  Many image processing problems involve identifying the region in the image
domain occupied by a given entity in the scene. Automatic solution of these
problems requires models that incorporate significant prior knowledge about the
shape of the region. Many methods for including such knowledge run into
difficulties when the topology of the region is unknown a priori, for example
when the entity is composed of an unknown number of similar objects.
Higher-order active contours (HOACs) represent one method for the modelling of
non-trivial prior knowledge about shape without necessarily constraining region
topology, via the inclusion of non-local interactions between region boundary
points in the energy defining the model. The case of an unknown number of
circular objects arises in a number of domains, e.g. medical, biological,
nanotechnological, and remote sensing imagery. Regions composed of an a priori
unknown number of circles may be referred to as a `gas of circles'. In this
report, we present a HOAC model of a `gas of circles'. In order to guarantee
stable circles, we conduct a stability analysis via a functional Taylor
expansion of the HOAC energy around a circular shape. This analysis fixes one
of the model parameters in terms of the others and constrains the rest. In
conjunction with a suitable likelihood energy, we apply the model to the
extraction of tree crowns from aerial imagery, and show that the new model
outperforms other techniques.
