Illusory smoke and dazzling fog

It is well known that a flat ellipse rotating in the frontoparallel plane appears, after brief inspection, as a rigid circular disc tilting back and forth in a 3-D space. We here report that rotation of a grey-shaded ellipse on a white or on a black background produces the compelling illusion of a dark smoke or a dazzling fog (depending on the conditions of the background) moving in front of a completely white or completely black tilting disc. The fog effect disappears when there is a luminance contrast all along the perimeter of the ellipse. An experiment is reported showing that the effect can be experienced in static conditions only to a limited extent and mostly in the `dazzling' version, and that relative movement between the contours of the figure and the shaded area is crucial to the occurrence of the effect, while the occurrence of a depth effect is not. Received: 25 February 1999 / Accepted: 11 April 2000     

On the minimal relative motion principle—the oscillating tilted bar

Beghi, Xausa, Tomat, and Zanforlin (J. Math. Psychol. 41(1997) 11) present a visual stereokinetic illusion. In the image plane, one end of an oblique bar moves horizontally back and forth, while the other end is stationary. Perceptually, this becomes a bar of a constant length rotating in depth around a vertical axis that passes through the stationary end of the bar. Beghi et al. (1997) provide a mathematical model of minimal relative motion to account for this percept. Here we show that the minimal relative motion principle cannot explain the perceptual phenomenon. Specifically, we raise two objections. (1) It is necessary to consider not only the length, but also the direction, of a vector when comparing vector fields. In fact, when directions are taken into consideration, Beghi et al.'s mathematical result diverges from their perceptual experimental result. (2) There is a mathematical inconsistency in Beghi et al. (1997): mixing absolute and relative velocities in their minimization is unwarranted, and does not lead to correct minimization.     

Stereokinetic phenomena revisited: The oscillating tilted bar, the swinging gate and the vertical contracting bar

We present here a revised version of our mathematical modelling of the stereokinetic phenomena known as the “oscillating tilted bar”, the “swinging gate” and the stereokinetic phenomenon elicited by a vertical, periodically contracting line segment simultaneously undergoing a lateral displacement from left to right and conversely in the frontal plane of an observer. The criticisms of Liu, Z. [(2004). On the minimal relative motion principle—The oscillating tilted bar. Journal of Mathematical Psychology, 48, 196-198] and Rokers and Liu [(2004). On the minimal relative motion principle—Lateral displacement of a contracting bar. Journal of Mathematical Psychology, 48, 292-295.] helped us in reformulating our models, eliminating discrepancies and ambiguities. Characteristic of the present modelling is the clear definition of a multi-stage mathematical procedure matching different requirements posed by the Visual System, as we know them from our experimental observations.     

Perception of the stereokinetic illusion by the common marmoset (<Emphasis Type="Italic">Callithrix jacchus</Emphasis>)

Stereokinetic illusions have never been investigated in non-human primates, nor in other mammalian species. These illusions consist in the perception of a 3D solid object when certain 2D stimuli are rotated slowly in the plane perpendicular to the line of sight. The ability to perceive the stereokinetic illusion was investigated in the common marmoset (Callithrix jacchus). Four adult marmosets were trained to discriminate between a solid cylinder and a solid cone for food reward. Once learning criterion was reached, the marmosets were tested in sets of eight probe trials in which the two solid objects used at training were replaced by two rotating 2D stimuli. Only one of these stimuli produced, at least to the human observer, the stereokinetic illusion corresponding to the solid object previously reinforced. At test, the general behaviour and the total time spent by the marmosets observing each stimulus were recorded. The subjects stayed longer near the stimulus producing the stereokinetic illusion corresponding to the solid object reinforced at training than they did near the illusion corresponding to the previously non-rewarded stimulus. Hence, the common marmosets behaved as if they could perceive stereokinetic illusions.     

On the minimal relative motion principle—lateral displacement of a contracting bar

Xausa, Beghi, and Zanforlin (J. Math. Psychol. 45(4) (2001) 635) provide an account of perceptual organization based on their ‘minimal relative motion’ principle. They claim that this principle can account for the percept generated by a contracting bar that is simultaneously translating laterally. We critique the mathematical analysis provided in the aforementioned paper. We conclude that the ‘minimal relative motion’ principle, in the form presented, cannot adequately explain the percept reported by observers.