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An exploration of plug-and-play flow matching for full-waveform inversion on breast imaging

Cristina Garcia-Cardona, Siting Liu, Fatma Terzioglu, Shanyin Tong, Lu Zhang

arXiv:2610.32185v1eess.IVcs.NAmath.NA

TL;DR

FWI can be difficult to stabilize while preserving relevant sound-speed structures, and learned priors may be unreliable when targets differ from their training distribution. This work alternates waveform-misfit updates with a pretrained flow prior; at matched update counts, it generally improves circular-model errors and breast lesion-contrast recovery over TV-FWI.

  • Problem

    FWI’s nonconvex inversion can be sensitive to initialization, and limited or imperfect measurements may weakly constrain features; learned priors can also favor training-distribution features under shift.

    Source: §1 · pp. 2–3

  • Method

    The method adapts PnP-Flow to nonlinear FWI by alternating blocks of waveform-misfit updates with updates from a pretrained flow-matching prior.

    Source: §1 · p. 3, §3.1 · p. 7

  • Results

    At matched waveform-misfit update counts, PnP-Flow generally lowers circular-model reconstruction errors and breast-phantom lesion-contrast errors than TV-FWI, while breast whole-image errors are comparable.

    Source: §5 · pp. 18–19

  • Takeaways & Limitations

    The breast transfer shows that lesion-contrast recovery can improve even when tissue texture remains inaccurate, supporting evaluation of local contrast alongside whole-image error.

    Source: §5 · p. 19

    Convergence analysis of the modified iteration for nonlinear waveform inversion remains open.

    Source: §3.2 · p. 10

Abstract

from arXiv · show

Full-waveform inversion (FWI) reconstructs sound-speed fields from measured waveforms, by solving an optimization problem that minimizes the discrepancy between measured waveforms and those predicted by a numerical wave-propagation model. However, choosing suitable regularization for this optimization is challenging because it must stabilize the inversion while preserving relevant structures, prompting us to investigate whether priors learned from representative sound-speed images can improve reconstruction. In this work, we adapt plug-and-play flow matching (PnP-Flow) to FWI by alternating waveform-misfit updates with updates from a pretrained flow prior. We explore the effectiveness of PnP-Flow reconstruction by comparing it with total-variation (TV) regularization using noiseless transmission-only plane-wave data. Separate flow priors are trained on two circular-inclusion model families with constant and spatially varying backgrounds. We evaluate the method both on these in-distribution circle models and under substantial distribution shift by transferring the Gaussian-circle prior and reconstruction settings to breast phantom patches without retraining or retuning. Using the same number of waveform-misfit updates, the proposed PnP-Flow framework generally improves circle reconstruction accuracy and recovers breast lesion contrast more accurately than TV baselines.

1 Introduction

FWI offers detailed acoustic-property reconstruction, but its nonlinear, nonconvex inversion can be unstable and sensitive to initialization, while limited data may weakly constrain some features. This work adapts PnP-Flow to FWI and evaluates learned priors on circular models and breast phantoms, including transfer under distribution shift.

1 Introduction

  • FWI has applications in breast ultrasound tomography and other medical imaging settings, building on broad prior work in quantitative acoustic reconstruction.Breast studies have examined efficiency, robustness, attenuation, regularization, and high-resolution three-dimensional reconstruction.Source: §1 · p. 2
  • FWI’s nonlinear, generally nonconvex objective can cause gradient methods to reach undesirable local minima through cycle skipping and makes reconstruction sensitive to initialization.Cycle skipping can occur when corresponding predicted and measured waveform features are misaligned by more than approximately half a cycle.Source: §1 · p. 2
  • Finite coverage, limited frequencies, noise, and modeling errors can leave medium features weakly constrained, motivating prior information in the inversion.Classical regularizers stabilize FWI but encode prescribed structural assumptions that may not capture broader wave-speed-field statistics.Source: §1 · p. 3
  • Learned priors extract statistical and structural information from representative data, while plug-and-play methods separate prior updates from data-consistency updates.Prior approaches include denoisers and deep generative models such as score-based, diffusion, and flow-matching models.Source: §1 · p. 3
  • A learned prior may favor training-set features and attenuate structures important to waveform fitting when the target distribution differs.This motivates testing transfer under substantial distribution shift as well as in-distribution reconstruction.Source: §1 · p. 3
  • The study adapts PnP-Flow to nonlinear FWI, alternating pretrained flow-prior updates with waveform-misfit updates and using the final iterate as the reconstruction.It compares two circular-inclusion model families and transfers the Gaussian-circle prior and settings to breast patches without breast-specific tuning.Source: §1 · pp. 3–4
  • At matched waveform-misfit update counts, PnP-Flow generally lowers circular-model reconstruction errors and breast-phantom lesion-contrast errors relative to TV-FWI.The breast transfer uses the Gaussian-circle prior and settings without retraining or retuning.Source: §1 · p. 4

2 Problem Formulation and Background

FWI estimates an admissible wave-speed field by matching modeled waveforms to measured data, typically with regularization to address nonconvexity and weakly constrained features. This section contrasts fixed penalties with learned priors that can be coupled to physics-based data consistency.

2.1 Full-waveform inversion

  • The forward model maps an admissible wave-speed field to detector-time waveform data across sources, with measurements affected by noise and modeling error.The observation operator samples wavefields at detector locations; the general formulation illustrates an interior source, while experiments place sources and receivers on the boundary.Source: §2.1 · pp. 4–5
  • FWI estimates the wave-speed field by minimizing the squared L2 discrepancy between predicted and observed waveforms because the nonlinear forward map generally has no explicit inverse.The measured data may be written as noiseless modeled data or modeled data plus a noise-and-modeling-error term.Source: §2.1 · p. 5
  • Each adjoint-state gradient evaluation avoids forming the full parameter-to-data Jacobian but requires forward and adjoint wave-equation solves, making repeated high-resolution optimization costly.The cost also reflects multiple source excitations.Source: §2.2 · p. 5

2.2 Regularization and prior modeling in FWI

  • Because waveform misfit is nonlinear and generally nonconvex, and practical data may weakly constrain parts of the field, data-only minimization may be unstable or implausible.Regularization incorporates prior information about the unknown wave-speed field.Source: §2.2 · pp. 5–6
  • The regularization term encodes prior assumptions about admissible fields, while λ controls the balance between data fidelity and regularization.Tikhonov and total-variation penalties are classical examples.Source: §2.2 · p. 6
  • Tikhonov regularization favors spatially smooth fields, whereas total variation promotes sparse spatial gradients and suits sharp interfaces and piecewise-smooth structures.After discretization, these yield familiar ℓ2- and ℓ1-type penalties, respectively.Source: §2.2 · p. 6
  • A generative model can learn statistical information about plausible wave-speed fields directly from representative fields rather than prescribing a fixed analytic penalty.The learned prior is incorporated through an operator, while the wave-equation data-fidelity term enforces consistency with observations.Source: §2.2 · p. 6
  • Unlike supervised inverse maps that typically require paired data and may depend on acquisition geometry, a generative prior can be trained on wave-speed fields alone and coupled to observed data through the forward model.This work uses a flow-matching model in a plug-and-play iterative reconstruction procedure.Source: §2.2 · p. 6

3 Proposed Methodology

PnP-Flow-FWI alternates waveform-driven data-consistency updates with a learned flow prior, using sound-speed models to train the prior independently of waveform data.

3.1 PnP-Flow formulation for FWI

  • PnP-Flow-FWI alternates PDE-constrained data-consistency updates with denoising updates from a pretrained flow-matching prior.The data-consistency block uses observed waveforms and the wave-equation model; the learned prior acts only in the denoising update.Source: §3.1 · pp. 7–8
  • The flow prior is trained offline on wave-speed models alone, so reconstruction measurements and acquisition geometry enter through the FWI objective rather than paired training examples.The paper notes that the prior can in principle be reused for different measurements associated with the same class of wave-speed models.Source: §3.1 · p. 8
  • The prior update replaces an explicit regularization proximal step with a flow-based denoiser, avoiding the need to specify a regularization functional R.The denoiser estimates the conditional mean of the clean target given an interpolated input; multiple independently noised inputs are denoised and averaged.Source: §3.1 · pp. 7–8

3.2 Algorithms

  • Multiple Adam iterations are performed between prior updates to enforce physics and data consistency before restricting the estimate toward the pretrained denoiser distribution.The flow-time schedule controls the prior's influence, with smaller tstart producing stronger prior regularization.Source: §3.1 · p. 8, §3.2 · pp. 9–10
  • A convergence analysis remains open for the modified iteration that uses multiple Adam updates between prior updates.The cited convergence result concerns the original PnP-Flow iteration under its stated assumptions.Source: §3.2 · p. 10

4 Numerical Experiments

The experiments compare PnP-Flow with TV-FWI at matched data-update budgets on two circular-inclusion families, then transfer the Gaussian-circle prior and reconstruction settings to breast phantoms without retraining or retuning (p43).

4.1 Experimental setup

  • Both methods start from m0 = 8; each run uses 2,500 data updates, while PnP-Flow adds 50 flow-prior updates and is summarized using medians over four runs.The primary experiments use noiseless data.Source: §4.1 · pp. 11–13

4.2 Circles on a constant background

  • At 2,500 updates, median relative L2 error on constant-background circles was 1.92% for PnP-Flow versus 2.65% for TV-FWI.PnP-Flow had lower relative L2 error on 39 of 50 images and lower median inclusion-contrast error, 0.051 versus 0.316.Source: §4.2 · p. 13, Table 1 · p. 14

4.3 Circles on spatially varying Gaussian background

  • At 2,500 updates, median relative L2 error on Gaussian-background circles was 3.67% for PnP-Flow versus 4.69% for TV-FWI.PnP-Flow had lower relative L2 error on 48 of 50 targets and lower median inclusion-contrast error, 0.239 versus 0.851.Source: §4.3 · p. 15, Table 2 · p. 16

4.4 Lesion identification in breast phantom images

  • On 32 breast patches, median lesion-contrast error was 0.214 for PnP-Flow versus 0.263 model sound-speed units for TV-FWI, while whole-image errors showed no clear overall advantage.PnP-Flow had lower lesion-contrast error on 28 of 32 patches; whole-image relative L2 error was lower on 17 of 32 patches.Source: Table 3 · p. 17, §4.4 · p. 17

5 Conclusion

At matched computational budgets, PnP-Flow lowers median whole-image and inclusion-contrast errors on both tested circle families. Under breast distribution shift, lesion-contrast recovery improves while whole-image errors remain comparable to TV-FWI, and tissue texture can remain inaccurate.

5 Conclusion

  • At matched computational budgets, PnP-Flow achieves lower median whole-image and inclusion-contrast errors than TV-FWI on both tested circular-inclusion model families.The families use constant and spatially varying Gaussian backgrounds.Source: §5 · pp. 18–19
  • With the Gaussian-circle prior and reconstruction settings transferred without breast-specific tuning, PnP-Flow improves lesion-contrast recovery while whole-image errors remain comparable to TV-FWI.The breast reconstructions also contain rounded structures resembling training inclusions, so tissue texture remains inaccurate.Source: §5 · p. 19
  • Future work includes a complete theoretical convergence analysis for nonlinear problems and evaluation of transfer learning for other experimental setups with limited data.Source: §5 · p. 19

Declarations

The declarations describe language-tool and software-development assistance, with the authors retaining responsibility for the manuscript and results; the code is available online.

Declarations

  • An AI language tool assisted with manuscript editing, and Claude Code assisted with software writing and debugging; the authors reviewed the work and take responsibility for the content and results.Source: Declarations · p. 19

Code availability.

Appendix A Adam optimization algorithm

Appendix A summarizes the clipped Adam update used for physics-based directions derived from PDE constraints and data misfit. Global ℓ2 clipping preserves direction while bounding the gradient norm.

Appendix A Adam optimization algorithm

  • The appendix summarizes clipped Adam for updating physics-based directions derived from PDE constraints and data misfit.Source: Appendix A · p. 19
  • The update takes a gradient, previous Adam state, β1, β2, ϵ, and clipping threshold G as inputs.Source: Appendix A · p. 20
  • Global ℓ2 clipping preserves the gradient direction while limiting its norm to G, and the experiments use β1 = 0.9, β2 = 0.999, and G = 10.The Adam moments are not bias-corrected, and the squared gradient denotes element-wise squaring.Source: Appendix A · p. 20

Appendix B Training data and learned priors

Appendix B constructs constant- and Gaussian-background circle families, then trains a separate conditional flow-matching prior for each. The Gaussian-background prior is also used for breast-phantom transfer.

Appendix B Training data and learned priors

  • Both training families use one to three nonoverlapping circular inclusions, with wave speeds in and radii from 0.06 to 0.14 times the domain width.The constant-background family uses background wave speed 8; the Gaussian family replaces it with 8 + σG, where σ ∼ U[0.15, 0.45].Source: Appendix B · p. 20
  • The fields are generated on 256 × 256 grids, downsampled to 64 × 64, and the families share 4,884 training, 100 validation, and 50 held-out test geometries.Gaussian-background realizations are generated independently across training, validation, and test sets.Source: Appendix B · p. 20
  • A separate optimal-transport conditional flow-matching prior is trained for each family using the same time-conditioned U-Net configuration for 300 epochs.The selected Gaussian-background prior is reused in the breast-phantom transfer experiment.Source: Appendix B · p. 20

Appendix C Sensitivity to measurement noise

Across the tested noise range, median errors changed only slightly from noiseless results; at 10 dB, PnP-Flow outperformed TV-FWI on most test images for both reported errors.

Appendix C Sensitivity to measurement noise

  • Median errors changed by at most 0.19 percentage points in relative L2 error and 0.014 in contrast error versus noiseless results.Both methods showed these small changes in the sensitivity tests.Source: Appendix C · p. 20
  • The sensitivity comparison added independent Gaussian noise at 30.5, 20, or 10 dB while keeping noiseless-data reconstruction settings fixed.Noise standard deviations were 0.03, 0.10, or 0.316 times each clean shot record’s RMS value; both methods used the same noise realization per image.Source: Table C1 · p. 21
  • At 10 dB, PnP-Flow had lower relative L2 error on 42 of 50 images and lower contrast error on 49 of 50 than TV-FWI.The comparison used the same 50 test images.Source: Appendix C · p. 21, Table C1 · p. 21

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