Recently, vision research has centred on both the extraction and organization of geometric features, and on geometric relations. It is largely assumed that topological structure, that is linked edgel chains and junctions, cannot be extracted reliably from image intensity data. In this paper we demonstrate that this view is overly pessimistic and that visual tasks, such as perceptual grouping, can be carried out much more efficiently and reliably if well-formed topological structures are available. The widespread assumption that edge detectors produce incomplete and erroneous topological relations, such as the image projection of polyhedral face-edge-vertex structures, is shown to be false by analyzing the causes for failure in traditional edge detectors. These deficiencies can largely be overcome, and we show that a good compromise between topological completeness and geometric accuracy can be achieved. Furthermore, edge detection should not be carried out in isolation. The resulting topological and geometric descriptions should be motivated by the subsequent representations to be built on such image features. Here, algorithm development is driven by general object recognition, which provides an objective evaluation of the resulting edgel structure. In effect, we test the design by examining the results of the entire recognition process. Although unconventional, we feel that visual algorithms can be developed more successfully when considered as part of a complete system.