Binocular Disparity as an Explanation for the Moon Illusion
Joseph Antonides and Toshiro Kubota
Department of Mathematics and Computer Science, Susquehanna University, 514 University Avenue, Selinsgrove, PA 17870
ABSTRACT
We present another explanation for the moon illusion, in which the
moon looks larger near the horizon than near the zenith. In our model, the
sky is considered spatially contiguous and geometrically smooth. When
an object (like the moon) breaks the contiguity of the surface, humans
perceive an occlusion of the surface rather than an object appearing
through a hole. Binocular vision dictates that the moon is distant, but this
perception model dictates that the moon is closer than the sky. To solve
the dilemma, the brain distorts the projections of the moon to increase the
binocular disparity, which results in an increase of the angular size of the
moon. The degree of the distortion depends on the apparent distance to
the sky, which is influenced by distance cues such as surrounding objects
and the condition of the sky. The closer the sky appears, the stronger the
illusion. At the zenith, few distance cues are present, causing difficulty
with distance estimation and weakening the illusion.
INTRODUCTION
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Humans puzzled by moon illusion for centuries
Countless theories proposed, dating back to Aristotle (Hershenson)
Moon illusion now researched more by psychological researchers
than mathematical and physical
No one theory has been accepted
Two purposes of our project: to introduce existing theories and to
propose a new theory
Our proposal: moon illusion caused by a contradiction between
binocular clues and occlusion clues
Binocular disparity indicates distance to the moon is greater than
perceived distance to the sky
TABLE 1: Summaries of the Apparent Distance, Size-Contrast, and Oculomotor
Micropsia/Macropsia theories and our rationales for rejection
THEORY
SUMMARY
Apparent Distance
Theory
RATIONALE FOR REJECTION
• First proposed by Alhazen in the • If the moon appears larger at
11th century
the horizon, the Apparent
• Humans perceive the sky as a
Distance theory requires that
two-dimensional surface
the moon also appears farther
• As objects move closer to the
away. This is contrary to what
horizon, their perceived angular
most claim to observe.
• Most claim moon appears
sizes decrease
closer or perhaps the same
distance at the horizon, but
not farther
• Perceived angular size of moon • Does not explain the
dependent upon angular sizes
occasional extreme degree of
of objects surrounding moon
expansion at the horizon
Size-Contrast
• Does not explain why the
(Restle)
Theory
• Theory likened to Ebbinghaus
illusion is not detected on
illusion (Figure 1): context
video camera
circles affect perceived sizes of
center circles
• Distance cues in the moon’s
• Poses similarity to the
vista cause eyes to experience
“flattened sky dome” model of
adjustments (McCready)
the Apparent Distance theory;
• Cues to “far” distances induce
results in the same sizemacropsia: objects appear
distance paradox.
Oculomotor
angularly larger than they
Micropsia/Macropsia
actually are
Theory
• Cues to “near” distances induce
micropsia: objects appear
angularly smaller than they
actually are
FIGURE 1: The Ebbinghaus illusion. Notice the center circles are actually the same size, but
surrounding context circles cause the center circles to appear larger (right) and smaller (left).
SURVEY
• Objective: To see if people, in general, perceive the moon as occluding the sky
• Question: Order the three dominant objects in the images (Figure 2), from
closest to farthest
• Results: 347 of the 450 participants (77.3%) perceived the moon as occluding
the sky
OUR PROPOSAL
First, we establish two rules about visual interpretation, using Hoffman’s Visual
Intelligence as a guide:
Rule 1: When humans see a spatially homogeneous area, they interpret it as a
spatially contiguous surface.
Rule 2: When humans see a small area disturbing the homogeneous area, they
interpret it as an object occluding the surface, rather than a hole in the surface.
With these rules we find:
• We perceive the sky as a spatially contiguous surface, and we interpret that the
moon occludes the sky
• The moon appears closer than the sky, but binocular vision dictates the moon is
distant, resulting in a contradiction
• To solve this contradiction: the brain distorts the visual projections of the moon,
causing the visual angle of the moon to increase
• Distortion is dependent upon perceived distance to the sky, which is influenced
by distance cues such as mountains, trees, and birds
• Distance cues are frequently available at the horizon but absent at the zenith,
causing perception of have difficulty estimating distance to the sky (Figure 3)
FIGURE 3: An illustration of the illusory phenomenon due to binocular disparity
We derived a function that models our brains’ distortion:
𝜃� = 2 tan−1
EXISTING THEORIES
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There are many existing theories to explain the moon illusion
Most prominent are Apparent-Distance theory, Size-Contrast theory, and
Oculomotor Micropsia/Macropsia theory
None of these theories sufficiently explain the cause of the illusory
phenomenon
Table 1 summarizes these theories and their shortcomings
𝜃
tan
2
𝛥𝛥
1−
𝑧
−1
where 𝜃̂ is the expanded angular size of the object, 𝜃 is the actual angular size of
the object, 𝑧 is the actual distance to the object, and ∆𝑧 is the desired displacement
of the object.
REFERENCES
FIGURE 2: The five images used in our survey to test distance perception (Arbelaez, et al.)
• Arbelaez, P., Fowlkes, C., and Martin, D., 2007. The Berkeley Segmentation Dataset and
Benchmark. [Online] . Available at:
http://www.eecs.berkeley.edu/Research/Projects/CS/vision/bsds/. [Accessed 27 July 2012]
• Hershenson, M. (1989). The Moon Illusion. Lawrence Erlbaum Associates, Hillsdale, NJ.
• Hoffman, D.D. (1998). Visual Intelligence: How We Create What We See. W. Norton &
Company Ltd., New York, NY.
• McCready, D. (1985). On size, distance, and visual angle perception. Perception &
Psychophysics 37, 323-334.
• Restle, F. (1970). Moon illusion explained on the basis of relative size. Science 167, 1092-1096.