Perceptual assumptions and projective distortions in a three-dimensional shape illusion.

We examine a shape illusion, in which the balconies of a building appear to tilt up or down, depending on the viewpoint. The balconies are actually level parallelogram shapes, but appear as tilted rectangles. We measured the illusory tilts observed when parallelogram shapes are viewed above the line of sight, using three-dimensional stimuli consisting of parallelograms of various tilts viewed at different orientations. Under perspective projection, parallelism and orthogonality are not preserved. However, perspective distortions alone cannot account for the perceived tilts measured in these experiments, since observers perceived illusory tilts even for stimuli in the frontoparallel plane. We introduce a model, based on the theory that observers assume ambiguously projected three-dimensional angles to be equal to 90 degrees, but revise their predictions on the basis of observation. In the model, perceived tilt is predicted as a weighted sum of the tilts predicted by the assumptions that the shape is rectangular, and that the shape is level (i.e. that the angle between the shape and the vertical backboard is equal to 90 degrees). We prove that it is mathematically impossible for a planar rectangle to share a projection with a nonrectangular parallelogram. A less restrictive assumption that just the two leading internal angles are equal to 90 degrees is suggested as an alternative, and it is further proven that this new configuration of angles leads to a unique perceived tilt. The relative weights in the model reflect the amount that each prediction is revised, and are shown to vary systematically with stimulus orientation. For some observers a better fit was found by replacing the level-tilt assumptions with an assumption that physical tilt was equal to the projected tilt.