectivity ground truth.The coherent PSF mixes returns before intensity detection, so neighboring intensity samples are correlated.Source: §I · p. 1, §II · p. 3
  • Blind-spot methods face a sonar-specific trade-off: masking neighborhoods can reduce correlated-noise leakage but removes local structure needed for restoration.Registered-frame methods also must account for viewpoint change, motion, registration error, speckle decorrelation, and the PSF.Source: §I · p. 2

II. PHYSICAL OBSERVATION MODEL

  • The coherent observation model explains correlated speckle and system blur jointly through coherent propagation and the sonar PSF, rather than imposing correlation with intensity-domain post-filtering.Source: §I · pp. 1–2

I. INTRODUCTION

  • The conditional-mean intensity objective provides a physics-consistent forward-prediction target for learning from detected intensities without clean reflectivity targets.The expected intensity relates scene reflectivity to the squared magnitude of the system PSF.Source: §I · p. 2
  • The proposed framework uses registered multi-frame observations for joint despeckling and deblurring, with stochastic coherent simulation and REI regularization for the ill-conditioned inverse problem.Source: §I · p. 2
  • PFPE provides a ground-truth-free measure of forward-prediction consistency under a fixed paired-frame protocol.Source: §I · p. 2

III. METHODOLOGY

The reconstruction network estimates latent reflectivity from detected sonar intensity without clean ground truth, but the inverse problem is ill-conditioned. Registered frame pairs provide observations of approximately the same scene with different speckle realizations.

A. Problem Formulation

  • The goal is to estimate latent reflectivity from coherent-model intensity observations without clean ground truth, despite stochastic fluctuations and PSF suppression of high-frequency structure.The measurement-domain constraint must be paired with regularization of the latent estimate.Source: A. Problem Formulation · p. 3

B. Paired Observation Construction

  • Selected frames from a short temporal window are registered as pairs that share approximately the same latent reflectivity but have different coherent speckle realizations.Short-interval motion is approximated by a two-dimensional translation estimated using phase correlation.Source: B. Paired Observation Construction · p. 3
  • After registration, the paired frames are assumed to have aligned conditional means; the second frame is denoted y2 for notational simplicity.Source: B. Paired Observation Construction · p. 3

C. Statistical Assumptions and Stochastic Forward Model

The statistical model connects latent reflectivity to expected intensity through a linear conditional-mean forward operator. Paired observations and stochastic coherent simulation use this mean relation to support self-supervised learning and regularization.

C. Statistical Assumptions and Stochastic Forward Model

  • The paired-loss analysis assumes a shared latent reflectivity and approximately aligned conditional means, while exact conditional independence is sufficient but stronger than this condition.Source: C. Statistical Assumptions and… · p. 3
  • Under the local shift-invariant approximation, the conditional mean is a linear function of reflectivity despite the stochastic observation model being nonlinear.The operator A is induced by the coherent observation model.Source: C. Statistical Assumptions and… · p. 4
  • A J-look coherent simulator preserves the conditional mean while averaging independent speckle fields; increasing J makes its output converge in mean square to A(z).J=1 gives a single-look simulated observation.Source: C. Statistical Assumptions and… · p. 4

D. Physics Consistency Learning

The physics-consistency loss trains the network through measurement-domain predictions because clean reflectivity samples are unavailable. Under an idealized inter-frame conditional-mean condition, it matches the projected conditional mean up to a network-independent constant, but does not uniquely determine reflectivity.

D. Physics Consistency Learning

  • The network predicts the registered measurement through Afθ(y1), allowing training without clean reflectivity samples.The objective uses the registered-pair assumptions and conditional-mean relation.Source: D. Physics Consistency Learning · p. 4
  • Under E[y2 | x, y1] = Ax, the paired loss equals the forward-prediction loss plus a network-independent constant.The condition makes the cross term in the squared-loss expansion vanish.Source: D. Physics Consistency Learning · pp. 4–5
  • Minimizing the loss matches the conditional mean projected into the measurement domain, rather than fitting one noisy observation.Source: D. Physics Consistency Learning · p. 5
  • The reflectivity solution need not be unique, and the blurred operator weakly determines high-frequency components, motivating equivariant regularization.This ambiguity remains even when A is full rank if it is severely ill-conditioned.Source: D. Physics Consistency Learning · p. 5

E. Robust Equivariant Imaging with Stochastic Forward Simulation

The method adapts robust equivariant imaging to coherent sonar speckle to further constrain latent reflectivity. Its REI branch uses stochastic coherent simulation, and its regularization strength is controlled in the final objective.

E. Robust Equivariant Imaging with Stochastic Forward Simulation

IV. EXPERIMENTS

Real-sonar restoration is evaluated without clean reflectivity ground truth using paired-frame forward-prediction consistency and complementary indicators of smoothing and edge preservation. These measures provide proxy evidence, not direct reconstruction-accuracy measurements.

IV. EXPERIMENTS

  • Two real multibeam sonar datasets are evaluated using paired-frame PFPE alongside ENL, EPI, and visual evidence.Deblurring is assessed indirectly through forward-model consistency and preservation of visible structures.Source: §IV · p. 5

A. Evaluation Metrics

  • PFPE scores held-out prediction of a registered paired observation under the same sonar operator for every method, not full-reference quality or each baseline’s native target.Under the idealized conditional-mean condition, expected PFPE equals expected conditional-mean measurement error plus a reconstruction-independent constant.Source: A. Evaluation Metrics · p. 5
  • ENL indicates homogeneous-region speckle suppression and may favor oversmoothing, while DEPI measures edge-strength deviation; neither measures reconstruction accuracy without ground truth.Source: A. Evaluation Metrics · pp. 5–6
  • The evaluation fixes homogeneous and edge ROIs from the input observation across methods, reducing output-dependent selection and manual-selection bias.Edge ROIs are selected separately by Sobel gradient energy.Source: A. Evaluation Metrics · pp. 5–6

B. Datasets

On the Boat dataset, the proposed method has the strongest reported forward-prediction agreement while balancing speckle suppression and edge preservation. The experiments use two Tritech Gemini 720ik datasets and a fixed registered-pair protocol.

B. Datasets

  • The datasets use Tritech Gemini 720ik intensity images: Boat contains 1455 frames of cluttered scenes, and Underwater Structure contains 1282 frames of more regular inspection scenes.The sonar outputs 512 beams with a 120° field of view.Source: B. Datasets · p. 6

C. Implementation Details

  • Training, validation, and test blocks are separated before patch extraction, and paired samples remain within each split; all paired-frame experiments use frame gap 1.The final protocol retains 1328 Boat pairs and 1137 Underwater Structure pairs; a four-scale U-Net is used.Source: C. Implementation Details · p. 6

D. Performance on Boat Dataset

  • The proposed method achieves the lowest PFPE and a highest aggregate PFPE improvement of 14.10% on Boat, while increasing ENL to 12.81 and keeping DEPI among the lowest restored results.It suppresses background speckle while preserving clearer separation of high-reflectivity structures; B2UB has a similar aggregate improvement of 13.39% but ENL falls from 8.18 to 7.74.Source: D. Performance on Boat Dataset · pp. 6–7, Figure 3 · p. 7

E. Performance on Underwater Structure Dataset

On Underwater Structure, the proposed method achieves the strongest aggregate forward-prediction result while balancing speckle suppression and edge preservation. Its inference cost matches PI-SSD’s model size and remains faster than both datasets’ acquisition rates; the conclusion notes limits in registration, PSF accuracy, and direct deblurring evaluation.

E. Performance on Underwater Structure Dataset

  • 4.36% aggregate PFPE improvement is the highest on Underwater Structure, with relatively low DEPI and clearer structural continuity than the compared methods.Higher-ENL methods such as SS-BSN, PI-SSD, and LG-BPN show much larger DEPI; the proposed method also suppresses background speckle while preserving distinguishable boundaries.Source: E. Performance on Underwater Structure… · p. 7, Table I · p. 8

F. Computational Cost

  • At 0.043 s per image, inference is approximately 23 frames per second, faster than the Boat and Underwater Structure acquisition rates.The proposed method has the same parameter count and MACs as PI-SSD; its paired-observation branch, simulator, and REI loss are training-only.Source: F. Computational Cost · p. 7

V. CONCLUSION

  • The framework formulates restoration as a physics-consistent self-supervised inverse problem and uses sonar-specific forward simulation for robust equivariant regularization.Experiments on Boat and Underwater Structure show better paired forward prediction and a balance between speckle suppression and edge preservation than representative baselines.Source: §V · pp. 7–8
  • The framework relies on registered pairs and an approximate local PSF, and real-sonar deblurring is evaluated indirectly without clean latent-reflectivity ground truth.Severe registration errors, dynamic scenes, and forward-model mismatch may affect it.Source: §V · p. 8

Supplementary Material

The supplementary material checks whether registered frame pairs are aligned and practically suitable for training and PFPE evaluation. Its diagnostics balance reduced inter-frame correlation against geometric reliability, while treating correlation as a suitability check rather than proof of independence.

S1. IMAGE REGISTRATION VALIDATION

  • Registration increases overlap of persistent scene structures in the Boat example, supporting registered pairs for training and PFPE evaluation.Fig. 6 encodes the reference frame in red, the paired frame in green, and overlap in yellow.Source: Figure 6 · p. 11, S1. IMAGE REGISTRATION VALIDATION · p. 11

S2. FRAME-PAIR CORRELATION ANALYSIS

  • The supplementary material reports displacement and overlap diagnostics alongside correlation because reduced correlation alone does not guarantee reliable paired observations.Larger frame gaps can reduce common valid support and increase apparent motion and registration uncertainty.Source: S2. FRAME-PAIR CORRELATION ANALYSIS · p. 11, Table IV · p. 12
  • Frame-gap selection trades reduced residual correlation against motion, scene mismatch, and registration uncertainty, so the main comparisons use a shared fixed evaluation gap.The reduced-correlation diagnostic uses the percentage of valid pairs with r < 0.5 and selects the smallest gap reaching at least 50%, subject to displacement bounds.Source: S2. FRAME-PAIR CORRELATION ANALYSIS · pp. 11–12
  • Registered-pair correlation is dataset-dependent and reflects motion, range, geometry, viewpoint, registration residuals, and ROI structure, not only speckle decorrelation.The authors therefore treat it as a practical check of pair suitability, not proof of statistical independence.Source: S2. FRAME-PAIR CORRELATION ANALYSIS · pp. 11–12

S3. RESULTS ON THE BOAT DATASET

The Boat results report improved aggregate forward-prediction agreement while maintaining observed edge strength, and the Underwater Structure results show a similar balance. The supplementary section identifies additional ablations and sensitivity analyses but provides no findings for them in these passages.

S3. RESULTS ON THE BOAT DATASET

  • On Boat, the proposed method achieves the highest aggregate PFPE improvement while keeping EPI close to 1.The additional metrics show that SS-BSN and LG-BPN can raise ENL while also producing large DEPI, consistent with edge attenuation or oversmoothing.Source: S3. RESULTS ON THE BOAT DATASET · p. 12

S4. RESULTS ON THE UNDERWATER STRUCTURE DATASET

  • On Underwater Structure, the proposed method has the highest aggregate PFPE improvement while maintaining EPI close to one.The section also includes a representative qualitative comparison in Fig. 8.Source: S4. RESULTS ON THE UNDERWATER… · p. 12

S5. ABLATION AND SENSITIVITY ANALYSIS

  • The additional Boat analyses cover component ablations and sensitivity to registration, REI transformation, regularization weight, rotation magnitude, simulated looks, and frame gaps.They are interpreted within each analysis block rather than as main cross-method rankings.Source: S5. ABLATION AND SENSITIVITY ANALYSIS · p. 12

A. Ablation Analysis

The ablation results indicate that physics-consistency loss alone can improve paired forward prediction but does not yield meaningful reflectivity reconstruction; REI regularization addresses the inverse problem’s ill-conditioning. Sensitivity analyses show trade-offs among prediction, smoothing, and structural preservation.

A. Ablation Analysis

  • Physics-consistency loss alone improves paired forward-prediction score but yields a degenerate reflectivity solution; adding REI regularization is needed to mitigate the blurred operator’s ill-conditioning.The loss constrains measurement-domain predictions but weakly determines high-frequency or weakly observed latent-reflectivity components; removing registration also performs poorly in PFPE.Source: A. Ablation Analysis · p. 12, Table V · p. 13

B. Sensitivity Analysis

  • Moderate geometric perturbations give the best balance, while very large rotations conflict more with the locally shift-invariant PSF approximation and sonar’s anisotropic range-azimuth structure.The transformation and rotation-angle blocks fix α = 0.05 and J = 8.Source: B. Sensitivity Analysis · p. 13
  • Increasing REI weight raises ENL, but excessive regularization reduces PFPE improvement and attenuates edges, as reflected by lower EPI and larger DEPI.The REI-weight block fixes J = 8 and the maximum rotation angle at 20°.Source: B. Sensitivity Analysis · p. 13
  • Very small simulated-look counts yield poorer PFPE and larger edge-strength deviation despite higher ENL; increasing J improves PFPE as the simulator approaches the conditional-mean model.The reported setting is a balance across PFPE, ENL, EPI, and DEPI rather than an optimum on one metric.Source: B. Sensitivity Analysis · p. 13

C. Frame Gap Analysis

The frame-gap analysis holds the model and optimization settings fixed while varying training gap under a common 90% pair-retention target. Gap 1 provides the most favorable balance in the reported analysis.

C. Frame Gap Analysis

  • The analysis fixes α = 0.05, J = 8, and maximum rotation angle 20◦, while setting gap-specific displacement tolerances for 90% pair retention.Model and optimization settings are held fixed across training gaps.Source: C. Frame Gap Analysis · p. 13, Table VIII · p. 16
  • Boat and Underwater Structure comparisons show speckle suppression alongside preservation of local structures, clearer transitions, and structural continuity.The Boat visuals highlight high-reflectivity structures and shadow transitions; the Underwater Structure visuals highlight boundaries and weak internal reflectors.Source: Figure 7 · p. 14, Figure 8 · p. 15
  • PFPE scores are not directly comparable as absolute image-quality measures across test gaps; the main comparisons use common test gap 1.At a fixed gap, common test pairs and image support preserve the ranking of mean PFPE.Source: C. Frame Gap Analysis · pp. 13–14
  • Table VIII shows empirically that the noisy PFPE baseline rises from 1.59×10−4 at test gap 1 to 2.35×10−4; imp is gap-dependent because it is normalized by noisy paired-frame discrepancy.Across test gaps, both the noisy baseline and paired prediction error can change.Source: C. Frame Gap Analysis · pp. 13–14
  • Increasing training gap from 1 to 2 and 3 raises ENL from 12.40 to 15.31 and 16.48, while DEPI rises from 5.17% to 15.82% and 20.91%.The passage interprets this as stronger smoothing with larger deviation from observed edge strength.Source: C. Frame Gap Analysis · p. 14
  • At test gap 3, train-gap 3 yields 24.98% aggregate improvement versus 24.50% for train-gap 1, but DEPI is 20.91% versus 5.17%.Across test gaps 1 and 2, the gap-1-trained model has the highest PFPEagg improvement and the lowest DEPI.Source: C. Frame Gap Analysis · p. 14, Table VIII · p. 16

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