A geometric model for perceived line orientation based on dissimilarity estimates and human VEP data

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• From a geometrical point of view the angle is a
  combination of two lines. It is in accordance
  with intuitive experience that angles are more
  complex perceptual stimuli than lines, and
  perception of angles is accomplished at a higher
  level of information processing than perception
  of lines. This claim is supported by both
  neurophysiologic (Hubel Wiesel, 1962, Shevelev
  et al., 2000) and psychophysical (Granovskaya,
  Bereznaya Grigoreva, 1981 Selfridge Neisser,
  1974) arguments.
• However, another data (Sillito Versiani, 1977
  Supin, 1981) permit to suggest an alternative
  idea, namely that lines and angles are equally
  specific percepts, and discrimination of angles
  is neither more complex nor simpler than
  discrimination of lines. Some computer programs
  which recognize three-dimensional objects
  represented by contour drawings (Guzman, 1968
  Waltz, 1975) have been based on a set of 8-10 key
  stimuli. The keys were combinations of two,
  three, four or five lines. This set of basic
  combinations can be considered as an alphabet of
  contour scenes. It may be assumed that the basic
  combinations of lines are perceptually
  equivalent, independently from geometrical
  complexity of stimuli.

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Psychophysical part of research consists from
four sets of experiments.
In first experiment 10 lines of different
orientations (from 0 up to 162 in relation to
visual horizontal axis) were used as a 45 pair
combination of lines.
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In the next experiment 30 stimuli (12 angles
from 23 up to 165 having different orientations
in a frontal plane) were presented on a TV
monitor.
Example of stimuli angles
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In the third experiment 24 angles were presented
as fingers on schematic watch dial, and in the
forth experiment 10 lines were presented as a
small finger on a schematic watch dial. One named
these stimuli as line segments because one of the
ends of line-finger was fixed.
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Fig. 3-13. Example of visual evoked potential
recorded from the occipital area of the humans
brain as response on abrupt changes color
stimuli.
N1(N87), N1(N87) P1(P120) - color components of
VEPD P1(P120) N2(N180), N2(N180), N2(N180)
P2(P230) pattern components of VEPD
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Fig. 3-15. Matrix of evoked potentials of
differences (DEP), recorded in response to abrupt
stimulus (line orientations) change. Rows and
columns of matrix represent the same set of
stimuli while the entries contain all pair-wise
dissimilarities from the cortical point of
view. Amplitudes of the component increasing
monotonically with increasing of differences
between stimuli can be used as measures of
dissimilarity.

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Spherical model of line orientations obtained by
multidimensional scaling of dissimilarities
estimates. Two dimensions of Euclidean plane
represent dual-channel neuronal network detecting
stimuli orientations on the frontal plane of
visual space. First axis corresponds
horizontal-vertical opponent channel, and second
one left-right declination channel.
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Spherical model of line-segments obtained by
multidimensional scaling of dissimilarity
estimates. Two dimensions of Euclidean plane
represent dual-channel neuronal network detecting
stimuli orientations on the frontal plane of
visual space. First axis corresponds
horizontal-vertical opponent channel, and second
one left-right declination channel.
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b) Psychophysical function of line-segments
orientation obtained from spherical model.
Abscissa axis represents orientation of small
fingers oh schematic watch dials, and axis of
ordinate represents perceived orientations
measured as angles of points on the plane X1X2..

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Location of points-stimuli (angles) in
two-dimensional space. The configuration of
points can be presented by semicircle on a plane.
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Psychophysical function of perceived angle
values obtained from spherical model. Abscissa
axis represents angles of stimuli, and axis of
ordinate represents perceived angles measured as
spherical coordinate of point on the X1X2 plane.

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Projections of point-stimuli on X1X2 plane of
two-dimensional space obtained by
multidimensional scaling of interpeaks amplitudes
of N180-P230, recorded on a stimuli-lines
differences. Two dimensions of Euclidean plane
represent dual-channel neuronal network detecting
stimuli orientations on the frontal plane of
visual space. The circle and the direct lines
represent spherical and city-block models of
points configurations. The both models
correspond to points configurations.
Occipital sites O1 and O2
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Projections of point-stimuli on X1X2 plane of
two-dimensional space obtained by
multidimensional scaling of interpeaks amplitudes
of N180-P240, recorded on a stimuli-lines
differences. Two dimensions of Euclidean plane
represent dual-channel neuronal network detecting
stimuli orientations on the frontal plane of
visual space. The circle and the direst lines
represent spherical and city-block models of
points configurations.
Temporal sites T5 and T6
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Projections of point-stimuli on X1X2 plane of
two-dimensional space obtained by
multidimensional scaling of early interpeaks
amplitudes of P120-N180, recorded on a
stimuli-lines differences. Only first dimension
of Euclidean plane (X1) represents a monotonic
correspondence with stimuli orientations on the
frontal plane of visual space. The second one
reflects a noise of neuronal environment.
Occipital sites O1 and O2
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Projections of point-stimuli on X1X2 plane of
two-dimensional space obtained by
multidimensional scaling of interpeaks amplitudes
of P120-N180, recorded on a stimuli-lines
differences. Only first dimension of Euclidean
plane (X1) represents a monotonic correspondence
with stimuli orientations on the frontal plane of
visual space. The second one reflects a noise
of neuronal environment.
Temporal sites T5 and T6
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In the third experiment 24 angles were presented
as fingers on schematic watch dial, and in the
forth experiment 10 lines were presented as a
small finger on a schematic watch dial. One named
these stimuli as line segments because one of the
ends of line-finger was fixed.
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Circular representation of stimuli in subjective
space
Dimension Schematic watch dials Schematic watch dials Schematic watch dials Schematic watch dials Line orientation Line orientation Line segment Angles
Dimension 1 2 3 4 2 2 2 2
Coefficient of correlation 0.63 0.77 0.92 0.93 0,97 0,97 0,97 0,98
Stress 0.46 0.23 0.10 0.08 0,06 0,03 0,08 0,06
Mean radius 1,79 1,93 2,0 1,0 1,0
Standard deviation 0,16 0,11 0,11 0,04 0,05
Coefficient of variation 8.9 6.0 5.5 4,3 5,2 8,1 9,0
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Projections of point-stimuli on X1X2 plane of
three-dimensional space obtained by
multidimensional scaling of dissimilarities
estimates. The stimuli were angles between
fingers on schematic diagrams of watch dials. Two
dimensions of Euclidean plane represent
dual-channel neuronal network detecting stimuli
orientations on the frontal plane of visual
space. First axis corresponds horizontal-vertical
opponent channel, and second one left-right
declination channel.
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Psychophysical function of perceived
orientations of angles obtained from spherical
model of schematic watch dials. Abscissa axis
represents orientation of bisectors of stimuli,
and axis of ordinate represents perceived
orientations of angles between fingers measured
as spherical coordinates of points on the X1X2
plane of three-dimensional space
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Projections of point-stimuli on X1X3 plane of
three-dimensional space obtained by
multidimensional scaling of dissimilarities
estimates. The stimuli were angles between
fingers on schematic diagrams of watch dials. Two
dimensions of Euclidean plane represent
dual-channel neuronal network detecting values of
stimuli-angles on the frontal plane of visual
space. First axis corresponds horizontal-vertical
opponent channel, and second one is a projection
of .twodimensional space of angle discrimination
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Three stage neuronal net of color vision
Light stimulus
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Fig. 3-13. Example of visual evoked potential
recorded from the occipital area of the humans
brain as response on abrupt changes color
stimuli.
N1(N87), N1(N87) P1(P120) - color components of
VEPD P1(P120) N2(N180), N2(N180), N2(N180)
P2(P230) pattern components of VEPD
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P120-N180 ? N180-P230 ???, ??????????????????? ?
?????????? ?????????? ?????? (??????? ???), ?
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P120-N180 (?????? ???????), ??????? ???????? N180
(??????? ???????) ? ?????????? ???????? N180-P230
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?1?2 ?????????????? ???????????? ?????????????
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??????? ???????? N180 (??????? ???????) ?
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The schematic diagram of neuronal network
detecting line orientations on a frontal visual
plane. Dual channels system of the pre-detectors
provides both a sine and cosine- transformation
of the stimulus for each of orientation
detectors. The detector is characterized by a
specific composition of synaptic inputs which
define its selectivity for a given orientation of
stimulus. This structure is a base of sphericity
of stimuli representation in subjective space.
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Circular representation of stimuli in the
cortical space
Indexes of spheroid Interpeaks amplitudes N2(N180) P2(P230) - Interpeaks amplitudes N2(N180) P2(P230) - Interpeaks amplitudes N2(N180) P2(P230) - Interpeaks amplitudes N2(N180) P2(P230) - Interpeaks amplitudes N2(N180) P2(P230) - Interpeaks amplitudes N2(N180) P2(P230) - Estimates of dissimilarities
Indexes of spheroid O1 O2 T5 T6 P3 P4 Estimates of dissimilarities
Stress 0.01 0.03 0.01 0.01 0.01 0.01 0.02
Coefficient of variation 5.8 9.5 12.5 8.0 9.3 12.9 5.2
Coefficient of correlation
Euclidean metrics 0.94 0.91 0.94 0.89 0.91 0.91 0.973
City-block metrics 0.91 0.88 0.94 0.85 0.93 0.90 0.912
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Circular representation of stimuli in the
cortical space
Indexes of spheroid Interpeaks amplitudes P1(P120) - N2(N180) Interpeaks amplitudes P1(P120) - N2(N180) Interpeaks amplitudes P1(P120) - N2(N180) Interpeaks amplitudes P1(P120) - N2(N180) Interpeaks amplitudes P1(P120) - N2(N180) Interpeaks amplitudes P1(P120) - N2(N180) Estimates of dissimilarities
Indexes of spheroid O1 O2 T5 T6 P3 P4 Estimates of dissimilarities
Stress 0.01 0.03 0.01 0.01 0.01 0.01 0.02
Coefficient of variation 8.5 8.6 21.0 13.9 17.4 13.8 5.2
Coefficient of correlation
Euclidean metrics 0.92 0.81 0.91 0.89 0.91 0.96 0.973
City-block metrics 0.93 0.77 0.91 0.84 0.93 0.92 0.912