Extraction of Near Circular Objects using Markov Random Fields: The Multilayer 'Gas of Circles' Model

A multi-layer binary Markov random field (MRF) model is developed which assigns high probability to object configurations in the image domain consisting of an unknown number of possibly touching or overlapping near-circular objects of approximately a given size. Each layer has an associated binary field that specifies a region corresponding to objects. Overlapping objects are represented by regions in different layers. Within each layer, long-range interactions favor connected components of approximately circular shape, while regions in different layers that overlap are penalized. Used as a prior coupled with a suitable data likelihood, the model can be used for object extraction from images, e.g. cells in biological images or densely-packed tree crowns in remote sensing images.

Stable configurations of the multi-layer MRF GOC model for different numbers of layers and values of κ

Results on noisy synthetic images (SNR= 0dB) containing two circles of radius 10 with different degrees of overlap.

Extraction of lipid drops from light microscope images using the multi-layer MRF GOC model.

Multi-layered phase field model

A phase field model represents a region by a function and a threshold. If we extend the model using multiple instances of the single layer phase field 'gas of circles' model, we can handle multiple touching or overlapping regions over the image domain.

An example of a low-energy configuration of the layered phase field model.

The model combined with a suitable data model and initialization is efficiently applicable to extract e.g. lipid droplets, cells, nuclei and other subcellular components.

Extraction of lipid droplets and cells using phase field 'gas of circles' model.