MIT Encyclopedia of the Cognitive Sciences - Cryptome

column and ocular dominance column structure of primate
V1, was introduced by Landau and Schwartz (1994), making
use of a new construct in computational geometry called
the “protocolumn.”
Global Map Function
One obvious functional advantage of using strongly spacevariant
(e.g., foveal) architecture in vision is data compression.
It has been estimated that a constant-resolution version
of visual cortex, were it to retain the full human visual field
and maximum human visual resolution, would require
roughly 10 4 as many cells as our actual cortex (and would
weigh, by inference, roughly 15,000 pounds; Rojer and
Schwartz 1990b). The problem of viewing a wide-angle
work space at high resolution would seem to be best performed
with space-variant visual architectures, an important
theme in MACHINE VISION (Schwartz, Greve, and Bonmassar
1995). The complex logarithmic mapping has special
properties with respect to size and rotation invariance. For a
given fixation point, changing the size or rotating a stimulus
causes its cortical representation to shift, but to otherwise
remain invariant (Schwartz 1977). This symmetry property
provides an excellent example of computational neuroanatomy:
simply by virtue of the spatial properties of cortical
topography, size and rotation symmetries may be converted
into the simpler symmetry of shift. One obvious problem
with this idea is that it only works for a given fixation direction.
As the eye scans an image, translation invariance is
badly broken. Recently, a computational solution to this
problem has been found, by generalizing the Fourier transform
to complex logarithmic coordinate systems, resulting
in a new form of spatial transform, called the “exponential
chirp transform” (Bonmassar and Schwartz forthcoming.)
The exponential chirp transform, unlike earlier attempts to
incorporate Fourier analysis in the context of human vision
(e.g., Cavanagh 1978), provides size, rotation, and shift
invariance properties, while retaining the fundamental
space-variant structure of the visual field.
Local Map Function
The ocular dominance column presents a binocular view of
the visual world in the form of thin “stripes,” alternating
between left- and right-eye representations. One question
that immediately arises is how this aspect of cortical anatomy
functionally relates to binocular stereopsis. Yeshurun
and Schwartz (1989) constructed a computational stereo
algorithm based on the assumption that the ocular dominance
column structure is a direct representation, as an anatomical
pattern, of the stereo percept. It was shown that the
power spectrum of the log power spectrum (also known as
the “cepstrum”) of the interlaced cortical “image” provided
a simple and direct measure of stereo disparity of objects in
the visual scene. This idea has been subsequently used in a
successful machine vision algorithm for stereo vision (Ballard,
Becker, and Brown 1988), and provides another excellent
illustration of computational neuroanatomy.
The regular local spatial map of orientation response in
cat and monkey, originally described by Hubel and Wiesel
Computational Neuroanatomy 165
(1974), suggested the hypothesis that a local analysis of
shape, in terms of periodic changes in orientation of a stimulus
outline, might provide a basis for shape analysis
(Schwartz 1984). A parametric set of shape descriptors,
based on shapes whose boundary curvature varied sinusoidally,
was used as a probe for the response properties of
neurons in infero-temporal cortex, which is one of the final
targets for V1, and which is widely believed to be an important
site for shape recognition. This work found that a subset
of the infero-temporal neurons examined were tuned to
stimuli with sinusoidal curvature variation (so-called Fourier
descriptors), and that these responses showed a significant
amount of size, rotation, and shift invariance (Schwartz
et al. 1983).
See also COLUMNS AND MODULES; COMPUTATIONAL
NEUROSCIENCE; COMPUTATIONAL VISION; COMPUTING IN
SINGLE NEURONS; STEREO AND MOTION PERCEPTION;
VISUAL ANATOMY AND PHYSIOLOGY
—Eric Schwartz
References
Ballard, D., T. Becker, and C. Brown. (1988). The Rochester robot.
Tech. Report University of Rochester Dept. of Computer Science
257: 1–65.
Bonmassar, G., and E. Schwartz. (Forthcoming). Space-variant
Fourier analysis: The exponential chirp transform. IEEE Pattern
Analysis and Machine Vision.
Cavanagh, P. (1978). Size and position invariance in the visual system.
Perception 7: 167–177.
Dow, B., R. G. Vautin, and R. Bauer. (1985). The mapping of
visual space onto foveal striate cortex in the macaque monkey.
J. Neuroscience 5: 890–902.
Hubel, D. H., and T. N. Wiesel. (1974). Sequence regularity and
geometry of orientation columns in the monkey striate cortex.
J. Comp. Neurol. 158: 267–293.
Landau, P., and E. L. Schwartz. (1994). Subset warping: Rubber
sheeting with cuts. Computer Vision, Graphics and Image Processing
56: 247–266.
Rojer, A., and E. L. Schwartz. (1990a). Cat and monkey cortical
columnar patterns modeled by bandpass-filtered 2D white
noise. Biological Cybernetics 62: 381-391.
Rojer, A., and E. L. Schwartz. (1990b). Design considerations for a
space-variant visual sensor with complex-logarithmic geometry.
In 10th International Conference on Pattern Recognition,
vol. 2. pp. 278–285.
Schwartz, E. L. (1977). Spatial mapping in primate sensory projection:
Analytic structure and relevance to perception. Biological
Cybernetics 25: 181–194.
Schwartz, E. L. (1980). Computational anatomy and functional
architecture of striate cortex: A spatial mapping approach to
perceptual coding. Vision Research 20: 645–669.
Schwartz, E. L. (1984). Anatomical and physiological correlates of
human visual perception. IEEE Trans. Systems, Man and
Cybernetics 14: 257-271.
Schwartz, E. L. (1994). Computational studies of the spatial architecture
of primate visual cortex: Columns, maps, and protomaps.
In A. Peters and K. Rocklund, Eds., Primary Visual
Cortex in Primates, Vol. 10 of Cerebral Cortex. New York: Plenum
Press.
Schwartz, E. L., R. Desimone, T. Albright, and C. G. Gross.
(1983). Shape recognition and inferior temporal neurons. Proceedings
of the National Academy of Sciences 80: 5776–5778.