A multi-layer phase field model for extracting multiple near-circular objects

This paper proposes a functional that assigns low `energy´ to sets of subsets of the image domain consisting of a number of possibly overlapping near-circular regions of approximately a given radius: a `gas of circles´. The model can be used as a prior for object extraction whenever the objects conform to the `gas of circles´ geometry, e.g. cells in biological images. Configurations are represented by a multi-layer phase field. Each layer has an associated function, regions being defined by thresholding. Intra-layer interactions assign low energy to configurations consisting of non-overlapping near-circular regions, while overlapping regions are represented in separate layers. Inter-layer interactions penalize overlaps. Here we present a theoretical and experimental analysis of the model.