The presence of speckle (a spatial stochastic process in an ultrasound image) makes ultrasound segmentation difficult. Speckle
introduces local minima in the MAP energy function of an active contour, and when evolving under gradient descent, the contour gets trapped in a spurious local minimum. In this paper, we propose an alternate technique for evolving a MAP active contour. The technique has two parts: a deterministic evolution strategy called <i>tunneling descent</i> which escapes from spurious local minima, and a <i>stopping rule</i> for terminating the evolution. The combination gives an algorithm that is robust and gives good segmentations. The algorithm also benefits from having only a few free parameters which do not require tweaking. We present the conceptual framework of the algorithm in this paper, and study the impact of different stopping rules on the performance of the algorithm. The algorithm is used to segment the endocardium in cardiac ultrasound images. We present segmentation results in this paper and an experimental evaluation of different stopping rules on the performance of the algorithm. Although the algorithm is presented as an ultrasound segmentation technique, in fact, it can be used to segment any first-order texture boundary.