Role of 3D rotational symmetry in visual perception.

Abstract

A 3D shape is N-fold rotational-symmetric if the shape is invariant for 360/N degree rotations about an axis. Mach pointed out that human observers are sensitive to rotational-symmetry in a 2D image, but less sensitive than to mirror-symmetry, which involves invariance to reflection across an axis. We have previously shown that observers are also sensitive to mirror-symmetry of a 3D shape, and the assumption of mirror-symmetry helps to perceive the veridical 3D shape from its 2D image. Now we examine whether rotational-symmetry of a 3D shape plays a role in visual perception. We compared the perceptual and geometrical properties of 3D rotational-symmetry to those of 3D mirror-symmetry. We found that these two types of symmetry have similar geometrical properties. Both types of symmetry, with an additional constraint (planarity of contours), provide invariants for a 3D to 2D projection. Namely, a relation between projections of a pair of contours with either type of symmetry can be represented by a subset of 2D Affine transformations. Consequently, we show formally that a 3D shape with either type of symmetry can be recovered from a single 2D image, by using the symmetry as an a priori constraint. Unlike for mirror-symmetry of a 3D shape, observers do not seem to reliably detect N-fold 3D rotational-symmetry unless N is roughly 20 or higher. When N is infinity, the N-fold rotational-symmetric 3D shape becomes a surface of revolution, and every 2D projection of a surface of revolution is itself 2D mirror-symmetrical. We will show how rotational-symmetry appears in a 3D shape as a secondary product of its mirror-symmetry by increasing the number of folds of its mirror-symmetry (Note that infinite-fold 3D mirror-symmetry is also a surface of revolution). These considerations suggest that the human visual system is sensitive to 3D rotational-symmetry only through its accompanying 3D mirror-symmetry. Meeting abstract presented at VSS 2015.