Perceptual Coherence : Hearing and Seeing

116 Perceptual Coherence
Field argued that a 1/f amplitude relationship would occur if the relative
contrasts were independent of viewing distance. Imagine taking a simple
picture of a black-and-white grating and moving it twice the distance away.
Now the black-and-white bars are half as wide, so that the frequency of the
grating is twice the original frequency. If the amplitude falls off at 1/f, each
octave will still have equal energy, so that the relative power in each doubled
frequency will not change. Moreover, each octave will have the same
variance (i.e., the same amount of information), so that if each neural unit
has the same bandwidth, each will transmit an equal amount of information.
Ruderman (1997) pointed out that the 1/f relationship for natural
scenes is remarkably constant even for striking changes in the brightness of
individual pixels. A strong test of the 1/f relationship can be done by first
averaging small blocks of pixels, say all 2 × 2 blocks, and then demonstrating
that the 1/f relationship still holds for the averaged signal. (Since mixing
pixels reduces the total contrast, the contrast would need to be
renormalized.) As an extreme case of blocking, Ruderman converted all
pixels below the average brightness to black and all pixels above average
brightness to white. All such conversions did not affect the 1/f scaling,
demonstrating that whatever statistical structure exists at one spatial (i.e.,
angular) grain exists at all levels of the grain.
Up to this point, the 1/f amplitude-scaling factor was calculated by measuring
the correlations among the pixels in one static image, although it is
clear that images change gradually in time just as they change in space.
Dong and Atick (1995) argued that in general it is impossible to separate
the spatial and temporal variation. However, in cases in which the spatial
and temporal regularities can be separated, the temporal power scaling factor
is 1/w 2 (in Hz) and the amplitude scaling factor is 1/f (also see Hateren,
1993). Thus the spatial and temporal scaling factors are roughly the same.
To explain this outcome, Dong and Atick suggested that there is a distribution
of static images at different distances and also that there is a distribution
of relative motions at different velocities (essentially the same type
of distribution found for relative contrast levels).
Implications of 1/f c Power Laws
Why is this important? Remember the basic argument in chapter 1 that contrast
provides the critical information for audition and vision. If the amplitude
falls off at 1/f, then there will be roughly equal contrast auditory and
visual energy in all octave bandwidths. If the visual system or the auditory
system is organized into sensors with octave bandwidths, which seemed to
be true for the visual and auditory space-time receptive fields described in
chapter 2, then each sensor will encode and transmit an equal amount of information
about the contrast in terms of the variance in its firing rate. As is