  We present a fully automatic, graph-based technique for extracting the
retinal vascular topology -- that is, how different vessels are connected to
each other -- given a single color fundus image. Determining this connectivity
is very challenging because vessels cross each other in a 2D image, obscuring
their true paths. We quantitatively validated the usefulness of our extraction
method by using it to achieve comparable state-of-the-art results in retinal
artery-vein classification. Our proposed approach works as follows: We first
segment the retinal vessels using our previously developed state-of-the-art
segmentation method. Then, we estimate an initial graph from the extracted
vessels and assign the most likely blood flow to each edge. We then use a
handful of high-level operations (HLOs) to fix errors in the graph. These HLOs
include detaching neighboring nodes, shifting the endpoints of an edge, and
reversing the estimated blood flow direction for a branch. We use a novel cost
function to find the optimal set of HLO operations for a given graph. Finally,
we show that our extracted vascular structure is correct by propagating
artery/vein labels along the branches. As our experiments show, our
topology-based artery-vein labeling achieved state-of-the-art results on three
datasets: DRIVE, AV-WIDE, and INSPIRE. We also performed several ablation
studies to separately verify the importance of the segmentation and AV labeling
steps of our proposed method. These ablation studies further confirmed that our
graph extraction pipeline correctly models the underlying vascular anatomy.
