Mathematical analysis of the Accordion Grating illusion: a differential geometry approach to introduce the 3D aperture problem

When an observer moves towards a square-wave grating display, a non-rigid distortion of the pattern occurs in which the stripes bulge and expand perpendicularly to their orientation; these effects reverse when the observer moves away. Such distortions present a new problem beyond the classical aperture problem faced by visual motion detectors, one we describe as a 3D aperture problem as it incorporates depth signals. We applied differential geometry to obtain a closed form solution to characterize the fluid distortion of the stripes. Our solution replicates the perceptual distortions and enabled us to design a nulling experiment to distinguish our 3D aperture solution from other candidate mechanisms (see Gori et al. (in this issue)). We suggest that our approach may generalize to other motion illusions visible in 2D displays.