Fig. 6: Object recognition and perceptual switching of nested structure-from-motion (SfM) displays.

a Cylindrical SfM stimulus. A random point cloud on the surface of a rotating, transparent cylinder (left) supports two possible percepts when viewed from the front without depth information (right). Humans perceive the structured motion of this 2D projection as a rotating 3D cylinder, albeit with bistable direction of the perceived rotation. b Top view illustration of how the generative model supports rotational motion. The rotational motion source, \({s}_{t}^{{{{{{{{\rm{rot}}}}}}}}}\), describes angular velocity about the vertical axis (s^rot > 0 for CCW rotation, by definition). In location-indexed experiments, observed velocities, v_t, at a (fixed) location with angle φ and radius R are a linear function of the rotational motion source. In the frontal view of SfM experiments, only the x-component, \({v}_{x}=-\!R\,\sin (\varphi )\,{s}_{t}^{{{{{{{{\rm{rot}}}}}}}}}\), and the vertical y-component, v_y = 0, are visible. c Motion tree and correspondence problem. The graph contains self-motion, rotational motion of the entire cylinder, and individual motion for every location. For any x-y coordinate, there exist two overlapping observed velocities which are ambiguous regarding their depth position (front or back). We performed the assignment of observations to their perceived depth (front or back) such that the prediction error, ϵ_t, in Eq. (2) is minimized. d 3D percept and perceptual bistability. Like humans, the model identifies rotation as the single motion component. The value of \({s}_{t}^{{{{{{{{\rm{rot}}}}}}}}}\) switches randomly between CW and CCW rotation with constant angular speed. e Distribution of perceptual switches. The distribution of duration-of-percepts closely follows a Gamma distribution, as commonly reported in human psychophysics. f Extension of the SfM display adding a smaller point cloud-cylinder, nested within the original cylinder. g Motion tree for the extended experiment. Three rotational components are provided: shared rotation of both cylinders, rotation of the outer cylinder, and rotation of the inner cylinder. The correspondence problem now demands assigning 4 observations where both cylinders overlap. h Perceived structure for identical angular speed of both cylinders. The model infers a single shared rotational component. i Fast inner cylinder. When increasing the angular speed of the inner cylinder by 50% (sketch on the left), the inferred structure is unaffected (right): the cylinders are perceived as having the same angular velocity. j Fast outer cylinder. In contrast, when increasing the angular speed of the outer cylinder by 50% (left), the cylinders’ speeds are perceived as separated (right). For visual clarity, the trees in panels c and g show only 5 and 3 receptive field locations for the outer and inner cylinder, respectively, while for the simulations, we used 7 and 5 locations. Source data are provided as a Source Data file.