Theory for variational area-based segmentation using non-quadratic penalty functions

In this paper a theory is developed for variational segmentation of images using area-based segmentation functionals with non-quadratic penalty functions in the fidelity term. Two small theorems, which we believe are new to the vision community, allow us to compute the Gateaux derivative of the considered functional, and to construct the corresponding gradient descent flow. The functional is minimized by evolving an initial curve using this gradient descent flow. If the penalty function is sub-quadratic, i.e. behaves like the p´th power of the error for p<2, the obtained segmentation model is more robust with respect to noise and outliers than the classical Chan-Vese model and the curve evolution has better convergence properties.