## Section: New Results

### Variational models

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

Keywords : higher-order active contour, phase field, flow, directed network, river network extraction.

Participants : Aymen El Ghoul, Ian H. Jermyn [ contact ] , Josiane Zerubia.

This work is funded by a PACA Region grant in collaboration with Thales Alenia Space, and by INRIA from a contract with the French Space Agency (CNES).

Previous work in the project developed phase field higher-order active
contour models for network regions (regions in the image domain consisting
of narrow branches joining together at junctions), and applied those models
to the extraction of road networks from medium and very high resolution
images. These networks are `undirected': the flow in them proceeds in both
directions. Many of the networks that appear in applications (*e.g. * river
networks in remote sensing, vascular networks in medical imaging) are,
however, directed. Each network branch has a `flow direction', and each
junction therefore has `incoming' and `outgoing' branches. The existence of
such a flow typically changes the geometry of the network, because often
the flow is in some sense conserved.

In this work, we have extended the models of undirected networks described in [5] [42] and [18] . The phase field still represents the region corresponding to the network, and still interacts nonlocally so as to favour network configurations. The novel element is a tangent vector field v representing the `flow' through the network branches [17] . The vector field is coupled to in such a way that it is strongly encouraged to be zero outside the region; to have unit magnitude inside the region; to have zero divergence; and, more weakly, to be smooth. The transition from unit magnitude inside the region to zero magnitude outside, coupled with small divergence, encourages the vector field to be parallel to the region boundary. Both the divergence and smoothness constraints then tend to propagate this parallelism to the interior of network branches. Small divergence and parallelism, coupled with the constraint on the magnitude, aids prolongation of network branches; allows a larger range of stable widths; controls rate of change of width along a branch; and encourages asymmetric junctions for which total incoming branch width equals total outgoing branch width. Figure 15 shows that the directed model outperforms the undirected model for both synthetic and real images.