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Psychophysical methods: method of constant stimuli
Hans du Buf
UALG Vision Laboratory
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Introduction
Threshold detection is a probabilistic process. An observer looks at a screen
with a certain background luminance and hence brightness. On this background
can be superimposed a stimulus pattern, for example a disk with a certain
size and luminance increment. A presentation of the disk needs a specification
of the temporal function. Because there are different subsystems called the
transient and sustained system, with quite different spatial characteristics,
either shortly pulsed temporal envelopes are used with different durations
(in this case the flanks of the pulse are very steep, a true step function),
or a quasistatic pulse that can consist of a plateau of 300 ms that is
flanked by two transitions of also 300 ms each: a gradual rise and a decay.
Both can consist of an errorfunction that is truncated at 1% of the left and
right asymptotic tails (polynomial approximations can be found in Abramowitz
and Stegun, Handbook of mathematical functions).
   The difference between threshold curves of briefly flashed and quasistatic
incremental and decremental disks (luminance increments or decrements) can be
seen in: du Buf, J.M.H. (1991) Detection symmetry and asymmetry. Spatial Vision 
5, pp. 189-203.
   Also note that our pupil can vary in diameter. It is best to observe through
an artificial pupil, for example 2 mm in diameter, equipped with a system to
check the centering of the two pupils.
   Now, let us suppose that we want to determine the threshold of an incremental
disk with a certain size and temporal envelope. The observer must concentrate
on the position where the disk is going to be presented, hence there must be
two or four points on the screen (can be coloured) around the centre of the
disk position. Also needed are a signal to warn the observer that within e.g.
half a second there will be a stimulus presentation, as is an instruction of
the observer like: respond with yes if you think you saw a distortion of the
background, else with no. Note also that observers need to be trained; never
use data from first experiments, let them train for a few hours...
   In this way, starting with a luminance increment of zero, increasing the
luminance increment a little bit at each stimulus presentation, the observer
will reply with a series of: no, no, no, ..., no, no, perhaps, perhaps, ...,
perhaps, yes, yes, yes. In reality the no's and perhaps's and yes's can be
mixed. By excluding the "perhaps" reply, the observer will reply with a certain
number of no's and yes's, and the threshold is defined at the luminance
increment with exactly 50% no's and 50% yes's. Actually, a well trained
observer has a very small transition interval between 100% no's and 0% no's.
By repeating stimulus presentations for the same luminance increment, and
doing so for different luminance increments, we can measure what is called
the psychometric function: the percentage of detection against the luminance
increment. This function looks like an errorfunction with a certain slope
at the 50% point.
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Psychometric function
If we repeat the above experiment one hour later or the next day, the threshold
will be different. This is due to the concentration of the observer and other
factors that influence his "neural state." The final threshold is defined as
the average (geometric mean) of at least 8 complete and independent experiments.
In the past there was a discussion about the slope of the psychometric function.
Some researchers systematically found very shallow functions (small slope),
whereas others found very steep functions. It turned out that a small slope
is caused by averaging over all experiments, and a large slope is obtained by
doing a very fast experiment. A repetition of the last approach will lead to
many steep functions that are displaced with a sort-of Gaussian distribution.
The latter explains the shallow function obtained with the other appraoch: if
you average over displaced steep functions you will obtain a shallow one.
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Method of constant stimuli
A well trained observer doing a fast experiment has a steep psychometric
function with a slope that is almost constant. Also, if we plot the function
on log paper it will be straight between 10% and 90%. This means that if we
do two fast experiments of 10 presentations with probabilities between these
values, we can connect the two points and the position of the 50% gives a
threshold value. This must be repeated at least 8 times to take the geometric
mean (it may be better to use more repetitions and to exclude outlyers assuming
a Gaussian distribution).
   But the slope of the well trained observer is constant! This means that once
we know his slope, we need only one series of 10 presentations with a probability
between 10% and 90% to determine a threshold value. Assuming 8 repetitions, this
means only a total of 80 presentations to determine the final threshold. This
is very efficient and the quality of the data is very good (see my paper ref-
erred to above).
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