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Deep Bayesian REFoCUS

Simon Penninga, Ruud van Sloun

arXiv:2610.03419v1cs.LG

TL;DR

Rank-deficient multistatic recovery leaves linear REFoCUS decoders unable to uniquely determine unobserved components. Deep Bayesian REFoCUS learns a generative prior and performs posterior sampling; it outperforms linear decoders in simulation and performs best when inversion is highly rank-deficient. The paper also reports promising in-vivo results despite the measurements being out of training distribution.

  • Problem

    When transmit events are fewer than array elements, the encoding has a nontrivial null space, while sequential single-element acquisition limits SNR and frame rate.

    Source: §I · p. 1, A. The multistatic data set and… · p. 3

  • Method

    Deep Bayesian REFoCUS learns a deep generative prior on multistatic data sets and uses it to recover plausible data through posterior sampling.

    Source: §I · p. 1

  • Results

    DB-REFoCUS outperformed classical linear REFoCUS decoders across simulation, performed best for extremely rank-deficient inversion, and regressed toward linear performance when inversion was exact.

    Source: §V · p. 7, §VII · p. 9

  • Takeaways & Limitations

    Despite in-vivo measurements being out of training distribution, the model structured the multistatic data sufficiently for image reconstruction.

    Source: §VII · p. 9

    The model is orders of magnitude slower and more expensive than linear alternatives, mainly because many transport steps are required.

    Source: §VI · p. 9

Abstract

from arXiv · show

In this work we formulate ultrasound multistatic recovery from arbitrary transmit sequences as a Bayesian inference problem. To that end, we train a deep generative prior on multistatic data sets to tackle the rank-deficient regime in which classical linear REFoCUS decoders fail. This appproach, which we term Deep Bayesian REFoCUS, outperforms the linear baselines for all regimes of rank-deficiency and noise levels, and regresses to linear decoding when inversion is exact. The model also expresses uncertainty in the null space of the acquisitions, whereas the linear REFoCUS decoders only provide point estimates. Finally, we analyze the impact of distribution shift between simulation and in-vivo acquisitions, showing remarkable generalization ability without any fine-tuning or adaptation.

I. INTRODUCTION

Ultrasound transmit sequences trade frame rate against image quality, while acquiring complete multistatic data directly is challenging. Deep Bayesian REFoCUS uses a learned generative prior and posterior sampling to address the ambiguity of rank-deficient recovery where linear decoders are limited.

I. INTRODUCTION

  • Ultrafast plane- and diverging-wave sequences raise frame rate but reduce contrast, lateral resolution, and SNR; coherent compounding recovers part of that loss while the trade-off remains.Unfocused compounding also faces tissue motion across transmits, and generating tissue harmonics for harmonic imaging is more difficult.Source: §I · p. 1
  • Complete multistatic recordings enable retrospective transmit and receive focusing, but sequential single-element firing limits SNR and frame rate.REFoCUS instead recovers the multistatic data set by inverting the linear encoding of single-element responses.Source: §I · p. 1

II. RELATED WORK

  • Linear REFoCUS estimates are exact only for noiseless, full-rank transmits, leaving rank-deficient recovery without unique solutions from measurements alone.When transmit events are fewer than array elements, the encoding has a nontrivial null space; existing approaches impose assumptions such as sparsity or a zero null-space component,.Source: §I · p. 1, §II · p. 2
  • DB-REFoCUS learns a generative prior over complete multistatic data sets and uses posterior sampling to infer plausible unobserved components from measurements.Unlike fixed-scheme discriminative decoders, it supplies the encoding at inference and repeated samples expose underdetermined components.Source: §I · p. 1, §II · p. 2

III. METHODS

The methods describe multistatic recovery as a per-frequency linear inverse problem. Adjoint and regularized linear estimators provide estimates, but rank deficiency leaves a null space that prevents unique recovery.

A. The multistatic data set and encodings

  • Each transmit event records a superposition of single-element responses, and the collection across transmit-receive element pairs forms the multistatic data set.Fourier transformation along fast time decouples recovery by frequency and receive element.Source: A. The multistatic data set and… · p. 2
  • Stacked events and receive channels form S_R(f) = H(f)U_R(f), making accurate recovery dependent on the encoding matrix pseudoinverse H†.The matrix has N_beams rows and N_elem columns.Source: A. The multistatic data set and… · p. 3
  • The adjoint gives a re-delay-and-sum estimate, while Tikhonov regularization equalizes invertible components and suppresses noise amplification where inversion is unstable.Both estimators remain linear, and their accuracy depends on the rank of H(f).Source: A. The multistatic data set and… · p. 3
  • When N_beams < N_elem, every frequency bin has at least an (N_elem − N_beams)-dimensional null space, so linear recovery cannot uniquely determine the multistatic responses.The rank constraint leaves components unconstrained by the measurements.Source: A. The multistatic data set and… · p. 3

B. Bayesian REFoCUS

Bayesian REFoCUS treats the multistatic data set as a random variable conditioned on measurements. It combines a learned generative prior with an adjoint-based measurement likelihood to produce posterior samples.

B. Bayesian REFoCUS

  • The method models the multistatic data set x as a sampleable random variable and the acquired channel recordings as measurements y.The posterior is conditioned on the measurements through Bayes’ rule.Source: B. Bayesian REFoCUS · p. 3
  • A continuous-time flow-matching model learns the multistatic prior by interpolating between data samples and Gaussian noise, then regressing the interpolation velocity.The learned velocity field can be used to invert the noising process into an unconditional multistatic estimate.Source: B. Bayesian REFoCUS · p. 3
  • The likelihood guidance backprojects the measurement residual with the adjoint transmit encoding, using the same adjoint operator as conventional REFoCUS for measurement consistency.The likelihood approximation uses the clean estimate in place of the noisy intermediate state.Source: B. Bayesian REFoCUS · p. 3
  • Posterior samples combine prior-distributed null-space components with measurement guidance, but are not exactly measurement-consistent by design because measurements are noisy.The guidance is normalized and weighted by a parameter that varies with diffusion time.Source: B. Bayesian REFoCUS · p. 3

A. Model prior

The prior is trained on idealized multistatic simulations derived from cardiac anatomy, with a generative model intended to recover structured multistatic data. The simulation omits several acquisition effects and carries distribution bias from processed B-mode inputs.

A. Model prior

  • EchoNet-LVH B-mode intensities determine point-scatterer sampling; zea simulates full-rank STA data, with a beamformed data set shown for reference.The simulator omits depth-wise attenuation, measurement noise, sound-speed variation, and out-of-plane clutter; the authors describe the simulations as idealized, not realistic, but sufficiently rich to construct a useful prior.Source: A. Model prior · p. 4, Figure 2 · p. 4
  • The simulator uses a linear frequency-domain model, while deriving scatterer density from processed B-mode images introduces distribution bias.Those images include noise, aberration effects, and clutter.Source: A. Model prior · p. 4
  • Multistatic data are compressed to IQ with 4 MHz demodulation and 3× axial decimation, then normalized and clipped to ±7 standard deviations.The resulting model input has dimensions 80×512×80×2; the corpus contains 107,880 training and 11,205 validation samples from held-out patients.Source: A. Model prior · p. 4
  • A 3D convolutional U-Net with residual blocks models the velocity field and is trained with the flow-matching objective, using an exponential moving average of its weights.The passage refers to objective (11) but does not provide its formula.Source: A. Model prior · p. 4

B. Evaluation

The evaluation compares DB-REFoCUS with three REFoCUS decoders across transmit families and measures recovery in channel and image domains.

B. Evaluation

  • The comparison includes adjoint, ramp-filtered adjoint, and Tikhonov-regularized REFoCUS; Tikhonov uses an oracle grid search over 26 log-spaced γ values per data point.The search selects the best-performing result for each data point, making it an upper-bound comparison.Source: B. Evaluation · p. 4
  • DB-REFoCUS performance is reported for the posterior mean using 8 samples.The method uses 1,000 sampling steps.Source: B. Evaluation · p. 5
  • Channel-domain recovery uses gain-compensated complex correlation and NMSE; image-domain recovery uses PSNR and SSIM after delay-and-sum beamforming and image processing.The rendered images use pressure-field weighting, envelope detection, 99.99th-percentile normalization, log compression, and a fixed 50 dB dynamic range.Source: B. Evaluation · p. 5
  • Six transmit families span different inverse-problem conditioning; focused sector beams use a 60 mm focal radius and wide beams use 150 mm.Both focused and wide beams include fundamental and harmonic modes in the in-vivo experiments.Source: B. Evaluation · p. 5
  • The evaluated families also include plane waves with linear delays and diverging waves from virtual point sources 30 mm behind the array.The diverging-wave sources are steered virtual point sources.Source: B. Evaluation · p. 5
  • Hadamard transmissions use rows of an order-80 Hadamard matrix, while random transmissions use zero-delay, i.i.d. Bernoulli ±1 apodizations.The random family is described as the incoherent compressed-sensing counterpart of Hadamard; steered families use interleaved subsampling to cover the sector.Source: B. Evaluation · p. 5

1) Simulation:

Simulation experiments match the in-vivo SNR profile with depth-dependent noise and test guidance selection, recovery quality, noise robustness, and posterior-variance calibration.

1) Simulation:

  • A global measurement SNR of 10 dB is used unless stated otherwise, with depth-dependent noise shaped by a power-normalized TGC profile.Across 32 blocks spaced 2.951 mm apart, the profile spans local SNRs from 35.95 to 0.94 dB.Source: 1) Simulation · p. 5
  • Guidance strength is swept on 50 test frames across transmit families and beam counts, then recovery is evaluated on 100 different frames in channel and image domains.The passage does not specify the selected guidance value.Source: 1) Simulation · p. 5
  • Noise robustness is evaluated for FOC and HAD transmit types at 32 and 80 beams across SNRs of −20, −10, 0, 10, 20, and ∞ dB.Source: 1) Simulation · p. 5
  • The study correlates multistatic posterior variance with channel-domain loss using Spearman ρ to assess uncertainty and rank-deficiency localization.The correlation is calculated for every transmit type and beam count.Source: 1) Simulation · p. 5

2) In vivo:

DB-REFoCUS improves recovery in rank-deficient simulations and in-vivo acquisitions, while distribution shift limits its in-vivo performance. Its posterior variance tracks reconstruction loss most clearly at high rank deficiency.

V. RESULTS

  • In-vivo image recovery fills radial-streak gaps but shifts toward EchoNet-LVH and artificially increases contrast; image-domain metrics diverge even though NMSE supports the correlation result.The authors argue that out-of-distribution image recovery is better evaluated visually.Source: §V · pp. 5, 8, Figure 3 · p. 6, Figure 11 · p. 9
  • Simulation results are insensitive to guidance strength across transmit types and measurement ranks; ζ=0.35 is selected for subsequent simulation experiments.The guidance-strength sweep is shown in Fig. 4.Source: Figure 4 · p. 6, §V · p. 7
  • With as few as 4 of 80 transmits, simulated DB-REFoCUS recovery often reaches complex correlation 1 and surpasses linear decoders across operating points.At higher transmit counts, the multistatic data are better populated and the performance gap narrows; 45° steering introduces grating lobes for steered families.Source: Figure 5 · p. 6, §V · p. 7
  • At 32 beams, DB-REFoCUS outperforms linear methods across noise levels; at 80 beams, linear methods win at sufficiently high SNR because inversion is exact while sampling remains stochastic.At low SNR with 80 beams, DB-REFoCUS still performs better.Source: §V · p. 7, Figure 7 · p. 7
  • Posterior variance and multistatic loss have strong monotonic correlation at high rank deficiencies, but correlation drops sharply for N_beams ≥16 as the posterior converges.At higher beam counts, fixed reconstruction gap and stochastic sample generation dominate the correlation.Source: §V · p. 8, Figure 9 · p. 8
  • In vivo, DB-REFoCUS outperforms linear decoders at low beam counts, including Hadamard acquisitions, but distribution shift and prior pull limit gains at higher counts.Evaluation uses held-out transmits; the in-vivo model was not trained on those acquisitions.Source: 2) In vivo · p. 5, §V · p. 8, Figure 10 · p. 8

VI. DISCUSSION

Simulation results support the potential of deep generative models for ultrasound channel data, while in-vivo performance and practical use remain bounded by training-data availability, distribution shift, and computational cost. The discussion connects the model’s uncertainty and arbitrary-transmit capabilities to future cognitive ultrasound agents.

VI. DISCUSSION

  • Simulation results indicate the potential of deep generative models for ultrasound channel-domain imaging.The authors present this as promising potential, not as an established universal solution.Source: §VI · p. 9
  • High-quality multistatic training data are scarce and difficult to obtain across anatomies, limiting the method’s development.The authors suggest ambient diffusion as a possible way to learn in-vivo multistatic structure from any transmit type.Source: §VI · p. 9
  • If simulation-trained priors face substantial in-vivo distribution shift, they risk hallucinating; incorporating artifacts into training could also diminish denoising and aberration-correction capabilities.The authors describe a trade-off between learning realistic in-vivo statistics and retaining capabilities learned from idealized simulation.Source: §VI · p. 9
  • The current U-Net limits the model’s operating domain, while a conditioned transformer is proposed to generalize across probe types, spatial dimensions, frequencies, and anatomies.The proposed conditioning features also include bandwidth, depth, and number of probe elements.Source: §VI · p. 9
  • Orders of magnitude slower and more expensive than linear alternatives, the current model needs computational improvements such as fewer transport steps, pruning, or better initialization.The authors identify the number of steps Γ needed to transport the noise initialization as the main limitation.Source: §VI · p. 9
  • Uncertainty maps and arbitrary transmit designs connect the method to cognitive ultrasound, where future agents could adapt transmit-receive sequences in a Bayesian reconstruction loop.The cited existing agent implementations use beamformed, envelope, or image-domain data and are limited to fixed transmit sets.Source: §VI · p. 9

VII. CONCLUSION

Deep Bayesian REFoCUS uses a channel-domain generative prior to recover multistatic data and obtain posterior distributions for transmit types. It outperforms linear REFoCUS decoders in simulation, shows promising in-vivo results despite distribution shift, and performs best under extreme rank deficiency.

VII. CONCLUSION

  • Deep Bayesian REFoCUS conditionally generates multistatic data sets and obtains posterior distributions for all transmit types using a channel-domain generative prior.The authors identify larger in-vivo data collection, architecture generalization, and computational speedups as directions for further improvement.Source: §VII · p. 9
  • The method outperformed classical linear REFoCUS decoders across the board in simulation and showed promising results for in-vivo data.It performed best on extremely rank-deficient inversion tasks and regressed toward linear-decoder performance when inversion was exact.Source: §VII · p. 9
  • Despite in-vivo measurements being out of training distribution, the model structured multistatic data for sufficient image reconstruction.Source: §VII · p. 9

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