Visual detection of 3D mirror-symmetry and 3D rotational-symmetry

ABSTRACT

Detecting 3D symmetry is important for the human visual system because many objects in our everyday life are 3D symmetrical. Many are 3D mirror-symmetrical and others are 3D rotational-symmetrical. But note that their retinal images are 2D symmetrical only in degenerate views. It has been suggested that a human observer can detect 3D mirror-symmetry even from a 2D retinal image of a 3D mirror-symmetrical pair of contours. There are model-based invariants of the 3D mirror-symmetrical pair of contours in the retinal image and there are additional invariant features when the contours are individually planar. There are also model-based invariants of a 3D rotational-symmetrical pair of contours. These invariant features of 3D mirror-symmetry and rotational-symmetry are analogous to one another but the features of 3D rotational-symmetry are computationally more difficult than the features of 3D mirror-symmetry. Experiment 1 showed that only 3D mirror-symmetry could be detected reliably while the detection of 3D rotational-symmetry was close to chance-level. Experiment 2 showed that the detection of 3D mirror-symmetry is partly based on the model-based invariants of 3D mirror-symmetry and the planarity of the contours. These results show that the visual system has evolved to favour the perception of 3D mirror-symmetry.

KEYWORDS:

• Mirror-symmetry
• rotational-symmetry
• 3D shape perception
• perceptual organization

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 There is a model-based invariant of 3D rotational-symmetry under a 2D perspective projection, but dealing with this would require us to consider the perspective of the symmetry line-segments and of the symmetry axis (see Sawada & Zaidi, 2018 for details).

2 This control of the viewing orientation of a 3D symmetrical object relative to an observer makes symmetrical pairs of feature points of the object often visible even when the object has an opaque surface. It has been shown that human performance in recovering the shape of a 3D mirror-symmetrical object is reliable when its orientation is within the range between 30 and 60 degrees (Li et al., 2009, 2011; see also Boutsen et al., 1998; Nonose et al., 2016 for relevant studies).

3 This t-test comparing the two types of symmetry was also conducted without data from TS (the first author) as a post-hoc test and its results were consistent with the statistical results reported in the main text (t₈ = 4.23, p = 0.003). The difference in the d′ between the types of symmetry was 1.28 ± 0.91 (average ± standard deviation) where its standard error and confidence interval (t_{0.05/2, 8} = 2.31) were ±0.30 and ±0.70. The effect size of the difference from 0 (one-sample Cohen’s d, Cohen, 1988) was 1.41.

4 This t-test comparing the two types of contours was also conducted without data from TS (the first author) as a post-hoc test and its results were consistent with the statistical results reported in the main text (t₈ = 8.76, p = 0.00002). The difference in the d′ between the types of contours was 1.07 ± 0.37 (average ± standard deviation) where its standard error and confidence interval (t_{0.05/2, 8} = 2.31) were ±0.12 and ±0.28. The effect size of the difference from 0 (one-sample Cohen’s d, Cohen, 1988) was 2.92.

5 Chen and Sio (2015) showed that planarity can also facilitate the perception of 2D mirror-symmetry in a frontoparallel plane. They tested human performance in detecting 2D mirror-symmetry of a random dot pattern with the depth distribution of the pattern serving as a control. The depth information of the distribution was geometrically irrelevant with respect to the task but the depth information affected the observer’s response to the 2D mirror-symmetry. The detection performance was the best when the depth distribution represented a symmetrical pair of planes. Farell (2015) studied the perception of 2D mirror-symmetry of a depth pattern composed of 4 dots and discussed “planarity” but his usage of the word “planarity” was different from the geometrical meaning of “planarity” that we used in our study. All of the patterns used in Farell (2015) were geometrically-planar configurations of 4 dots. The plane of the dots was perpendicular to the frontoparallel plane (see also Minkov & Sawada, 2021 for a concern about this configuration of the visual stimuli). Farell (2015) did not refer them as being “planar”.

6 This includes composition of coplanar figures, like a planar ground and a planar bottom face of a volumetric object resting on the ground (Gibson, 1979, p. 163).

7 Gibson (1979, p. 89) also wrote that an invariant structure can specify the layout of edges and corners but this invariant may or may not be a formless invariant for his radical hypothesis.

8 Based on the fact that the many objects, or their components, are 3D symmetrical, 3D symmetry itself can be regarded as an “invariant” (high-frequency, see above) across the objects and their components in real environments.