We address the problem of contour inference from partial data, as obtained from state-of-the-art edge detectors. We argue that in order to obtain more perceptually salient contours, it is necessary to impose generic constraints such as continuity and co-curvilinearity. The implementation is in the form of a convolution with a mask which encodes both the orientation and the strength of the possible continuations. We first show how the mask, called the ``Extension field'' is derived, then how the contributions from different sites are collected to produce a saliency map. We show that the scheme can handle a variety of input data, from dot patterns to oriented edgels in a unified manner, and demonstrate results on a variety of input stimuli. We also present a similar approach to the problem of inferring contours formed by end points. IN both cases, the scheme is non-linear, non iterative, and unified in the sense that all types of input tokens are handled in the same manner.