Submitted to Nature. 20 November 2007
A visual sense of number
David Burr
1,2
and John Ross
2
1.
Dipartimento di Psicologia, Università Degli Studi di Firenze, Via S. Nicolò 89,
Florence, Italy.
2.
Department of Psychology, University of Western Australia, Perth WA, Australia.
Key words: numerosity vision adaptation perception
Evidence exists for a non-verbal capacity to apprehend number, in humans
1
(including infants
2, 3
) and in other primates
4-6
. Here we show that perceived
numerosity is susceptible to adaptation, along with primary visual properties of a
scene like colour, contrast, size and speed. Apparent numerosity was decreased by
adapting to large numbers of dots and increased by adapting to small numbers, the
effect depended entirely on the numerosity of the adapter, not on contrast, size,
orientation or pixel density, and occurred with very low adapter contrasts. We
suggest that numerosity is also an independent primary visual property, not
reducible to others like spatial frequency or density of texture
7
.
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2
Jevons
8
, a 19
th
Century economist, rather than counting beans, assessed his accuracy in
estimating the number of beans in a box at a single glance. He made no errors at four or
below but became increasingly inaccurate as the number of beans increased beyond four.
Subsequent studies have confirmed his findings and the lack of errors below five has led
to the concept of subitizing
9-12
, usually presumed to be a separate process allowing
immediate apprehension of the numerosity of collections containing fewer than five
objects. The perception of larger numbers is usually assumed to involve other more
cognitive processes, like counting.
All primary visual properties are susceptible to adaptation, sometimes giving
rise to dramatic aftereffects, like the waterfall illusion
13
, and changes in colour, size,
distance, spatial frequency and orientation. If numerosity were a primary property, like
colour or motion, it too should be prone to adaptation. The on-line demonstration shows
that it is. After 30 seconds adaptation to the two adapter patches, the two subsequent
patches appear to differ considerably in numerosity (while inspection or counting after
adaptation wears off shows that they both comprise 30 dots). We quantified adaptation
effects by asking subjects whether a test stimulus (of variable numerosity), presented to
the region that had been adapted, appeared more or less numerous than a probe stimulus
(of fixed numerosity), presented to a different position a little later. The proportion of
trials where the test appeared more numerous than the probe was plotted against test
numerosity, and fitted with Gaussian functions whose mean estimates the point of
subjective equality (PSE) between test and probe, and standard deviation the threshold
for discriminating between the two (the just-noticeable difference: jnd). Fig. 1B shows
sample psychometric functions for a 30-element probe, with and without adaptation to a
400-element stimulus. The PSE of the test increases from a veridical 30 with no
adaptation to more than 100 after adaptation (the test number increased to compensate for
the reduction in apparent numerosity). Note also that that after adaptation the
psychometric function is steeper (on logarithmic coordinates), implying a smaller jnd.
We first measured the effect of adapting to a large number (400) of dots as a
function of number of dots in the probe (Fig. 1B). The amount of adaptation was fairly
constant with probe numerosity down to about 12 dots, then decreased as the probe
approached the subitizing range. The precision of the match, given by the jnd or Weber
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3
fraction (jnd) expressed as a fraction of dot number), did not deteriorate during
adaptation, average percentage Weber fractions for unadapted and adapted conditions,
being on average 28% for unadapted and 26% for the adapted conditions, similar to
published Weber fractions for numerosity
14, 15
.
We next investigated whether adapting to small numbers can cause an increase in
apparent numerosity. The red circles of Fig. 2 show that adaptation occurred in both
directions: adapting to small numbers increased apparent numerosity (so the matched
number decreased), and adapting to large numbers decreased apparent numerosity.
Adapting to 50 dots (the number of the probe) had no effect, with the amount of
adaptation increasing with the difference between adapt and probe number. The curves of
both subjects were well fit by linear regression on log coordinates, with a slope around
0.25.
In order to test whether adaptation depends on numerosity per se, or is derived
from other factors, like texture density
7
we performed a number of controls. We firstly
varied the size of the adapter and test dots, in order to vary pixel density. In the above-
described study (red circles of Fig. 2), both adapter and test dots were circles of 6 pixel
(20 arcmin) diameter (28 pixels area). We repeated the experiment with square adapter
stimuli of 8 X 8 pixels (64 pixels) and test stimuli of 3X3 pixels (9 pixels, 1/7 as many as
the adapter). If pixel density were the relevant attribute, the curves of Fig. 2 should shift
leftwards by a factor of 7, so the null point occurs when adapter and test pixel density are
matched (for adaptation dot number of 7). This clearly does not occur. For naïve observer
PB the curves remain superimposed, for DB there is a slight shift in the opposite
direction.
We also examined the effect of adapter contrast. As Fig. 2C shows, contrast of
adapter dots had little effect on the magnitude of adaptation. At contrasts as low as 12%,
the adaptation effect is still nearly two-fold, dropping only near detection threshold. It
appears that the only factor that affects adaptation is numerosity, not density, orientation,
or contrast.
As a direct control for the effects of texture we next adapted to vertical elements
and tested either vertical or horizontal elements. As the bar graphs of Fig. 3A show, there
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4
was little difference in the magnitude of the effects. If texture or spatial frequency were
being adapted, one would expect some specificity for orientation. We also performed
discriminations (without adaptation) for patterns of completely different pixel density,
orientation, Fourier transform etc. An example is shown in the psychometric functions of
Fig 3B, where the test or probe could be either small 5 X 5 pixel (16.5 X 16.5 arcmin)
squares or 20 X 5 pixel rectangles, randomly vertical or horizontal. Neither the PSE nor
the width of the curves depended on the type of stimuli being compared, even though the
stimuli were visually completely different, varied by a factor of 4 in pixel density and
Michelson contrast, and had completely different Fourier power spectra. All these results
agree with a recent study
16
showing that apparent numerosity of a field of dots can be
reduced by adding links between some dots: the linked pairs contribute to the numerosity
as single entities, rather as two separate dots.
We propose that just as we have a direct visual sense of the reddishness of half a
dozen ripe cherries so we do of their sixishness. In other words there are distinct qualia
17
for numerosity, as there are for colour, brightness and contrast. One distinctive feature of
the numerosity sense is that the Weber fraction (jnd expressed as a fraction of dot
number) is considerably higher for numerosity (around 25%) than, for example,
luminance (near 0.2%), possibly because of high prior uncertainty about the stimulus and
the informational limitations of the visual system
18
. The high Weber value accounts for
subitizing, without having to postulate a separate mechanism, as for numbers below 4 the
quantal leap to the next number is at least 25%, more than the Weber fraction (supporting
several recent studies that fail to find evidence for separate mechanisms for the subitizing
and counting range of numbers
12
).
One of the more fascinating aspects of this study readers can verify it for
themselves with the on-line demonstration is that although the total apparent number of
dots is greatly reduced after adaptation, one would be hard pressed to know which dots
disappear. This reinforces much recent evidence
18-20
in suggesting that the perceived
richness of our perceptual world is very much an illusion. Although we seem to perceive
30 or 50 or 100 individual dots occupying very specific positions, this cannot be the case,
as adaptation could not reduce or increase the total number of dots without annihilating or
creating them. Rather, it would seem that what is encoded is a statistical description of
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5
the scene, where some aspects of the elements (colour, shape, contrast etc), together with
a rough (±30%) estimate of their numerosity.
Recent studies have demonstrated the existence of neurons broadly tuned for
number in the parietal cortex of macaque monkeys
4, 21, 22
. fMRI studies also point to their
existence in a the intraparietal sulcus in humans, both for symbolic
12, 23, 24
and non-
symbolic
25
representation of numbers. These neurones are likely candidates for the
physiological substrate of the visual sense of number and, like most neurones, they are
probably adaptable. Vision has formidable in-built computational powers, correcting for
variation in image size with distance, in image shape with tilt and in image spectral
composition with changes in illuminant to allow approximately constant perception of
object size, shape and colour; it can also segment images, a difficult computational task
26
.
It should occasion little surprise that it can provide approximate estimates of number.
Methods
Stimuli Stimuli were generated by a framestore (Cambridge Research Systems VSG
Visage) and displayed on the face of a Hitachi Accuvue monitor at 170 Hz framerate,
with a resolution of 640 X 480 pixels and luminance of 18 cd.m
-2
. The 37 X 28 cm screen
subtended 35 X 26.5 deg at the viewing distance of 60 cm (each pixel 3.3 arcmin wide).
The stimuli were fields of small disks (of 6° diameter, unless otherwise stated), randomly
positioned within a circle of 10° diameter (similar in appearance to those of Fig. 1A). The
disks were half bright half dark, of 100% contrast (unless otherwise stated).
Experimental Procedure Subjects fixated a fixation spot at the centre of the screen. The
adaptation stimuli were centred 7° away from fixation, above left for half the sessions,
below right for the others. The test stimulus was displayed in the same position as the
adapter for 600 ms, then the probe for 600 ms, directly below or below the test (all
stimuli separated by a pause of 400 ms). Subjects adapted for 30 sec at the beginning of
each session, with 7 sec top-up adaptation between trials. On each trial subjects were
required to report whether the probe appeared more or less numerous than the test,
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6
guessing if unsure. After each trial, an adaptive algorithm (QUEST
27
) estimated the PSE
which, after addition of a random quantity (drawn from a log Gaussian distribution of
standard deviation 0.15 log-units) determined the probe number for the following trial.
The technique ensured an approximately equal number of right and left button presses, as
well as placing most trials at a numerosity to estimate best PSE and curve slope. The
proportion of "greater" trials was plotted against the logarithm of probe numerosity, and
fit with a cumulative Gaussian function (see Fig. 1B), whose mean yielded an estimate of
PSE and standard deviation an estimate of jnd.
Contrast thresholds (reported in Fig. 2C) were measured by a two-alterative
forced choice procedure. Half the dots (above or below a diagonal line radiating from
fixation) were removed, and subjects were required to identify in which half the dots
were confined. Again the QUEST
27
algorithm homed in near threshold, and threshold
was calculated by Gaussian fit (allowing for guessing).
Subjects Four subjects were measured systematically for most conditions, the two authors
and two others naïve to the goals of the study (PB and ED). Sample results are shown in
the figures.
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7
Fig. 1
The effect of adaptation on numerosity. A Sample psychometric functions plotting the
proportion of trials where the probe seemed more numerous, as a function of number of
test dots. The vertical dashed lines indicate the PSE of the match, about three times
higher than the probe number (indicated by the arrow) after adaptation. B Magnitude of
adaptation (test/probe dot number at PSE) as a function of the number of dots in the
probe. For a wide range of numerosities, adaptation caused a doubling of the matched
number.
10
100
400
0.0
0.2
0.4
0.6
0.8
1.0
P(gr
eater
)
Matched dot number
Control
Adapt to
400 dots
Probe
3
10
100 300
0.5
1.0
1.5
2.0
2.5
3.0
10
100
0.5
1.0
1.5
2.0
2.5
3.0
Test/Probe Dot Number
Probe dot number
DB
JR
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8
10
100
10
100
200
Matched d
o
t number
Contrast of Adaptor
No adaptation
detection
thresholds
10
100
500
20
50
100
Ma
tched
do
t n
u
mbe
r
Adaptor dot number
Large-small
Med-med
DB
10
100
500
20
50
100
Matched dot number
Adapt dot number
Large-small
Med-med
PB
Fig. 2
Effect of numerosity and contrast of the
adapter. A & B Effect of adapter
numerosity and density on apparent
numerosity of a 50-dot probe. The red
circles refer to adapter and test dots of
6 pixel (20 arcmin) diameter, the
squares to adapters of 8 X 8 and tests of
3 X 3 pixels (7 times more adapt than
test pixels for matched numerosity). In
all cases the adapters were of 50%
Michelson contrast, the tests 100%.
Adaptation occurs for both high and
low adaptation numbers, and is
independent of pixel density. C Effect
of adapter contrast on apparent
numerosity of a 30-dot probe (red
symbols DB, blue PB). The vertical
dashed lines indicate the contrast
threshold for detecting the patterns (see
methods), the horizontal lines the
matches with no adaptation. Adaptation
effects were pronounced down to near-
threshold contrasts.
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9
Fig. 3
Effect of element size and shape. A Effect of adapter orientation. Subjects adapted to a
field of 200 vertical elements (3 X 10 pixels), and matched a field of either vertical or
horizontal same-sized elements to a probe (same orientation as test). The effects of
orthogonal and parallel adapters were little different. B psychometric curves for matching
numerosity of element arrays that were either the same (5 X 5, or 5 X 20 pixels), or small
with large, or large with small. Element size and shape has very little effect of either PSE
or Weber fraction (given by the function width), suggesting that the matches were based
solely on number of elements.
10
100
0.0
0.5
1.0
P(more)
Dot number
Small-small
Big-big
Big-small
Small-big
Control
Parallel
Orthog
0
20
40
60
Matc
hed dot number
DB
ED
PB
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