In Press: Perception

Natural images have a tendency towards purely 1-D local variation Lewis Griffin (Computer Science, University College London, London WC1E 6BT, UK; email: [email protected]) To test the hypothesis that natural images have a tendency towards purely 1-D local variation (e.g. straight edges or lines), we use derivatives-of-Gaussians filters to probe image structure. Determining whether or not an image patch is locally 1-D requires the six derivative filters up to 2nd order – their outputs represent local structure as a point in the ‘jet space’ J2. Having six unbounded dimensions, J2 is difficult to work with. So we factor out the effect of the local-structure-preserving transformations (T) of the image i.e. linear transformations of intensity, rotation, and reflection. The result – J2/T – is a bounded 3-D manifold. It is intrinsically curved but we have found an embedding of it into Euclidean 3-space that is volume-preserving and only mildly distorting. The embedding has allowed us to visualize the local structure histograms for different image classes. The histograms for Gaussian and Brownian noise matched our analytical predictions. Gaussian noise shows no preference for locally 1-D image structure, while Brownian noise shows a weak preference e.g. 7% of patches are in the 3% of J2/T closest to the locally 1-D forms, and 40% of patches are in the closest 23%. The histogram for natural images is well-modelled as a mixture of the Brownian density (80% of the mixture) and a ridge of high density, tightly clustered around the 1-D forms (20% of the mixture). The preference of natural images for 1-D forms can be quantified as 15% of patches are in the 3% of J2/T closest to the locally 1-D forms, and 50% patches are in the closest 23%.