Unsupervised methods for automatic vessel segmentation from retinal images are attractive when only small datasets, with associated ground truth markings, are available. We present an unsupervised, curvature-based method for segmenting the complete vessel tree from colour retinal images. The vessels are modeled as trenches and the medial lines of the trenches are extracted using the curvature information derived from a novel curvature estimate. The complete vessel structure is then extracted using a modified region growing method. Testresults of the algorithm using the DRIVE dataset are superior to previously reported unsupervised methods and comparable to those obtained with the supervised methods in [1],[2].