In this paper we consider region-based variational segmentation of two-and three-dimensional images by the minimization of functionals whose fidelity term is the quotient of two integrals. Users often refrain from quotient functionals, even when they seem to be the most natural choice, probably because the corresponding gradient descent PDEs are nonlocal and hence require the computation of global properties. Here it is shown how this problem may be overcome by employing the structure of the

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... er-Lagrange equation of the fidelity term to construct a good initialization for the gradient descent PDE, which will then converge rapidly to the desired (local) minimum. The initializer is found by making a one-dimensional search among the level sets of a function related to the fidelity term, picking the level set which minimizes the segmentation functional. This partial extremal initialization is tested on a medical segmentation problem with velocity-and intensity data from MR images. In this particular application, the partial extremal initialization speeds up the segmentation by two orders of magnitude compared to straight forward gradient descent.