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

Vector flow mapping (VFM) reconstructs left-ventricular (LV) blood velocity from color-Doppler echocardiography by combining the measured beamwise component with physical and regularizing constraints. Analysis of the discrete VFM formulation shows that the inverse problem is intrinsically ill posed: the occurrence of singular modes can be predicted from the geometry of the segmented blood-pool domain, the imposed boundary conditions, and the degree of smoothing. These modes can propagate uncertainty along entire transverse bands of the reconstructed velocity field, yet conventional VFM neither quantifies this uncertainty nor allows for correcting the blood-pool segmentation. We introduce Bayesian VFM (B--VFM), a hierarchical framework that jointly infers radial and transverse velocities, a probabilistic blood-pool mask, their spatially resolved uncertainties, and hyperparameters weighting Doppler and segmentation fidelity, mass conservation, boundary conditions, and smoothness. The discretized posterior admits a closed-form gradient and exact Hessian, enabling computationally efficient, gradient-based MAP estimation, sampling, and direct analysis of ill-posed modes. Posterior inference combines Gibbs sampling of conjugate Gamma-distributed hyperparameters with conditional maximum-a-posteriori estimation and a Laplace approximation for the high-dimensional velocity and mask fields. To accommodate systematic departures from planar mass conservation, B-VFM can learn the covariance of the planar divergence residual from an ensemble of flows and incorporate it as a structured model-discrepancy prior. Independent chains converged reproducibly, while covariance priors learned from flow ensembles illustrated how model discrepancies can be incorporated into the inference. B--VFM was evaluated using Lamb-Chaplygin dipoles under ideal conditions and with Doppler corruption, Doppler voids, and segmentation defects, and using the Hicks-Moffatt family of spherical vortices to assess violations of planar mass conservation. The method produced smooth reconstructions, localized uncertainty near unreliable measurements and regions of model inconsistency, and used flow information to correct segmentation errors. Within the tested vortex family, the data-informed planar divergence prior reduced velocity bias and mask distortion. B--VFM thus provides an uncertainty-aware reconstruction method and a flexible foundation for future VFM formulations incorporating additional priors, observations, and physical models. Future work will evaluate the method using clinical data and more complex three-dimensional benchmark flows.