Many visual tasks depend upon the interpretation of visual structures that are flow fields, such as optical flow, oriented texture, and shading. The computation of these visual flows involves a delicate tradeo off: imaging imperfections lead to noisy and sparse initial flow measurements, necessitating further processing to infer dense coherent flows; this processing typically entails interpolation and smoothing, both of which are prone to destroy visual flow discontinuities. However, discontinuities in visual flows signal corresponding discontinuities in the physical world, thus it is critical to preserve them while processing the flow. In this paper we present a computational approach motivated by the architecture of primary visual cortex that directly incorporates boundary information into a flow relaxation network. The result is a robust computation of visual flows with the capacity to handle noisy or sparse data sets while providing stability along flow boundaries. We demonstrate the effectiveness of our approach by computing shading flows in images with intensity edges.