We propose a novel iterative algorithm for multiphase image segmentation by curve evolution. Specifically, we address a multiphase version of the Chan-Vese piecewise constant segmentation energy. Our algorithm is efficient: it is based on an explicit Lagrangian representation of the curves and it converges in a relatively small number of iterations. We devise a stable curvature-free semi-implicit velocity computation scheme. This enables us to take large steps to achieve sharp decreases in the multiphase segmentation energy when possible. The velocity and curve computations are linear with respect to the number of nodes on the curves, thanks to a finite element discretization of the curve and the gradient descent equations, yielding essentially tridiagonal linear systems. The step size at each iteration is selected using a non-monotone line search algorithm ensuring rapid progress and convergence. Thus, the user does not need to specify fixed step sizes or iteration numbers. We also introduce a novel dynamic stopping criterion, robust to various imaging conditions, to decide when to stop the iterations. Our implementation can handle topological changes of curves, such as merging and splitting as well. This is a distinct advantage of our approach, because we do not need to know the number of phases in advance. The curves can merge and split during the evolution to detect the correct regions, especially the number of phases.
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