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.
Figures & tables
Fig. 1: Schematic of the Deep Bayesian REFoCUS sampling process. A multistatic data set is initialized with random Gaussian noise. The diffusion process is guided by the residual of the measurements, which we obtain through transmit beamforming. The neural network steps introduce the multistatic prior statistics, and the adjoint guidance steps transport the generation process towards the measurements.
Fig. 2: Multistatic data generation example for an EchoNet-LVH frame. Point scatterers are sampled based on pixel intensities, full-rank STA firings are simulated with zea. For reference, the multistatic data set is beamformed.
Stage
Output (tx × ax × rx)
Channels
Blocks
Attn.
Stem
80×512×20
96
—
—
Encoder stage 1
80×512×20
96
4
—
Encoder stage 2
40×256×10
192
3
✓
Encoder stage 3
20×128×5
384
3
✓
Bottleneck
10×64×5
384
3
✓
Decoder stage 3
20×128×5
384
3
✓
TABLE I: U-Net and training configuration.
Fig. 3: Comparison of reconstruction quality of linear REFoCUS decoders and Deep Bayesian REFoCUS. For the simulation a single plane wave is fired and the target is the simulation multistatic ground truth. For the in vivo 40 harmonic wide beams are fired, and the target shows 80 harmonic wide beams.
Fig. 4: Complex correlation of DB-REFoCUS recovered multistatic data sets against the ground truth. Panels show different transmit schemes, the lines show different numbers of transmit events. We sweep guidance strength to find the global operating point ζ=0.35 .
Fig. 5: Complex correlation of DB-REFoCUS compared with the linear baselines for a range of Nbeams at 10dB measurement SNR. The noiseless Tikhonov method shows that linear recovery is limited by rank deficiency of the inversion.
Fig. 6: Heatmap showing NMSE, PSNR and SSIM of DB-REFoCUS relative to the best performing linear decoder, oracle Tikhonov.
Figure 8
Fig. 10: In vivo complex correlation for different Nbeams for all transmit types. The complex correlation is calculated on a held-out set of the remaining 80−Nbeams transmits that belong to the same frame. The recovered multistatic data set is transmit beamformed to these acquisitions to compare.
Fig. 11: In vivo NMSE, PSNR and SSIM of DB-REFoCUS relative to the oracle Tikhonov linear decoder. NMSE is calculated through transmit beamforming to the held-out set, the image-domain metrics are calculated with respect to the Nbeams=80 image.
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