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DP-JMRNet: A Deep Unfolding Network for Differential Phase Preservation in Sparse Bitemporal SAR Reconstruction

Juncheng Bao, Zhen Zhang, George P. Petropoulos

arXiv:2608.26605v1eess.SPeess.IV

TL;DR

Sparse bitemporal SAR reconstruction often evaluates magnitude or single-epoch fidelity, although differential phase is central to InSAR deformation retrieval. DP-JMRNet jointly reconstructs masked acquisitions under a differential-phase objective and achieves the lowest reported differential-phase error across sampling rates and Sentinel-1 scenes while preserving competitive amplitude and complex-image fidelity.

  • Problem

    Sparse SAR reconstruction is typically assessed by magnitude or single-epoch fidelity, despite differential phase driving bitemporal InSAR deformation retrieval.

  • Method

    DP-JMRNet jointly reconstructs masked bitemporal complex SAR data using deep unfolding, exchange-equivariant interaction, and coherence-aware cross-epoch information sharing.

  • Results

    47.5%–51.3%: DP-JMRNet reduced W-Diff RMSE versus the best baseline and achieved the lowest W-Diff across three sampling rates and three Sentinel-1 scenes.

  • Takeaways & Limitations

    Shared aperture support is necessary for differential-phase fidelity, whereas fine-grained sampling-mask optimization does not appreciably change the differential phase.

  • Takeaways & Limitations

    Timing measurements used different batching and scopes, so the paper does not convert them into a speed–quality comparison.

Abstract

from arXiv · show

Complex SAR imagery is usually visualized and evaluated mainly through its magnitude. Phase is retained in the complex data but is rarely treated as a direct image-quality objective. Existing sparse reconstruction methods typically focus on magnitude fidelity and single-epoch complex reconstruction accuracy. However, the phase difference between two acquisitions is what drives line-of-sight deformation retrieval in InSAR, from ground subsidence monitoring to earthquake deformation mapping. This paper proposes the Differential-Phase-Oriented Joint Masked Reconstruction Network (DP-JMRNet), which uses deep unfolding to reconstruct the two epochs jointly from masked observations under a differential-phase objective. An exchange-equivariant interaction module makes the reconstruction independent of epoch ordering. A coherence-aware gate opens cross-epoch sharing in coherent regions and closes it where the two epochs disagree. On simulated bitemporal SAR data, DP-JMRNet attains the lowest differential-phase RMSE at 30\%, 40\%, and 50\% sampling rate, while maintaining competitive amplitude and complex-image fidelity. This corresponds to a 47.5\%--51.3\% reduction over the best baseline, achieved with one third of its parameters. The same trend is validated on three Sentinel-1 scenes. A systematic study of acquisition design further shows that sharing the same aperture support across epochs is necessary for phase fidelity, whereas optimizing the sampling mask does not improve the differential phase. The code and data are available at https://github.com/JasonBao05/coherent-sar-unfolding.

I. INTRODUCTION … A. Bitemporal SAR Observation Model

DP-JMRNet addresses sparse bitemporal SAR reconstruction as a differential-phase preservation problem, jointly modeling two masked epochs with shared aperture support. Its exchange-equivariant, coherence-aware unfolding architecture targets interferometric fidelity while retaining competitive single-epoch reconstruction quality.

  • I. INTRODUCTION: Sparse aperture acquisition arises because operational constraints limit recorded pulses, producing nonuniform, gapped, or incomplete measurements.Such missing or irregular samples can violate conventional focusing assumptions and create severe reconstruction artifacts.
  • I. INTRODUCTION: Differential phase, rather than magnitude fidelity alone, is the direct objective for evaluating sparse bitemporal SAR reconstruction.Separate epoch-wise magnitude or complex metrics can miss spatially varying phase bias that harms coherence and deformation estimates.
  • 1) Differential-phase-oriented bitemporal reconstruction:: DP-JMRNet jointly reconstructs two masked SAR epochs using a shared aperture mask and a differential-phase training objective.The framework is physics-guided and designed around the final interferometric task rather than single-epoch criteria alone.
  • 2) Reliable bitemporal unfolding with selective interaction:: The network alternates matrix-free SAR data-consistency updates with learned complex priors for joint bitemporal reconstruction.Cross-epoch information flow is governed by an exchange-equivariant interaction module and a coherence-aware reliability gate.
  • 2) Reliable bitemporal unfolding with selective interaction:: 10−9 swap error verifies that exchanging the two epochs exactly exchanges the network outputs.The reliability gate promotes sharing in coherent regions and suppresses interaction where coherence is low or epochs conflict.
  • A. Bitemporal SAR Observation Model: The bitemporal observation model applies one SAR forward operator to both epochs because they share nominal acquisition geometry.A binary aperture mask retains all frequency samples at selected aperture positions and defines the undersampled measurements.
  • A. Bitemporal SAR Observation Model: Shared aperture support is necessary for differential-phase fidelity, because breaking epoch pairing substantially degrades phase despite an unchanged sampling budget.The two epochs nevertheless retain different complex scattering states, measurements, and noise realizations.

B. Reconstruction and Differential-Phase Task

The task jointly reconstructs two complex SAR epochs while preserving both magnitude and phase, then evaluates their differential phase using a wrapped, reliability-weighted circular error. Because differential-phase error depends on the unequal components of the two single-epoch phase errors, shared errors cancel while mismatched errors remain.

  • The bitemporal model produces two complex-valued estimates from undersampled observations, preserving each image’s scattering magnitude and phase.
  • Differential phase is defined as the wrapped phase difference ϕ2[p] − ϕ1[p], with the first-to-second epoch convention determining its sign.The reconstructed differential phase is computed analogously from the two estimates.
  • Interferometric coherence measures local differential-phase reliability, and reconstructed coherence replaces the original images in the coherence estimate.The coherence formulation uses a neighborhood with normalized spatial weights and ε > 0 for numerical stability.
  • Phase error in either reconstruction directly affects the interferogram, while common errors cancel and unequal or spatially varying errors remain.The same cancellation mechanism applies to orbital, topographic, and atmospheric phase terms in repeat-pass interferometry.
  • Reliability-weighted circular RMSE evaluates differential-phase error using weights wp ∈[0, 1], while wrapped errors avoid the discontinuity between −π and π.The reliability map can incorporate coherence, amplitude support, and other validity information.

C. Feasible Sampling and Reconstruction Objective

The section defines feasible aperture masks using both a measurement-budget and aperture-continuity constraint, then optimizes a reliability-weighted objective combining differential-phase, single-epoch phase, complex, and magnitude fidelity. The same feasible mask is shared across both epochs, while mask design itself is evaluated separately rather than optimized.

  • Feasible Sampling: Feasible masks satisfy the measurement budget and bound the maximum number of consecutive missing aperture positions by a prescribed gap g.This aperture-continuity constraint prevents long unsampled intervals and excludes strictly periodic decimation that can cause azimuth ambiguities.
  • Reconstruction Objective: The reconstruction objective combines complex, magnitude, single-epoch phase, and differential-phase losses to preserve both epochs’ physical content.Nonnegative coefficients balance these terms, with λ∆ > 0 explicitly incorporating the final interferometric quantity into optimization.
  • Reconstruction Objective: Both phase losses are circular and reliability weighted, reducing the influence of low-amplitude or low-coherence pixels relative to reliable interferometric pixels.The additional complex and magnitude terms prevent phase-only optimization from sacrificing reconstructed SAR-image quality.
  • Feasible Sampling: The mask is drawn from a feasible standard aperture-pattern family, is shared by both epochs, and is not optimized during training or evaluation.The effect of pattern design within MK,g is examined empirically in Section V.

III. PROPOSED METHOD … 2) Paired reconstruction path:

DP-JMRNet reconstructs paired complex SAR acquisitions using a fixed shared aperture mask and a five-stage unfolding path with cross-epoch interaction. A shared, bounded phase refiner then makes small phase-only corrections without introducing epoch-specific parameters.

  • A. Overall Framework of DP-JMRNet: DP-JMRNet trains and infers with the same fixed, physically feasible aperture mask and matched reconstructor for both temporal tracks.This makes the training and inference operators identical and gives both epochs the same aperture support.
  • 1) Fixed shared-mask setting:: The shared mask M is stored as a frozen binary array, excluded from optimization, and supplied to reconstruction modules only as a conditioning variable.No gradient or optimizer update is applied to M for each sampling budget.
  • 1) Fixed shared-mask setting:: All trainable parameters are collected in Θ, while the sensing operator A and mask M remain fixed physical conditions rather than optimized variables.Θ includes unfolding priors, selective interaction modules, step sizes, the mask conditioner, and the final phase refiner.
  • 2) Paired reconstruction path:: Each temporal track begins with an independent matched-filtering initialization before the paired reconstruction path processes the two acquisitions jointly.The initial paired state contains the two independently initialized track states.
  • 2) Paired reconstruction path:: Five unfolding stages apply independent pre-prior data consistency, a shared complex prior, coherence-aware selective interaction, and independent post-interaction data consistency.The two channels are indexed by acquisition and stage, while the selective interaction construction is detailed in Section III-B and Fig. 2.
  • 2) Paired reconstruction path:: After stage five, the same phase-refinement network Fϕ and mask conditioner are reused for both tracks, avoiding epoch-specific refinement parameters.The conditioner takes the fixed mask as input without modifying it.
  • 2) Paired reconstruction path:: The final refinement restricts phase correction to |δ_t| ≤ 0.08 rad, changes only phase, preserves amplitude, and cannot override the unfolding-stage phase.Only Θ is optimized under objective (19), with the two gate regularizers added.

3) Training objective: … 2) Coherence-aware selective gate:

DP-JMRNet trains a fixed-mask, jointly unfolded reconstruction with shared temporal priors, measurement-consistent updates, and a symmetric coherence-aware gate that selectively exchanges information between epochs.

  • 3) Training objective:: Fixed-mask learning keeps the observation operator, aperture support, and reconstruction path identical during training and inference, without updating M.No gradient or optimizer update is applied to M.
  • B. Exchange-Equivariant Selective Reconstruction: Each unfolding stage processes the paired temporal state through pre-prior data consistency, a shared complex prior, selective interaction, and post-interaction data consistency.The stage contains five blocks, with stage-specific learned operators and parameters shared between temporal tracks.
  • 1) Data consistency and shared complex prior:: The two temporal tracks share prior parameters while retaining separate measurement-consistency updates before and after interaction.The shared prior does not force temporal states to be identical, and information exchange does not replace measurement consistency.
  • 2) Coherence-aware selective gate:: The coherence-aware gate generates one spatial gate for both tracks from local coherence magnitude and normalized image-domain data-consistency gradient evidence.These are magnitude-valued sources of evidence computed at each unfolding stage.
  • 2) Coherence-aware selective gate:: The coherence term favors reliable common structure, while the disagreement term suppresses conflicting track-wise evidence.The gate also attenuates interaction where the mean residual is large through an exponential term.
  • 2) Coherence-aware selective gate:: Symmetric evidence makes the gate invariant to exchanging the two epochs.This provides exchange-equivariant interaction at the gating level.
  • 2) Coherence-aware selective gate:: The first-stage gate is analytically anchored, whereas later stages learn positive pℓ, qℓ, and sℓ values.The learned positive values control the remaining-stage gate behavior.

3) Exchange-equivariant interaction:

The exchange-equivariant interaction block jointly processes both epochs with shared components and paired complex corrections. Symmetric descriptors and opposite endpoint ordering enforce exact output exchange under epoch swapping, while gated residual updates preserve controllable cross-epoch interaction.

  • Exchange-equivariant interaction: A single shared module takes two epoch inputs and produces two paired complex corrections, followed by gated residual updates on each track.The interaction block is designed as a joint two-input, two-output operation.
  • Exchange-equivariant interaction: Exchange equivariance is enforced by evaluating the shared interaction operator in opposite endpoint orders rather than through an auxiliary loss.The construction preserves temporal exchange equivariance directly in the architecture.
  • Exchange-equivariant interaction: Symmetric magnitude statistics form a permutation-invariant pair descriptor, while shared endpoint encoders and decoders operate in both ordered arrangements.Shared components remain order independent, while the decoder retains endpoint order.
  • Exchange-equivariant interaction: Exchanging the two inputs exchanges the two outputs exactly, up to floating-point precision.This is the resulting behavioral guarantee of the shared, oppositely ordered construction.

4) Gate regularization and stage symmetry:

DP-JMRNet regularizes its coherence-aware gate to remain spatially smooth and active where evidence is favorable. Shared two-track operators and exchange-equivariant stages preserve symmetry under swapping epochs.

  • Gate regularization: Gate training enforces spatial smoothness and prevents a trivial all-zero solution in regions supported by favorable evidence.The evidence-weighted mean activation elig is computed over high-coherence, low-residual locations.
  • Stage symmetry: Data-consistency maps and learned operators are shared across the two tracks, while the gate is symmetric.
  • Stage symmetry: Swapping the two inputs and observations only swaps the stage outputs, and this exchange-equivariance persists across all L stages.

IV. EXPERIMENTAL SETUP … B. Sampling and Observation Protocol

The experiments use controlled bitemporal simulated SAR pairs and three real Sentinel-1 repeat-pass pairs, evaluated under matched phase-history sampling protocols. Sampling budgets, fixed versus learned masks, and complex Gaussian noise are specified to assess reconstruction and cross-domain transfer.

  • 1) Bitemporal simulated SAR dataset:: The simulated dataset contains 4000 pairs of 256×256 complex-valued images generated directly from 1024×1024 parent scenes.Each parent scene combines distributed background, extended scattering regions, linear structures, and point-scatterer clusters.
  • 1) Bitemporal simulated SAR dataset:: The second epoch introduces 2–5 smooth deformation regions, amplitude modulation, scatterer changes, and 1–3 reduced-coherence regions across easy, medium, and hard cases.The difficulty probabilities are 25%, 50%, and 25%, respectively.
  • A. Datasets: The simulated data are split into 3000 training, 500 validation, and 500 sealed pairs without shared parent scenes or overlapping crops.Stored files include complex references, differential-phase labels, reliability weights, validity masks, crop coordinates, and scene metadata.
  • 2) Sentinel-1 external evaluation:: Three public Sentinel-1 C-band Level-1 IW SLC repeat-pass pairs provide cross-domain evaluation without external-domain training.All use VV polarization, the IW1 subswath, and a 12-day same-track interval; 30 non-overlapping 256 × 256 patch pairs are retained per scene.
  • 2) Sentinel-1 external evaluation:: Registered SLC patches serve as complex references, while undersampled observations use the same frozen phase-history operator as the simulated data.This preserves real-SAR scene statistics, including speckle, terrain-induced phase, and natural coherence structure.
  • B. Sampling and Observation Protocol: All methods use a matrix-free phase-history operator with exact-count budgets K ∈ {80, 104, 128}, corresponding to 31.25%, 40.625%, and 50% sampling rates.These are referred to below as 30%, 40%, and 50%.
  • B. Sampling and Observation Protocol: DP-JMRNet uses a fixed standard pattern, whereas J-MoDL-SAR, LOUPE-SAR, and MF-JMoDL-Net learn sampling patterns jointly with reconstruction.Complex-FISTA also uses a fixed uniform mask, and the comparison includes those four baselines.
  • B. Sampling and Observation Protocol: The nominal evaluation SNR is 20 dB, with circular complex Gaussian noise independent between epochs; robustness is separately tested at 10, 15, 20, 25, and 30 dB.The training SNR range is 15–25 dB.

C. Evaluation Metrics … B. Full-System Performance and Robustness

DP-JMRNet is evaluated with differential-phase-centered metrics and a specified unfolding training protocol, then tested across aperture budgets and complete acquisition–reconstruction systems. It achieves the lowest differential-phase error across budgets while preserving competitive amplitude and complex fidelity and remaining robust across operating conditions.

  • C. Evaluation Metrics: W-Diff, amplitude NMSE, and complex NMSE are the three primary metrics, with W-Diff measuring bitemporal phase performance and NMSEs guarding amplitude and complex fidelity.W-Diff is reported in radians and degrees; NMSE values are also reported in dB, where more negative values are better.
  • C. Evaluation Metrics: Secondary metrics include Hard-Diff, W-Raw, PSNR, SSIM, correlation, PSLR, ISLR, and an AASR proxy, while sidelobe diagnostics are not used for checkpoint selection.PSLR, ISLR, and the AASR proxy characterize sampling-pattern sidelobe and azimuth-ambiguity behavior using a fixed 3 × 3 mainlobe.
  • D. Training Protocol: DP-JMRNet uses five unfolding stages and 16 hidden channels, with AdamW training at learning rate 3 × 10−4, weight decay 10−6, batch size 8, and gradient clipping 2.0.Loss terms and detailed stopping rules, comparison settings, environment, and timing scope are provided in Appendix A.
  • V. EXPERIMENTAL RESULTS AND DISCUSSION: Each complete system uses its own sampling mask and matched reconstructor, while both epochs share that mask within the system.The evaluation first varies aperture budget, then compares complete acquisition–reconstruction systems.
  • V. EXPERIMENTAL RESULTS AND DISCUSSION: The comparison includes Complex-FISTA, J-MoDL-SAR, LOUPE-SAR, and MF-JMoDL-Net, whose learned designs are supervised in the magnitude domain rather than optimized for bitemporal differential phase.Complex-FISTA uses a fixed uniform mask; the other listed methods learn sampling jointly with unrolled or feed-forward reconstruction.
  • A. Behavior Across Aperture Budgets: 9.0221° is DP-JMRNet’s W-Diff RMSE at 50% sampling rate, improving from 10.3355° at 30% sampling rate while amplitude NMSE changes from −5.0620 to −7.4919 dB.Complex NMSE likewise improves from −1.2503 to −2.6632 dB across the same aperture range, without degrading magnitude or complex fidelity.
  • B. Full-System Performance and Robustness: 47.9%, 47.5%, and 51.3% are DP-JMRNet’s W-Diff reductions over the best qualifying baseline at 30%, 40%, and 50% sampling rate.Relative to J-MoDL-SAR, the corresponding reductions are 52.3%, 52.8%, and 58.4%.
  • B. Full-System Performance and Robustness: W-Diff decreases as SNR rises from 10 to 30 dB, while harder scenes and lower coherence increase difficulty without changing the overall performance ordering.The relative behavior across aperture budgets remains stable, indicating robustness beyond the nominal 20 dB setting and a particular scene regime.

C. Qualitative Reconstruction Results · D. Ablation and Mechanism Analysis

DP-JMRNet produces cleaner differential-phase reconstructions while preserving amplitude structure, and its selective, stage-adaptive interaction improves robustness and exchange consistency. Ablations show that coherence-aware sharing benefits phase and amplitude reconstruction, especially in difficult cases.

  • C. Qualitative Reconstruction Results: At 40% sampling, DP-JMRNet yields cleaner differential-phase fields with fewer severe outliers and more localized errors than competing methods.It preserves large-scale amplitude structure in a representative medium-coherence deformation case.
  • C. Qualitative Reconstruction Results: −6.03 dB is DP-JMRNet’s best amplitude NMSE in the representative case, while its W-Diff RMSE reaches 12.35◦.J-MoDL-SAR attains a slightly lower complex NMSE, showing that complex fidelity and differential-phase preservation can diverge.
  • D. Ablation and Mechanism Analysis: The learned gate is largest when local coherence is high and normalized residual conflict is low, enabling stronger inter-epoch information exchange.High-conflict regions receive suppressed coupling relative to low-conflict regions.
  • D. Ablation and Mechanism Analysis: The gate’s temporary reduction during the second unfolding stage shows that selective interaction is stage adaptive rather than monotonically increased by construction.Mean gate values were evaluated across five stages at 40% sampling over eight audited validation cases.
  • D. Ablation and Mechanism Analysis: 19.3409◦ is the W-Diff RMSE after disabling exchange, compared with 10.0488◦ for the full trained gate at 40% sampling.The intervention compares g = 0, g = 1, and the spatially varying continuous gate without changing learned parameters.
  • D. Ablation and Mechanism Analysis: 1.80 dB is the amplitude NMSE improvement from enabling exchange, reducing error from −3.98 dB to −5.78 dB.This gain exceeds DP-JMRNet’s 0.79 dB amplitude margin over the best baseline.
  • D. Ablation and Mechanism Analysis: 0.36◦ is the paired W-Diff reduction for hard cases, versus 0.04◦ for easy cases, while 95th-percentile error decreases from 23.50◦ to 21.57◦.Selective interaction primarily improves robustness in difficult, high-error cases.

E. Role and Limits of Aperture Pattern Design … VI. CONCLUSIONS

DP-JMRNet preserves differential phase across simulated and Sentinel-1 sparse bitemporal SAR reconstruction, while acquisition results show that shared aperture support matters far more than fine-grained mask optimization. The method achieves this with a compact implementation, although severe decorrelation and the absence of full-chain InSAR evaluation remain limitations.

  • E. Role and Limits of Aperture Pattern Design: The five aperture pattern families produce nearly indistinguishable curves across 20%–60% sampling rates, with a maximum spread of 0.15◦.All patterns satisfy the prescribed sample count and gap constraint.
  • E. Role and Limits of Aperture Pattern Design: 9.48◦ and 3.34◦ contribute to W-Diff from scene coherence and measurement noise, compared with less than 0.15◦ from family choice and about 0.008◦ from within-family refinement.Aperture design effects are therefore two to three orders of magnitude smaller than the dominant uncontrollable factors.
  • F. Cross-Domain Evaluation on Sentinel-1 Data: DP-JMRNet achieves the lowest W-Diff at all three sampling rates on three Sentinel-1 scenes, reducing W-Diff versus J-MoDL-SAR by 2.108◦, 0.904◦, and 1.580◦.Its absolute reductions versus Complex-FISTA, LOUPE-SAR, and MF-JMoDL-Net range from 4.23◦ to 14.08◦.
  • F. Cross-Domain Evaluation on Sentinel-1 Data: DP-JMRNet leads W-Diff at every sampling rate without retraining, while J-MoDL-SAR retains amplitude and complex NMSE advantages below 1.1 dB and 0.7 dB.Absolute W-Diff values remain high, between 64◦ and 81◦, because of low reference coherence in real scenes.
  • G. Computational Characteristics and Limitations: 122,533 trainable parameters make DP-JMRNet approximately one third the size of J-MoDL-SAR and MF-JMoDL-Net, and substantially smaller than LOUPE-SAR’s 31.39 million.Its dedicated 40%-sampling, batch-1 latency is 105.95 ms per sample; amortized validation times are 20.3, 43.2, and 20.4 ms at 30%, 40%, and 50%.
  • VI. CONCLUSIONS: The framework combines matrix-free SAR data consistency, complex-domain deep unfolding, and exchange-equivariant selective interaction without dependence on input epoch order.This architecture enables information sharing between epochs while preserving exchange equivariance.
  • VI. CONCLUSIONS: 47.5%–51.3% lower W-Diff RMSE is achieved at 30%, 40%, and 50% sampling while maintaining competitive amplitude and complex-image fidelity.Selective interaction improves single-epoch amplitude reconstruction by 1.80 dB, but severe noise and low temporal coherence still challenge phase recovery, and complete InSAR deformation retrieval is not evaluated.
  • E. Role and Limits of Aperture Pattern Design: Sharing the same aperture support across epochs is necessary for phase fidelity, whereas pattern-family choice and task-driven refinement scarcely change differential phase within feasible constraints.Breaking paired support degrades W-Diff by more than 50%, while pattern-induced errors cancel as common mode in the interferogram.

APPENDIX DETAILED EXPERIMENTAL PARAMETERS

The appendix provides experimental parameters omitted from Section IV, covering SAR acquisition, implementation and runtime, and model, loss, and training termination settings. For learned methods, reported results use the best validation checkpoint under early stopping rather than the final training update.

  • Experimental parameters: Tables VII–IX list parameters not stated in Section IV.The tables cover SAR acquisition and observation, implementation and runtime, and model, loss, and training termination parameters.
  • Training termination: Maximum update values serve only as training caps.They do not identify the update used for the reported results.
  • Training termination: Reported learned-method results use the checkpoint with the best validation performance under the corresponding early stopping criterion, not the final training update.This defines how the reported checkpoint is selected.
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