The Concept of Symmetry and the Theory of Perception

Abstract

Perceptual constancy refers to the fact that the perceived geometrical and physical characteristics of objects remain constant despite transformations of the objects such as rigid motion. Perceptual constancy is essential in everything we do, like recognition of familiar objects and scenes, planning and executing visual navigation, visuomotor coordination, and many more. Perceptual constancy would not exist without the geometrical and physical permanence of objects: their shape, size, and weight. Formally, perceptual constancy and permanence of objects are invariants, also known in mathematics and physics as symmetries. Symmetries of the Laws of Physics received a central status due to mathematical theorems of Emmy Noether formulated and proved over 100 years ago. These theorems connected symmetries of the physical laws to conservation laws through the least-action principle. We show how Noether's theorem is applied to mirror-symmetrical objects and establishes mental shape representation (perceptual conservation) through the application of a simplicity (least-action) principle. This way, the formalism of Noether's theorem provides a computational explanation of the relation between the physical world and its mental representation.

Keywords: Noether's theorem; conservation laws; human perception; invariance and symmetry; least-action principle; perceptual constancy.

Figure 1

(A) shows a transparent cube that is perceived as a 3D cube because of the cube's multiple symmetries. (B) shows a polygonal line connecting the vertices of a cube. (B) is not likely to be perceived as a 3D polygonal line because of the absence of 3D symmetries (from Pizlo, , MIT Press).

Figure 2

The transformation Θ is a symmetry of the dynamical evolution N if, for any initial state u, the final state when first evolved and then transformed is the same as the one obtained by first transforming and then evolving, i.e., NΘu = ΘNu.

Figure 3

A 3D object is invariant under rigid motions, and the perception of this object is invariant under the same group of rigid motion. This invariance implies that the observer perceives the permanent characteristics of this object. We used a label “mental rotation” instead of “mental rigid motion” because the former has already been used in the cognitive literature.

Figure 4

Illustration of the path (dotted line) formed in a plane by connecting midpoints of pairs of mirror-symmetrical points.

Figure 5

The two curves, φ and ψ, are 2D perspective images of a pair of 3D mirror-symmetrical curves Φ and Ψ that are reconstructed from this single 2D image. F is the center of perspective projection, and its coordinates are (0, 0, z_F). The principal point is at coordinates (0, 0, 0) (Sawada et al., 2011).

Figure 6

Stereoscopic images (crossed fusion) of the six types of stimuli used in Li and Pizlo (2011) study. (A) Polyhedron with one symmetry plane and planar surfaces. (B) 16 vertices which were obtained by removing the edges from the stimulus of type (A). (C) Polygonal line. The 16 vertices were connected randomly. (D) Partially non-planar and symmetric polyhedron. (E) Planar and asymmetric polyhedron. (F) Non-planar and asymmetric polyhedron.

Figure 7

Three subjects' performance and the averaged performance (d') across the six types of objects in Li and Pizlo (2011) study.