Joint Vector Flow Mapping and Segmentation: Ill-Posedness,Differentiable Bayesian Inference, and Synthetic Vortex-FlowBenchmarks

The research paper titled "Joint Vector Flow Mapping and Segmentation: Ill-Posedness, Differentiable Bayesian Inference, and Synthetic Vortex-Flow Benchmarks" discusses the challenges and advancements in reconstructing left-ventricular blood velocity using color-Doppler echocardiography. The inverse problem in vector flow mapping (VFM) is inherently ill-posed, meaning it can lead to singular modes and uncertainty propagation along reconstructed velocity fields.

Conventional VFM lacks uncertainty quantification and fails to correct blood-pool segmentation errors.

The authors introduce a hierarchical framework called Bayesian VFM (B-VFM), which jointly infers radial and transverse velocities, a probabilistic blood-pool mask, spatially resolved uncertainties, and hyperparameters. By incorporating Doppler fidelity, mass conservation, boundary conditions, and smoothness, B-VFM provides a comprehensive approach to ill-posed VFM problems.

The discretized posterior distribution of B-VFM allows for closed-form gradients and Hessians, enabling efficient gradient-based maximum-a-posteriori estimation, sampling, and direct analysis of ill-posed modes. The inference process utilizes Gibbs sampling for conjugate Gamma-distributed hyperparameters, conditional maximum-a-posteriori estimation, and a Laplace approximation for high-dimensional velocity and mask fields.

To address systematic deviations from planar mass conservation, B-VFM learns the covariance of the planar divergence residual from an ensemble of flows. This covariance is then incorporated as a structured model-discrepancy prior. Independent chains converged reproducibly, with covariance priors learned from flow ensembles illustrating how model discrepancies can be integrated into the inference process.

B-VFM was evaluated using Lamb-Chaplygin dipoles, Doppler-corrupted flows, Doppler voids, segmentation defects, and the Hicks-Moffatt family of spherical vortices. The method demonstrated the ability to produce smooth reconstructions, localize uncertainty near unreliable measurements, and correct segmentation errors. Additionally, the data-informed planar divergence prior reduced velocity bias and mask distortion within the tested vortex family.

By providing an uncertainty-aware reconstruction method and a flexible foundation for future VFM formulations, B-VFM offers a significant advancement in the field. Future work will focus on evaluating the method using clinical data and more complex three-dimensional benchmark flows.