Analysis and interpretation of a visual scene to extract its category, such as whether it contains an animal, is typically assumed to involve higher-level associative brain areas. Previous proposals have been based on a series of processing steps organized in a multi-level hierarchy that would progressively analyze the scene at increasing levels of abstraction, from contour extraction to low-level object recognition and finally to object categorization (Serre, PNAS 2007). We explore here an alternative hypothesis that the statistics of edge co-occurences are sufficient to perform a rough yet robust (translation, scale, and rotation invariant) scene categorization. The method is based on a realistic model of image analysis in the primary visual cortex that extends previous work from Geisler et al. (Vis. Res. 2001). Using a scale-space analysis coupled with a sparse coding algorithm, we achieved detailed and robust extraction of edges in different sets of natural images. This edge-based representation allows for a simple characterization of the ``association field'' of edges by computing the statistics of co-occurrences. We show that the geometry of angles made between edges is sufficient to distinguish between different sets of natural images taken in a variety of environments (natural, man-made, or containing an animal). Specifically, a simple classifier, working solely on the basis of this geometry, gives performance similar to that of hierarchical models and of humans in rapid-categorization tasks. Such results call attention to the importance of the relative geometry of local image patches in visual computation, with implications for designing efficient image analysis systems. Most importantly, they challenge assumptions about the flow of computations in the visual system and emphasize the relative importance in this process of associative connections, and in particular of intra-areal lateral connections.