Perceptual Coherence : Hearing and Seeing

132 Perceptual Coherence
light-sensitive units. The receptors learn not by being right or wrong, but
by maximizing mutual information under the sparseness constraint. What
we are mathematically trying to find is a set of independent basis functions
that when added together reproduce the image of the natural scene. 11
We impose the sparse-coding restriction that the output of each basis
function is zero for most inputs but strongly excitatory or inhibitory for a
small number of inputs. In terms of the mathematical analysis (principal
components), the basis functions are the causes of the visual scenes so that
if added together they will reproduce the natural scenes. If the causal basis
functions resemble the space-time receptive fields of retinal or cortical receptor
cells that decode the brightness patterning of visual scenes, then it is
possible to argue, somewhat indirectly, that the physiological receptive
fields have evolved to maximize mutual information by matching the
causes.
Field (1987), following such a strategy, found that the basis functions
indeed resembled Gabor functions similar to the receptive fields of cortical
neurons (see figures 2.7 and 2.14 for cortical receptive fields). Compared to
normal (Gaussian) probability distributions, the Gabor-like functions had
sharper probability peaks at zero response and higher probabilities of extreme
response rates. Field (1987) further demonstrated that by maximizing
the sparseness of the distribution, he was able to match the bandwidth
(approximately 0.8–1.5 octaves) and vertical-horizontal aspect ratio (2 to1)
of cortical cells. This outcome supported the notion of a sparse code: Individual
neurons would be active only when a particular brightness pattern
(i.e., a causal basis) occurred at a specific location and orientation. Any single
scene would stimulate only a small set of these receptors, but each possible
scene would activate a different set of receptors.
Olshausen and Field (1996, 1998) “trained” a set of basis functions using
image patches randomly selected from natural scenes to minimize the
difference between the image patch and the reconstruction using the addition
of basis functions. Minimizing the number of basis functions by
adding a cost for each additional basis function imposed sparseness. The
resulting set of basis functions was overcomplete, so that there were more
basis functions than the dimensionality of the input. 12 The theoretical advantage
of an overcomplete basis set is that the visual (and auditory) worlds
11. The set of sine and cosine functions for the Fourier analysis is one basis set. A set of
Gaussian functions, as illustrated in figure 3.6, is another basis set. We can produce any arbitrary
curve by multiplying the height of each Gaussian by a constant and then summing the resulting
Gaussian functions together point by point.
12. For example, an overcomplete set might have 200 basis functions to describe a 12
pixel × 12 pixel image that contains only 144 independent points. Because the set is overcomplete,
there are an infinite number of solutions to describing the 144 image points. The sparseness
criterion is necessary to force a single solution.