Perceptual Coherence : Hearing and Seeing

Characteristics of Auditory and Visual Scenes 111
happens on all time and frequency scales. The relationship between
amplitude and frequency is invariant. 7
For a random walk process, the amplitudes of the Fourier waves decrease
at the rate of 1/f 2 so that for each doubling of frequency (an octave),
the amplitude of the wave decreases fourfold. The brightness or pressure of
any next point is equal to the value of the first point plus a random increment.
Thus, there will be just slow changes in the amplitudes and there
will be high correlations between the values of adjacent points. As the distance
between the two points increases, the correlation gradually decreases
due to summing more and more random increments. The autocorrelation
between points separated by a constant distance will always be the same.
The difference between successive values is a series of random values and
therefore creates white noise (1/f 0 ).
The most interesting outcomes occur when the amplitudes decrease at
the rate of 1/f. Every time frequency is doubled the amplitude is decreased
by one half, so that the energy in each octave remains identical. Many
physical processes follow this 1/f relationship, including nerve membrane
potentials, traffic flow, sunspot activity, and flood levels of the Nile River.
Schroder (1991) pointed out that 1/f functions can be conceptualized as the
combination of several processes with different distances and timings and
that the end result is that there is a varying correlation among the fluctuating
levels across the frequency range. Consider a simple simulation suggested
by Voss (Gardner, 1978) based on the sum of three dice. We start by
throwing all three and recording the sum. Then only one die is selected,
rethrown, and the new sum recorded. On the next trial two dice are selected,
both are rethrown, and the new sum recorded. On the third trial, only one
die is selected, rethrown, and the sum recorded. On the fourth trial, all three
dice are selected and thrown and the sum recorded. All the following trials
follow the identical sequence. In this simulation, the correlation between
adjacent points is not constant. The correlation between points in which
one die changed will be 2/3 because two of the three dice remain the same;
the correlation between adjacent points in which two dice changed will be
1/3; and the correlation between adjacent points in which all three dice
changed will be zero. Thus, there will be regions in time or space of correlated
sums. The number of independent processes (i.e., the number of dice)
will determine the size of the correlated regions. All 1/f processes can be
represented by the linear sum of short-range processes that have different
time scales (Wagenmakers, Farrell, & Ratcliff, 2004). For example, heart
7. Power laws have been used to describe an extraordinary range of phenomena: the frequency
of earthquakes of different magnitudes, the distribution of income among individuals,
the energy for metabolism for animals with different body masses, and the density of animals
with different body masses (Marquet, 2002).