Divergence Prior and Vessel-Tree Reconstruction

Abstract

We propose a new geometric regularization principle for reconstructing vector fields based on prior knowledge about their divergence. As one important example of this general idea, we focus on vector fields modelling blood flow pattern that should be divergent in arteries and convergent in veins. We show that this previously ignored regularization constraint can significantly improve the quality of vessel tree reconstruction particularly around bifurcations where nonzero divergence is concentrated. Our divergence prior is critical for resolving (binary) sign ambiguity in flow orientations produced by standard vessel filters, e.g. Frangi. Our vessel tree centerline reconstruction combines divergence constraints with robust curvature regularization. Our unsupervised method can reconstruct complete vessel trees with near-capillary details on synthetic and real 3D volumes.

Confluent Vessel Trees with Accurate Bifurcations

“…Dense local vessel detections can be denoised using curvature regularization [24,20]. Prior knowledge about divergence or convergence of the vessel tree (arteries vs veins) can also be exploited to estimate an oriented flow pattern [34], see Fig. 2(b).…”

“…Well-formulated segmentation of thin structures requires Gaussian-or min-curvature surface regularization that has no known practical algorithms. Segmen-arXiv:2103.14268v1 [cs.CV] 26 Mar 2021 (a) Frangi filtering [9] (b) oriented flow pattern [34] Figure 2: Low-level vessel estimation: True centerline is black. Blue voxels in (a) are local maxima of some tubularity measure [9,18,28,10] in the direction orthogonal to the estimated centerline tangents (red).…”

“…Blue voxels in (a) are local maxima of some tubularity measure [9,18,28,10] in the direction orthogonal to the estimated centerline tangents (red). Regularization [24,20] can estimate subpixel centerline points (b) and oriented tangents [34] (red flow field).…”

“…We propose a simple flow-extrapolating circular arc construction that guarantees ε-confluence, which approximates confluence. Our confluence constraint implies directed tubular graph with asymmetric edge weights, which is in contrast to standard undirected geodesic tubular graphs [16,13,32,28,23,30,20,29,34].…”

“…estimation method [34] based on non-negative divergence constraint and regularization over the voxel-grid neighborhood. Since 80% of our large vasculature volumes are near-capillary vessels, the weak sub-voxel signal often results in missing data points and grid-based regularization fails to produce consistent flow orientations, particularly at the tree periphery.…”