Planar motion flows can induce the illusory appearance of a volume rotating in depth (“depth from motion”; G. Sperling, & B. A. Dosher 1994). This appearance changes spontaneously from time to time, reversing simultaneously its depth and its direction of rotation. We investigated *asymmetric* illusory volumes, which reverse more frequently at some angles of view than at others. In three experiments, we studied spontaneous joint reversals of depth and motion, as well as induced reversals of either motion or depth alone. We find that depth reversals occur exclusively when the illusory volume is depth symmetric (so that the shape of the volume remains unchanged). In contrast, motion reversals occur at all view angles, but their frequency varies with the motion speed. The probability of joint reversals is well approximated by the product of the individual reversal probabilities, suggestive of two independent random processes. We hypothesize that reversals of illusory volumes are conditioned by prior experience of physical transformations in the visual world.

^{2}. For dichoptic viewing (mirror stereoscope), the viewing distance was 87.5 cm (1 pixel subtending 0.014°) and the background luminance was 35 cd/m

^{2}.

^{2}, infinite lifetime) were distributed over (all or part of) the front and back surfaces of a virtual sphere rotating about its vertical axis and were projected orthographically onto the image plane. To create single, double, or quadruple rings, dots were placed on one, two, or four circumpolar bands spaced evenly along the equator, each with a width of 1/16 of a circumference (Figure 3, Movies 1–5 and 8). Depending on the fraction of the spherical surface covered by dots, local dot density differed between shapes. In addition, to favor a unitary illusion, local dot density was higher near the poles. The virtual sphere was centered on fixation and measured 4.7° in radius. The frequency of its rotation about the vertical axis was 0.25 Hz.

*left*for leftward rotation,

*right*for rightward, and

*down*for mixed percept; <1% of all reports). Mean duration of dominance phases and correlation with cumulative history (see Pastukhov & Braun, 2011 for details) are presented in Figure 4.

*Space*key, thereby removing the main display and stopping the clock dot. Thereafter, they used

*arrow*keys to return the clock to the position it had occupied at the moment of the reversal. Five stimuli—uniform sphere, four-band, double-band, single-band, and color-band—were used in this experiment and observers performed 120 trials with each stimulus. Trial duration depended on the average dominance phase duration for a given observer and stimulus (range of 3 to 15 s plus response interval).

*γ*= 0.05 compared to

*γ*= 1.1 for RT), has a smaller variance (

*σ*= 156.4 ms compared to

*σ*= 250 ms for RT), and has an almost zero mean (

*μ*= 31 ms compared to

*μ*= 636 ms for RT).

*no change*was perceived in the illusory depth (Movie 6). Symmetrically, no change in illusory rotation implied that the physical discontinuity of motion was compensated by a change in illusory depth (Movie 7, red dots were not present in the original display and were added only to make detection of depth reversal easier). Only double-band and single-band stimuli were used. Observers performed 315 trials for each stimulus. Trial duration was 1500 ms plus response interval (unspeeded response).

*saw no change*), perceived the change but it was not accompanied by illusory motion reversal (

*saw change*), or saw an illusory motion reversal (

*saw motion reversal*).

*arrow*keys. Only double-band and single-band stimuli were used.

*either*illusory motion

*or*illusory depth. Thus, perception faced a “forced choice” and necessary had to alter one aspect (and only one aspect) of the illusory percept. Here, we investigated how often one alternative was chosen over the other, as a function of the rotational angle of the display. Note that this design reveals the relative probability of two alternative events, not the spontaneous rate of reversal events as the previous experiment.

*p*

_{D}(

*α*) (as opposed to illusory motion), as a function of phase angle

*α*. Reversals of illusory depth occurred exclusively when the illusory shape was depth-symmetric, that is, for fully frontal, edge-on, and exactly diagonal (two-band only) viewing angles. This corroborates our earlier conclusion about depth symmetry as a necessary condition for depth reversals.

Stimulus | Angle of reversal | Report | ||
---|---|---|---|---|

Saw no change | Saw change | Saw motion reversal | ||

Single-band | 0° | 65% | 3% | 32% |

90° | 4% | 93% | 3% | |

Catch trial | 98% | 2% | 0% | |

Two-band | 45° | 8.8% | 89.5% | 1.7% |

22.5° | 0% | 1% | 99% | |

Catch trial | 100% | 0% | 0% |

*opposite*to the perceived illusory depth, thus inverting the latter. With depth transiently inverted, the further evolution largely reflected the ease (or difficulty) of reversing illusory motion. If motion reversed as well, the transition between illusory percepts was complete. If motion failed to reverse, the illusory percept remained unchanged. Phenomenally, the two reversal paths were again distinct: A completed reversal of illusory rotation was phenomenally prominent, while an abortive reversal appeared (at best) as a momentary “fuzziness” in the display.

*p*

_{M}(

*α*) (completed transition of illusory percept), as a function of phase angle

*α*. The probability of motion reversals exhibits a moderate dependency on phase angle but remains finite at all angles (i.e., there are no “forbidden” angles). For one-band displays, this probability is maximal near 0°/180° and minimal near 90°/270°. For two-band displays, it is maximal near diagonal view angles (45°/135°/225°/315°) and minimal near axial view angles (0°/90°/270°/180°).

*α*. Experiment 1 established the joint probability of spontaneous reversals of illusory motion

*and*depth,

*P*

_{MD}(

*α*). Experiment 2 measured the individual probability of a reversal of illusory depth

*P*

_{D}(

*α*), and Experiment 3 revealed the individual probability of a reversal of illusory motion

*P*

_{M}(

*α*).

*ρ*

_{MD},

*ρ*

_{D}, and

*ρ*

_{M}are the maximal rates and

*p*

_{MD}(

*α*),

*p*

_{D}(

*α*), and

*p*

_{M}(

*α*) are the observed angular dependencies.

*minimal*distance

*D*(

*α*) over all pairs of left- and right-moving dots and use this measure as a “penalty” in an exponential function

*f*(

*α*):

*a*and

*κ*are constants. To fit results of Experiment 2, we subtracted

*f*

_{D}(

*α*) from the observed function

*p*

_{M}(

*α*) of Experiment 3. The best (least-squares) fit of

*p*

_{M}(

*α*) −

*f*

_{D}(

*α*) to the observed function

*p*

_{D}(

*α*) of Experiment 2 was obtained with

*a*= 3.2 and

*κ*= 0.03 (Figure 9A).

*β*

_{ i }of all dots

*i*and use this measure as a “penalty” in a linear function

*f*

_{M}(

*α*):

*x*∣ is the absolute value of

*x*. The sum includes all dots except near stationary ones (planar speed below 0.12°/s; Figure 9B).

*p*

_{MD}(

*α*) with the products

*p*

_{M}(

*α*)

*p*

_{D}(

*α*) and

*f*

_{M}(

*α*)

*f*

_{D}(

*α*). Thus, the independence assumption

*R*

^{2}= 0.72.