Analysis of two- and three-dimensional rigid and nonrigid motions in the stereokinetic effect

Abstract

Continued viewing of a rotating ellipse induces three types of percept: the veridical percept (rigid, flat, rotating ellipse), the amoeboid percept (nonrigid, flat, deforming ellipse), and the stereokinetic percept (rigid circular disk rolling in three-dimensional space). A mathematical analysis of the nonveridical percepts is presented, based on the aperture problem and the assumption that the vector field N (the normal component of the rotational velocity field R of the ellipse) is the effective stimulus. Perceived nonrigidity of the amoeboid percept is assessed by an analysis of the geometrical nonrigidity of N. Two analytical tests of geometrical rigidity of velocity fields are described. The first is based on the decomposition of the velocity gradient matrix, and the second is based on an analysis of the temporal derivative of the curvature of moving plane curves. Both tests confirm the geometrical nonrigidity of N, on which the perceived nonrigidity might be based. A limitation of both approaches with respect to the analysis of shape-preserving nonrigidity is noted. For an understanding of the stereokinetic percept and its relation to N, a detailed kinematic analysis is presented of the type of motion that a real circular disk (the physical stereokinetic disk) would have to perform in order to duplicate the perceived motion of the perceptual stereokinetic disk induced by the rotating ellipse. The analysis is based on the Eulerian decomposition of rigid-body rotation into precession, nutation, and spin. It is concluded that the stereokinetic effect cannot be directly based on N because the projected velocity field of the physical stereo-kinetic disk, P differs from N. However, a subsequent transformation with N as the input vector field produces an output vector field whose structure is similar to P, except for scale. This transformation is based on the vector interaction algorithm, which is motivated by psychophysical data on contextual effects in the perception of motion direction. It is also demonstrated that this algorithm predicts qualitative aspects of percepts in several other relevant examples, such as the barber pole effect, translating nonoccluded line segments, and eccentrically rotating circles. An additional analysis shows that, although there is an infinity of figures projectively equivalent to a static ellipse, the added dynamical constraint of projective equivalence of the velocity vector field of a moving figure to P implies that the only rigid figure consistent with stimulation and processing conditions is a circular disk. The bases of the three types of percept induced by the rotating ellipse are assumed to be the neural counterparts of three vector fields: R N, and P.