Motion parallax is a difference in image velocity that is due to a
difference in depth. One interesting example of motion parallax is
the motion of falling snowflakes during a snowfall. Snowflakes that
are close to the observer fall with a greater image velocity than
snowflakes that are far from the observer. This speed/distance effect
correlates with a size/distance effect, namely closer objects appear
larger. The two effects together suggest a formal relationship
between the spatial and temporal frequency components of a motion
parallax stimulus such as falling snow. Following the motion plane
model of (Watson and Ahumada 1985, Journal of the Optical Society of
America A 2 322-342) that w_t = speed * w_y for a vertically
moving stimulus, we observe that speed is inversely related to spatial
frequency sqrt{w_x^2 + w_y^2} for a motion parallax stimulus,
and that the two equations combine to yield a non-planar
surface: w_t = w_y / sqrt{w_x^2 + w_y^2} in the 3D
frequency. When such a surface is constructed using 1/f amplitudes
and random phases, and the inverse Fourier transform is computed, the
resulting image sequence produces a motion in depth effect that looks
remarkably like falling snow. Besides being an interesting
application for computer graphics, this method can also be used to
create psychophysical stimuli for studying human motion perception.
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