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.
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