Fuzzy Region Competition: A Convex Two-Phase Segmentation Framework

We describe a novel framework for two-phase image segmentation, namely the Fuzzy Region Competition. The functional involved in several existing models related to the idea of Region Competition is extended by the introduction of a fuzzy membership function. The new problem is convex and the set of its global solutions turns out to be stable under thresholding, operation that also provides solutions to the corresponding classical formulations. The advantages are then shown in the piecewise-constant case. Finally, motivated by medical applications such as angiography, we derive a fast algorithm for segmenting images into two non-overlapping smooth regions. Compared to existing piecewise-smooth approaches, this last model has the unique advantage of featuring closedform solutions for the approximation functions in each region based on normalized convolutions. Results are shown on synthetic 2D images and real 3D volumes.