Anisotropy of object nonrigidity: High-level perceptual consequences of cortical anisotropy

Abstract

We demonstrate an unexpected anisotropy in perceived object non-rigidity, a higher-level perceptual phenomenon, and explain it by the population distribution of low-level neuronal properties in primary visual cortex. We measured the visual interpretation of two rigidly connected rotating circular rings. In videos where observers predominantly perceived rigidly-connected horizontally rotating rings, they predominantly perceived non-rigid independently wobbling rings if the video was rotated by 90°. Additionally, vertically rotating rings appeared narrower and longer than horizontally rotating counterparts. We decoded these perceived shape changes from V1 outputs incorporating documented cortical anisotropies in orientation selectivity: more cells and narrower tuning for the horizontal orientation than for vertical. Even when shapes were matched, the non-rigidity anisotropy persisted, suggesting uneven distributions of motion-direction mechanisms. When cortical anisotropies were incorporated into optic flow computations, the kinematic gradients (Divergence, Curl, Deformation) for vertical rotations aligned more with derived gradients for physical non-rigidity, while those for horizontal rotations aligned closer to rigidity. Our results reveal how high-level non-rigidity percepts can be shaped by hardwired cortical anisotropies. Cortical anisotropies are claimed to promote efficient encoding of statistical properties of natural images, but their surprising contribution to failures of shape constancy and object rigidity raise questions about their evolutionary function.

Keywords: Object nonrigidity; cortical anisotropy; cortical model; kinematic invariants; motion perception; shape constancy.

Figure 1.. Anisotropy of a rotating ring illusion videos.

A: Two Styrofoam rings glued together at an angle are seen to rotate together on the turntable. B: When rotated 90°, one ring appears to move independently and wobble against the ring on the turntable. C & D: Two circular (C) and octagonal (D)rings physically rotate together with a fixed connection at the junction. E & F: The wo pairs of rings physically move independently and wobble against each other. For the circular rings, E is indistinguishable from C. However, with trackable vertices, F is discernable from D. G & H are the same as C & D except that they are rotated 90° and they both look nonrigidly connected.

Figure 2.. Anisotropy of the shape of the ring.

A and B correspond to snapshots from Figure 1E and 1F respectively. Despite being physically identical shapes, B is perceived as vertically elongated and narrower in comparison to A. C and D: The shape illusion is similar for two elongated diamonds and can be explained by differences in perceived angles: ${\varphi }_h$ is perceived to be wider than ${\varphi }_v$ despite being physically equal, and ${\psi }_h$ is perceived to be wider than ${\psi }_v$.

Figure 3.. Cortical anisotropy.

A: Orientation tuning curves of V1 simple cells were simulated by von Mises distributions to match the anisotropic and isotropic cortices and weighted by the number of cells (See text for references). B: Detailed parameters for the von Mises distributions are shown on the left two panels for the anisotropic (red) and isotropic (black) cortex. Decoded angles around the vertical and horizontal diverging axes respectively are shown on the right two panels. C: The decoded angle difference $\left({\widehat{\gamma }}_H-{\widehat{\gamma }}_V\right)$ from the anisotropic cortex (red) and the isotropic cortex (dotted black). For the isotropic cortex, there is no difference in the decoded angles for ${\gamma }_V$ and ${\gamma }_H$. However, for the anisotropic cortex, ${\gamma }_H$ is decoded to be broader than ${\gamma }_V$.

Figure 4.. Effect of image and physical stretch on perceived nonrigidity.

A: Examples of rings elongated to match the shape of orthogonally oriented circular rings. Subjects were allowed to stretch rings in image domain (first and third rows) or physically (second and fourth rows), ranging from 0% (left column) to 50% (right column). B: Histograms representing the extent of horizontal stretch in the image domain (red) and physical domain (blue) for vertically rotating rings adjusted to match the shape of physically circular horizontally rotating rings. C: Histograms representing the extent of horizontal stretch in the image domain (red) and physical domain (blue) for horizontally rotating rings adjusted to match the shape of physically circular vertically rotating rings. D: Probability of observers reporting vertically rotating rings as more nonrigid. Original: Two pairs of circular rings with identical dimensions. $H_i$ & $H_p$: Horizontally rotating rings were stretched to match the shape of the vertically rotating rings, with the stretch applied either to the image (i) or physically before projection (p). $V_i$ & $V_p$: Vertically rotating rings were stretched to match the shape of the horizontally rotating rings, with the stretch applied either to the image (i) or physically before projection (p). Max: Both vertical and horizontal rings were maximally stretched, as shown in the last column of Panel A, causing horizontally rotating rings to appear narrower and longer compared to vertically rotating rings.

Figure 5.. Comparing optic flows from isotropic cortex and anisotropic cortex to templates for rotation and wobbling.

A: Tuning curves for direction selective cells reflecting documented anisotropies in width and number of cells. B: Tuning curves for direction selective cells for an isotropic corticex. C: Optic flow fields generated by isotropic cortex (top row) and anisotropic cortex (bottom row) for physical horizontal rotation (left column) and vertical rotation (right column). Anisotropic cortex generates vectors pointing horizontally. D: Depiction of rotation and wobbling axes. E: Physical velocity field for horizontal (top) and vertical (bottom) rotations mixed with physical wobbling (Weights $k=0-1$). F & G Cosine similarity of ME optic flows from isotropic (F) and anisotropic (G) cortex to different templates ($k=0$: physical rotation, $k=1$: physical wobbling, shown in Fig. S6 (a)). Isotropic cortex leads to no difference between best fitting $k$ for horizontal (blue) and vertical (red) rotation. However, cortical anisotropy results in different best fitting $k$: more wobbling for vertical rotation.

Figure 6.. Gradients of ME flow from the rotating ring align with percepts.

A: Examples of differential invariants. B: Gradients of rotation to wobbling, divergence (left), curl (middle), and deformatiosn1 (right) as a function of rotational phase (0°−180°: half a cycle). These gradients illustrate that with increasing emphasis on wobbling from k=0 (blue curves) to k=1 (red curves), variability increases across all three gradients, while rotational fields consistently exhibit higher values of curl and deformation. C: Isotropic Tuning Gradients for Vertical (left) and Horizontal (right) Rotation: Similar gradients are observed for both orientations, with optimal fitting by curves at a wobbling weight of k=0.78. D: Upon adjusting filter numbers and tuning widths to match V1 anisotropies before velocity field calculations, gradients for vertical rotation closely correspond to physical wobbling (k=0.91), while those for horizontal rotation indicate lower levels of wobbling (k=0.48). E & F : Best $k$ as a function of image stretch. E for isotropic cortex and F for anisotropic cortex. A greater amount of stretch leads to increased wobbling during horizontal rotation (red) and reduced wobbling during vertical rotation (blue). For an anisotropic cortex, the higher amount of wobbling during vertical rotation is maintained, consistent with experimental results. Furthermore, at maximum stretch, the isotropic and anisotropic cortices predict different outcomes: the isotropic cortex suggests that horizontal rotation should be perceived as more nonrigid, while the anisotropic cortex suggests that vertical rotation should appear more nonrigid.