Fourier and Non-Fourier Motion
Nonlinear preprocessing in short-range motion
Ethan Taub, Jonathan D. Victor, and Mary M. Conte
Vision Research 37, 1459-1477 (1997)
Abstract
The phenomenon of non-Fourier motion
(visually-perceived motion that cannot be explained simply
on the basis of the autocorrelation structure of the visual
stimulus)
is well-recognized, and is generally
considered to be due to nonlinear preprocessing of
the visual stimulus prior to a stage of standard motion
analysis. We devised a
sequence of novel visual stimuli
in which the availability of a motion stimulus
depends on the nature of the nonlinear preprocessing:
an nth order stimulus Pn will
generate a perception of
motion if it is preprocessed by a nonlinearity of polynomial
order n or greater, but not if preprocessed by a
nonlinearity of polynomial order less than n.
We found that unambiguous motion direction was perceived for
P2, P3, and P4,
but not for higher-order stimuli, and we measured the contrast
thresholds for direction
discrimination with superimposed noise.
We found that an asymmetric compressive nonlinearity can, in a
unified fashion, account for these results,
while a purely quadratic nonlinearity or a rectification of the
form
T(p) = |p| cannot.
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Definition of nth-order motion stimuli
These definitions relate to the stimuli themselves,
not how they are perceived.
- An 1st-order motion stimulus is a stimulus whose
spatiotemporal
contrast function has directional motion energy, and thus, can
be identified as moving by a Reichardt motion detector.
- For n = 2, 3, 4,..., an nth-order motion stimulus is a
stimulus
whose spatiotemporal contrast function has directional motion
energy only if it is first passed through a polynomial of order n
(and not
of lower order).
Other than the examples in this paper, all previous examples of
"second-order motion stimuli" are second-order by this definition
as well.
Note that this definition of third-order stimuli diverges from
that used by
Lu & Sperling (Vision Research 35, 2697-2772 (1995))
who use the term to describe feature-based motion.
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Construction of nth-order motion stimuli
The following equation provides a construction for the
spatiotemporal
contrast function Pn
of an nth-order motion stimulus. Here,
(x1,x2) is pixel position,
t is time,
(k1,k2) is spatial frequency, w is temporal
frequency,
and R(x1,x2) is a number chosen randomly
from {0,1,..., n-1},
independently at each pixel (x1,x2).
-
P1 is an ordinary drifting grating.
-
P2 is a contrast-modulated drifting grating, whose
motion can be detected
by local squaring or rectification, followed by standard motion
analysis.
-
P3 is similar to a contrast-modulated drifting
grating,
but the bars
consist (roughly) of (a) a small number of very dark pixels in a
sea of moderately bright pixels,
and (b) a small number of very light pixels in a sea of
moderately dark pixels.
Its motion cannot be detected
by local squaring, but can be detected by asymmetric
rectification,
followed by standard motion analysis.
-
P4 is similar to a contrast-modulated drifting
grating,
but the bars
consist (roughly) of (a) a small number of very dark and very
bright pixels in a sea of
pixels at mean illumination,
and (b) a large number of moderately light and dark pixels.
Its motion cannot be detected
by local squaring or even by a local nonlinearity of
polynomial order 3,
but can be detected by rectification,
followed by standard motion analysis.
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Controversy
See Lu and Sperling, JOSA A18, 2331-2370 (Three-systems theory of human visual motion perception: review and update) for a spirited defense of the separation of
first- and second-order motion pathways. They point out that the Taub/Victor/Conte model cannot account for certain aspects of motion perception in stimuli that are contain mixtures of first- and second-order signals. This is likely correct, but it should also be pointed out that the Taub/Victor/Conte model is grossly oversimplified, has no free parameters, and makes no pretense to account for how stimuli that contain motion energy at more than one spatial frequency might be integrated within one kind of motion pathway.
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