Perceptual Coherence : Hearing and Seeing

110 Perceptual Coherence
along a whitewashed wall or speech rhythms. Higher-frequency waves represent
more rapid changes (e.g., tree trunks in a sunny forest, picket fences,
or a rapid musical trill), and even higher frequencies are seen and heard as
uniform and continuous (e.g., musical tones) due to limitations in neural
resolution. The Fourier representation is most appropriate for repeating
scenes that do not vary across space or time. Each component sine or cosine
wave is invariant.
If the amplitudes of the Fourier waves are plotted against their frequencies,
there are several prototypical outcomes:
1. The amplitudes of each component sine wave are equal, and the
phases are random. 6 The amplitude of one point in the scene does not
predict the brightness of another point.
2. There is a nonmonotonic relationship between frequency and amplitude.
For example, the sound of a musical instrument or the sound
of a vowel can be portrayed by the amplitude of two or more frequency
components. In most cases, the frequencies of the components
are multiples of the lowest fundamental frequency, and to
some degree the differences between instrumental sounds or spoken
vowels are due to the pattern of the amplitudes of those harmonic
frequencies. For example, the amplitudes of the odd harmonics of a
clarinet note are high, but the amplitudes of the even harmonics are
low due to the resonances of an open tube. If the instrument note or
vowel is recorded at one speed (or sampling rate) and replayed at another,
the note or vowel will sound different.
3. There are monotonic relationships between amplitude and frequency
such that amplitude is proportional to ∼1/f c where the exponent c can
range from 0 to 3 or 4. These functions are termed power laws and
have the important property of being self-similar. They are termed
fractal processes: The same power law relationship holds on all
scales. The relationship between amplitude and frequency would be
identical whether a camera is zoomed in on a small location or
zoomed out to encompass the entire field. When viewing a mountain
range, the pattern of altitude variation within one peak is similar to
the pattern of variation across peaks. As one zooms in, the same level
of complexity emerges; the global and local contours are similar in
the geometric sense. When listening to sounds that obey a power law
function, the pitch of the sound would not change if the speed of
playback was increased or decreased. There are no inherent time or
frequency scales—whatever happens in one time or frequency range
6. The Fourier representation of a narrow edge or sound click also contains equal amplitudes,
but the phases are also equal.