Speed gradients and the perception of surface slant: analysis is two-dimensional not one-dimensional

Motion parallax provides cues to the three-dimensional layout of a viewed scene and, in particular, to surface tilt and slant. For example, as a textured surface, inclined around a horizontal axis, translates horizontally relative to an observer's view point, then, in the absence of head and eye movements, the observer's retinal flow will contain a one-dimensional (1D) vertical speed gradient. The direction of this gradient indicates the direction of surface tilt, and its magnitude and sign can be used in calculating the magnitude and sign of the surface slant. Alternatively, the same retinal flow contains a 1D translating component, plus a two-dimensional (2D) component of rotation (curl), and a 2D component of deformation (def). On this view, the direction of surface tilt is related to the orientation of def and the magnitude and sign of the surface slant is related to the magnitude and sign of def. We used computer generated random dot patterns as stimuli to determine whether the human visual system employs a 1D analysis (i.e. 1D speed gradients) or a 2D analysis (i.e. deformation) of surface slant from motion parallax. Using a matching technique we found compelling impressions of slant when we vector summed a translation field with (i) vertical shear, horizontal shear or deformation (made from vertical and horizontal shear), but not rotation; and (ii) vertical compression, horizontal compression or deformation (made from vertical and horizontal compression), but much less so for expansion. In both cases, the first three conditions contain def, but the fourth does not, and the last three conditions contain 1D speed gradients orthogonal to the perceived axis of inclination, but the first one does not. Therefore, the results from the first and fourth conditions distinguish between the two processing strategies. They support the idea that surface slant is coded by combining both horizontal and vertical speed gradients in a way similar to the 2D differential invariant def and oppose the view that surface slant is encoded by a 1D analysis of motion in a direction orthogonal to the perceived axis of inclination. In a further experiment, we found essentially no effect of reducing the field size from 18 to 9 deg.