What role does symmetry play in the perception of 3D objects?

By Zygmunt Pizlo, Yunfeng Li, Tadamasa Sawada, and Robert M. Steinman

The most general definition of symmetry is self-similarity: that one part of an object, pattern, signal, or process is similar, or more-or-less identical to another. According to this definition, the complete absence of symmetry is equivalent to perfect randomness, so symmetry is another name for redundancy. This makes the connection between symmetry and Shannon’s information theory explicit. The presence of symmetry also means that engineering and biological signals can be “compressed”; redundancy inherent in them can be reduced or even removed.

Symmetry is ubiquitous, as well as important, in our natural environments. There are several types of symmetry. The human body is mirror-symmetrical, one half is a reflection of the other. The two halves are never perfectly identical, but they usually are nearly so. The same is true of the bodies of almost all animals simply because mirror symmetry facilitates effective locomotion. A person could not walk and run along a straight line if his body were not mirror-symmetrical. A bird could not fly along a straight trajectory, and a fish or reptile could not swim along a straight trajectory if it were not mirror-symmetrical. Flowers are characterized by rotational-symmetry and many plants are characterized by translational-symmetry as well as by rotational symmetry. Man-made objects are usually made symmetrical because of the function they serve. A typical chair is mirror-symmetrical and a screw driver is rotationally-symmetrical. A completely asymmetrical object would most-likely be dysfunctional. Considering the fact that most things in our environment are symmetrical, one would think that our visual system should, at the very least, “know” about symmetry, and hopefully make good use of it. Symmetry is important not only because “it is there”, but also because the presence of symmetry implies that objects have shape and that scenes have structure.

Recently, we have been able to collect empirical evidence showing that the human visual system (one can also say, the human brain) uses symmetry to see 3D objects and scenes veridically (as they are). Symmetry is a natural, powerful predilection of our mind. It forms a large part of our a priori knowledge about the animate and inanimate things in the world around us. We are born with the concept of symmetry already in our minds. Why not? Symmetry is a mathematical concept, something that exists without any experience with the physical world. If our DNA contains information about the symmetry of our brain, why shouldn’t the brain know about symmetry, whether it is its own symmetry, or the symmetry of the real 3D objects and 3D scenes with which the brain’s owner’s will interact?

Our computational models show that symmetry is indispensable for veridical vision. It is also indispensable for avoiding the horrendous curse called computational intractability. Recovering a 3D shape from a single 2D retinal image would, without symmetry, require examining what are often called an “astronomically” large number of possibilities. How large? How about 10^10,000,000, a number starting with 1 followed by 10 million zeros. Considering the fact that the number of atoms in the entire Universe is estimated to be 10⁸⁰, a 1 followed by only eighty zeros, astronomers should probably start calling exceptionally large numbers “visually” rather than “astronomically” large. The visual system, by using symmetry, does not need to explore even a miniscule fraction of this huge number of possible 3D interpretations. Symmetry, and only symmetry allows a human, or a robot, to select the right 3D interpretation on its first attempt.

Zygmunt Pizlo, Yunfeng Li, Tadamasa Sawada, and Robert M. Steinman are the authors of Making a Machine That Sees Like Us. Zygmunt Pizlo is a professor of Psychological Sciences and of Electrical and Computer Engineering at Purdue University. Yunfeng Li is a postdoctoral fellow at Purdue University. Tadamasa Sawada is a postdoctoral researcher in the Graduate Center for Vision Research at SUNY College of Optometry. Robert M. Steinman devoted most of his scientific career, which began in 1964, to sensory and perceptual process, heading this specialty area in the Department of Psychology at the University of Maryland in College Park until his retirement in 2008.

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Animations created and provided by Yunfeng Li and Tadamasa Sawada.