Self-Supervised Multibeam Sonar Restoration via Coherent Forward Modeling

Zixin Wang; James Hopgood; Mike Davies

eess.IVOct 22, 2026

Self-Supervised Multibeam Sonar Restoration via Coherent Forward Modeling

Authors: Zixin Wang, James Hopgood, Mike Davies

Organizations: School of Engineering, University of Edinburgh, Edinburgh EH9 3FG, U.K.

Abstract

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.

Figures & tables

Fig. 1 : Empirical spatial correlation of detected sonar intensity measured from homogeneous ROIs in a representative Boat frame. Correlation is computed along the range and azimuth directions; offsets start from one pixel to exclude the trivial zero-offset self-correlation. Curves show the mean across ROIs, with shaded bands indicating one standard deviation.

Fig. 2 : Overview of the proposed framework. The framework contains four components: a coherent physical observation model, an image-registration module that constructs registered paired observations, a physics-consistency branch that compares the forward projection of the reconstruction with a registered frame, and a robust equivariant imaging branch that regularizes the latent reconstruction. At inference time, only the trained reconstruction network is used and it operates on a single sonar image.

	Boat				Underwater Structure
Method	$\mathrm{PFPE}$ $\downarrow$	$\mathrm{PFPE}_{\mathrm{imp}}^{\mathrm{agg}}$ $\uparrow$	ENL $\uparrow$	$D_{\mathrm{EPI}}$ $\downarrow$	$\mathrm{PFPE}$ $\downarrow$	$\mathrm{PFPE}_{\mathrm{imp}}^{\mathrm{agg}}$ $\uparrow$	ENL $\uparrow$	$D_{\mathrm{EPI}}$ $\downarrow$
Noisy	$1.59\pm 0.36$	$0.00$	$8.18\pm 1.12$	$0.00\pm 0.00$	$6.72\pm 3.06$	$0.00$	$3.38\pm 0.36$	$0.00\pm 0.00$
PPB [ 18 ]	$1.79\pm 0.26$	$-13.06$	$12.73\pm 2.71$	$19.32\pm 2.28$	$7.33\pm 2.34$	$-9.01$	$6.98\pm 1.36$	$27.17\pm 4.16$
SAR-BM3D [ 19 ]	$1.40\pm 0.29$	$11.53$	$12.14\pm 2.40$	$29.87\pm 1.60$	$6.72\pm 2.18$	$-0.02$	$4.71\pm 0.59$	$25.42\pm 1.05$
Noise2Void [ 29 ]	$1.77\pm 0.40$	$-11.59$	$11.03\pm 1.62$	$26.61\pm 1.36$	$6.82\pm 2.24$	$-1.49$	$4.09\pm 0.46$	$17.20\pm 0.56$
B2UB [ 35 ]	$1.37\pm 0.29$	$13.39$	$7.74\pm 1.03$	$4.68\pm 0.36$	$6.57\pm 2.18$	$2.25$	$3.36\pm 0.35$	$1.11\pm 0.13$

TABLE I : Quantitative comparison on the Boat and Underwater Structure datasets. PFPE is reported in units of

10^{-4}

, and

D_{\mathrm{EPI}}

is reported in percent.

\mathrm{PFPE}_{\mathrm{imp}}^{\mathrm{agg}}

denotes the aggregate percentage improvement over the noisy-frame baseline. The lowest PFPE and highest

\mathrm{PFPE}_{\mathrm{imp}}^{\mathrm{agg}}

are highlighted in bold. ENL and

D_{\mathrm{EPI}}

are complementary indicators of speckle suppression and edge-strength deviation, respectively.

Fig. 3 : The top panel shows

\mathrm{PFPE}_{\mathrm{imp}}^{\mathrm{agg}}

relative to the noisy baseline; the remaining three panels show distributions of per-pair PFPE and per-image ENL and EPI on the Boat dataset. In the three boxplots, boxes show interquartile ranges, horizontal lines indicate medians, and dots denote individual test images. Dashed gray lines and shaded gray bands indicate the corresponding noisy means and mean

\pm

one standard deviation, respectively.

Panel of Figure 5 — Fig. 4 : Qualitative comparison on the Boat dataset. Red and blue boxes show zoomed regions around high-reflectivity structures and background-shadow transitions. The proposed method reduces speckle while preserving separable local structures and clearer object-background transitions compared with representative model-based and self-supervised methods.

Panel of Figure 6 — Fig. 5 : Qualitative comparison on the Underwater Structure dataset. Red and blue boxes show zoomed regions around structural boundaries and background regions. The proposed method suppresses background speckle while preserving clearer structural continuity and weak local responses compared with representative baselines.

Method	Params (M)	MACs (G)	Time/img (s) $\downarrow$
PPB	–	–	$151.219$
SAR-BM3D	–	–	$324.225$
Noise2Void	$0.80$	$52.26$	$0.147$
B2UB	$1.10$	$73.94$	$0.018$
LG-BPN	$6.16$	$1605.23$	$13.789$
SS-BSN	$6.41$	$1757.44$	$7.334$

TABLE II : Computational cost on Boat images. MACs are reported using the model-level protocol, and time per image is measured using the actual inference path.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 8 — Fig. 6 : Representative registration check on the Boat dataset. The reference frame $y_{1}$ is shown in red and the paired frame $y_{2}$ is shown in green; yellow indicates spatial overlap between the two frames. Registration increases the overlap of persistent scene structures, supporting the use of the pair for self-supervised training and PFPE evaluation.

Frame gap	Boat		Underwater Structure
Frame gap	Pearson $r$	$r<0.5$	Pearson $r$	$r<0.5$
1	$0.485\pm 0.138$	$56.59\%$	$0.269\pm 0.174$	$88.98\%$
2	$0.370\pm 0.151$	$79.29\%$	$0.191\pm 0.121$	$96.71\%$
3	$0.294\pm 0.156$	$87.46\%$	$0.156\pm 0.100$	$98.90\%$
4	$0.240\pm 0.157$	$94.15\%$	$0.133\pm 0.089$	$99.61\%$
5	$0.208\pm 0.153$	$96.87\%$	$0.118\pm 0.083$	$99.92\%$

Appendix

TABLE III : Empirical ROI Pearson correlation between registered frame pairs at different frame gaps. The reduced-correlation ratio denotes the percentage of valid registered pairs with

r<0.5

Gap	Boat			Underwater Structure
Gap	$\mathrm{Med}_{\Delta\rho}$	$P_{s}(\%)$	$P_{o}(\%)$	$\mathrm{Med}_{\Delta\rho}$	$P_{s}(\%)$	$P_{o}(\%)$
1	$(3,0)$	91.78	99.17	$(3,0)$	88.83	99.84
2	$(5,0)$	68.24	94.48	$(5,0)$	69.56	93.82
3	$(8,1)$	47.73	83.89	$(6,0)$	60.66	85.58
4	$(10,1)$	38.89	72.35	$(6,0)$	56.75	80.85
5	$(14,1)$	30.17	64.03	$(6,0)$	54.25	75.94

Appendix

TABLE IV : Frame-pair displacement and overlap diagnostics for the Boat and Underwater Structure datasets.

\mathrm{Med}_{\Delta\rho}

denotes the median absolute displacement pair

(\mathrm{median}|\Delta\rho_{1}|,\mathrm{median}|\Delta\rho_{2}|)

P_{s}

denotes the percentage of candidate pairs satisfying

|\Delta\rho_{1}|\leq 10

and

|\Delta\rho_{2}|\leq 5

P_{o}

denotes the percentage of candidate pairs with valid-overlap ratio

q_{i,\ell}>0.95

after translation.

Category	Method	$\mathrm{PFPE}$ ( $\times 10^{-4}$ ) $\downarrow$	$\mathrm{PFPE}_{\mathrm{imp}}^{\mathrm{agg}}$ (%) $\uparrow$	ENL $\uparrow$	EPI $\rightarrow 1$	$D_{\mathrm{EPI}}$ (%) $\downarrow$
Baseline	Noisy	$1.59\pm 0.36$	$0.00$	$8.18\pm 1.12$	$1.000\pm 0.000$	$0.00\pm 0.00$
Model-based	PPB	$1.79\pm 0.26$	$-13.06$	$12.73\pm 2.71$	$0.807\pm 0.023$	$19.32\pm 2.28$
Model-based	SAR-BM3D	$1.40\pm 0.29$	$11.53$	$12.14\pm 2.40$	$0.701\pm 0.016$	$29.87\pm 1.60$
Self-supervised	Noise2Void	$1.77\pm 0.40$	$-11.59$	$11.03\pm 1.62$	$0.734\pm 0.014$	$26.61\pm 1.36$
	Blind2Unblind	$1.37\pm 0.29$	$13.39$	$7.74\pm 1.03$	$1.047\pm 0.004$	$4.68\pm 0.36$
	LG-BPN	$1.57\pm 0.31$	$1.19$	$18.57\pm 5.60$	$0.541\pm 0.023$	$45.89\pm 2.27$

Appendix

TABLE V : Complete quantitative comparison on the Boat dataset.

\mathrm{PFPE}_{\mathrm{imp}}^{\mathrm{agg}}

is computed from the ratio of summed errors, which is relative to the noisy paired-frame baseline. ENL measures homogeneous-region smoothing, while EPI and

D_{\mathrm{EPI}}

characterize edge-strength preservation relative to the observed image. High ENL should therefore be interpreted together with PFPE and

D_{\mathrm{EPI}}

, since it can also result from oversmoothing.

Panel of Figure 12 — Fig. 7 : Additional qualitative comparison on the Boat dataset. Red and blue boxes indicate zoomed regions containing high-reflectivity structures and background-shadow transitions. The proposed method suppresses speckle while preserving separable local structures and clearer transition boundaries. This visual behavior is consistent with its higher PFPE improvement and lower $D_{\mathrm{EPI}}$ in Table V .

Category	Method	$\mathrm{PFPE}$ ( $\times 10^{-4}$ ) $\downarrow$	$\mathrm{PFPE}_{\mathrm{imp}}^{\mathrm{agg}}$ (%) $\uparrow$	ENL $\uparrow$	EPI $\rightarrow 1$	$D_{\mathrm{EPI}}$ (%) $\downarrow$
Baseline	Noisy	$6.72\pm 3.06$	$0.00$	$3.38\pm 0.36$	$1.000\pm 0.000$	$0.00\pm 0.00$
Model-based	PPB	$7.33\pm 2.34$	$-9.01$	$6.98\pm 1.36$	$0.728\pm 0.042$	$27.17\pm 4.16$
Model-based	SAR-BM3D	$6.72\pm 2.18$	$-0.02$	$4.71\pm 0.59$	$0.746\pm 0.011$	$25.42\pm 1.05$
Self-supervised	Noise2Void	$6.82\pm 2.24$	$-1.49$	$4.09\pm 0.46$	$0.828\pm 0.006$	$17.20\pm 0.56$
	Blind2Unblind	$6.57\pm 2.18$	$2.25$	$3.36\pm 0.35$	$1.011\pm 0.001$	$1.11\pm 0.13$
	LG-BPN	$7.80\pm 2.35$	$-16.05$	$7.11\pm 1.35$	$0.858\pm 0.022$	$14.19\pm 2.16$

Appendix

TABLE VI : Complete quantitative comparison on the Underwater Structure dataset.

\mathrm{PFPE}_{\mathrm{imp}}^{\mathrm{agg}}

is computed from the ratio of summed errors, which is relative to the noisy paired-frame baseline. ENL measures homogeneous-region smoothing, while EPI and

D_{\mathrm{EPI}}

measure edge-strength preservation relative to the observed image.

Panel of Figure 14 — Fig. 8 : Additional qualitative comparison on the Underwater Structure dataset. The zoomed regions highlight structural boundaries, weak internal reflectors, and background speckle. The proposed method suppresses speckle while preserving clearer structural continuity than representative baselines, consistent with its lower PFPE and relatively small $D_{\mathrm{EPI}}$ in Table VI .

Loss and registration
Variant	$\mathrm{PFPE}$ ( $\times 10^{-4}$ ) $\downarrow$	$\mathrm{PFPE}_{\mathrm{imp}}^{\mathrm{agg}}$ (%) $\uparrow$	ENL $\uparrow$	EPI $\rightarrow 1$	$D_{\mathrm{EPI}}$ (%) $\downarrow$
$\mathcal{L}_{\mathrm{PC}}$ -only	$1.24\pm 0.29$	$22.02$	$0.02\pm 0.00$	$23.093\pm 1.419$	$2209.35\pm 141.89$
No registration	$1.62\pm 0.29$	$-2.32$	$14.88\pm 3.80$	$0.549\pm 0.026$	$45.09\pm 2.59$
Transformation
Scale	$1.38\pm 0.28$	$13.07$	$12.74\pm 2.53$	$0.956\pm 0.054$	$5.69\pm 3.96$
Shift	$1.40\pm 0.30$	$11.97$	$14.29\pm 3.47$	$0.926\pm 0.076$	$8.79\pm 5.95$

Appendix

TABLE VII : Ablation and sensitivity analysis on the Boat dataset.

\mathrm{PFPE}_{\mathrm{imp}}^{\mathrm{agg}}

is computed from the ratio of summed errors. ENL measures homogeneous-region smoothing, while

D_{\mathrm{EPI}}

measures mean per-image deviation from unity in relative edge strength. In each block, one design factor is varied while the remaining settings are kept fixed whenever possible. The results are interpreted jointly in terms of paired-frame forward prediction, speckle suppression, and structural preservation.

Variant	PFPE $\downarrow$	Imp. $\uparrow$	PFPE $\downarrow$	Imp. $\uparrow$	PFPE $\downarrow$	Imp. $\uparrow$	ENL $\uparrow$	EPI $\rightarrow 1$	$D_{\mathrm{EPI}}$ (%) $\downarrow$
	Test gap 1		Test gap 2		Test gap 3
Noisy	$1.59\pm 0.36$	0.00	$2.35\pm 0.58$	0.00	$2.94\pm 0.82$	0.00	$8.18\pm 1.12$	$1.000\pm 0.000$	$0.00\pm 0.00$
Train gap 1	$1.37\pm 0.30$	13.37	$1.82\pm 0.48$	22.20	$2.22\pm 0.75$	24.50	$12.40\pm 2.73$	$0.968\pm 0.058$	$5.17\pm 4.04$
Train gap 2	$1.52\pm 0.30$	4.04	$1.91\pm 0.46$	18.58	$2.27\pm 0.71$	22.60	$15.31\pm 4.67$	$0.842\pm 0.066$	$15.82\pm 6.60$
Train gap 3	$1.55\pm 0.26$	2.30	$1.89\pm 0.38$	19.45	$2.20\pm 0.59$	24.98	$16.48\pm 3.90$	$0.791\pm 0.058$	$20.91\pm 5.79$

Appendix

TABLE VIII : Effect of the training frame gap on Boat test performance. Model and optimization settings are fixed, while gap-specific displacement tolerances are set to meet a common 90% pair-retention target. PFPE is reported in units of

10^{-4}

, and Imp. denotes

\mathrm{PFPE}_{\mathrm{imp}}^{\mathrm{agg}}

in percent. PFPE and Imp. are evaluated using fixed PFPE test gaps 1, 2, and 3. For each trained checkpoint, ENL, EPI, and

D_{\mathrm{EPI}}

are reported once because they are computed from the restored single-frame output.

Image Restoration

Image restoration uses computational methods to recover useful visual content from noisy, blurred, low-resolution, or otherwise degraded images.

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Physics-Guided Self-Supervised Statistical Residual Learning for Sonar Despeckling with Improved Generalization

Swapna Pillai, Siddharth Singh Savner, Sujit Kumar Sahoo

School of Electrical Sciences, Indian Institute of Technology Goa, Goa 403401, India · Inria, Sophia Antipolis, France

Self-Supervised Image Denoising Self-Supervised Learning Underwater Imaging Residual Learning Physics-Guided Image Restoration Physics-Informed ML+1

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This letter introduces a physics-informed self-supervised framework for sonar image despeckling that reformulates despeckling as residual consistency in the homomorphic log domain. By constraining the log-ratio residual to obey multiplicative speckle statistics, the proposed method eliminates the need for clean supervision while preventing degenerate identity solutions. A variance-targeted statistical loss combined with edge-aware structural regularization and median-guided curriculum stabilization enables effective speckle suppression with preserved structural fidelity. This formulation along with a lightweight neural network achieves state-of-the-art performance across multiple real sonar datasets and demonstrates excellent cross-dataset robustness, while remaining suitable for real-time deployment.

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