A phase field higher-order active contour model of directed networks

The segmentation of directed networks is an important problem in many domains, e.g. medical imaging (vascular networks) and remote sensing (river networks). Directed networks carry a unidirectional flow in each branch, which leads to characteristic geometric properties. In this paper, we present a nonlocal phase field model of directed networks. In addition to a scalar field representing a region by its smoothed characteristic function and interacting non-locally so as to favour network configurations, the model contains a vector field representing the `flow´ through the network branches. The vector field is strongly encouraged to be zero outside, and of unit magnitude inside the region; and to have zero divergence. This prolongs network branches; controls width variation along a branch; and produces asymmetric junctions for which total incoming branch width approximately equals total outgoing branch width. In conjunction with a new interaction function, it also allows a broad range of stable branch widths. We analyse the energy to constrain the parameters, and show geometric experiments confirming the above behaviour. We also show a segmentation result on a synthetic river image.