Figure 1. Reduced symmetry, co-circularity and co-circularity with offset.

A In the natural world objects can be thought of as a group of line segments whose orientations are coupled to one another and to their positions in space (the retinotopy). Following Lee et al. (2003) we refer to coupling between orientation and spatial position as reduced symmetry. B This structure is modified if orientations are rotated independent of position. C When the particular reduced symmetry of A is maintained, by rotating the positions of the segments together with the rotation of the individual segments, the image retains its structure (Lee et al., 2003). D Coupling between the position and angle of oriented segments or orientation preferences can be described by α, the difference in orientation of the two segments, and β, the direction of one segment with respect to the orientation of the other segment. A feature of natural images is that segments tend to be tangent to a common circle (dotted line) which occurs when β = α/2 as shown here. E Co-circularity with offset, when β = α/2 − τ, is a more general form of coupling between relative positions and angles of oriented segments. Here τ is −20°. F A “set” diagram showing the relationship between full symmetry, reduced symmetry and co-circularity (with and without offset τ). The meaning of P(α, β) and τ is explained in the Methods section.