2.4. PSYCHOPHYSICS 41
Ideally, the subject can differentiate precisely between perceived and not perceived stimuli, as in this case, the function’s gradient is as large as possible.
0 0.025 0.05 0.075 0.1 0.125 0.15
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
50% threshold
99% threshold
Contrast
Detectionprobability
FitData
Figure 2.12: Detection probability related to contrast, fitted S-shaped psychometric function (solid line) to data points (blue), 50% as well as 99% detection probabilities are marked with dashed lines.
It is always possible that a subject responses with an incorrect evaluation. An incorrect evaluation is taken into account with a failure rate γ and expresses the amount of wrong answers. Thus, it represents a limitation factor of the function.
The psychometric function can be expressed as follows:
F(x) =γ+ (1−γ−λ)· 1
1 +xα−β (2.5)
with
• γ, failure rate
• λ, probability rate
• 1
1+(αx)−β, logistic function, α corresponds to the 50% threshold, β is propor-tional to the gradient
• x, intensity, in this investigation x=K, x=∆L
Based on the gradient of the psychometric function it can be seen to what extent an increase in detection probability can be achieved with increasing contrast. The threshold of the psychometric function is denoted at 50% and describes the point at which bright and dark are perceived as equivalent.
2.4. PSYCHOPHYSICS 43
2.4.2 Determination of the threshold contrast
There are several methods to determine the just noticeable difference and subse-quently, the psychometric function. It is important to note that biological systems are variable in their responses, that is to say, a participant possibly can make a mistake. Adaptive methods are particularly efficient, since they adapt the stimulus of each test run to the result of the previous test run.
The contrast sensitivity is determined by means of the threshold contrast. In Ta-ble 2.2 adaptive methods that can be used for the determination of the threshold contrast are presented.
In the following, the methods mentioned in Table 2.2 will be explained in more detail.
• Method of average error. The participant has to adjust a variable stimulus so that it is just perceived (absolute threshold) repeatedly or coincides with a predetermined standard stimulus (difference threshold). From the partici-pant’s responses, the mean value and difference threshold are calculated. The difference between mean and difference threshold is called constant error.
• Limits method. A stimulus series is presented in which the reactivity is alter-nately increased or decreased. For example, a stimulus is increased in intensity in small steps until the stimulus is perceived. Then the stimuli presentation is continued with an above-threshold stimulus, which is lowered until the thresh-old is reached. With this method as many as possible series of stimuli have to be carried out, until at the end the mean value is assumed as an estimate for the threshold value.
• Method of constant stimuli. A further method for threshold determination is the so-called method of constant stimuli. In this method, previously defined stimuli are presented randomly one after the other. Since each stimulus is repeated n-times, the proportion of the “yes” responses (object perceived) can be calculated as a probability. Subsequent, the psychometric function is determined from these data points.
As already mentioned, adaptive methods are particularly suitable to determine the psychometric function, since the sensitivity of the subject increases with the stimulus level. In this work the method of constant stimuli was used for contrast determination in the laboratory investigations.
In Figure 2.13 a flow diagram that describes the determination of the psycho-metric function is illustrated. To determine the threshold value, a whole series of stimuli is presented as a data set. Additionally, for each experiment, the factor of the participant’s response is taken into account, this means that the stimulus needs to be changed constantly.
The threshold value is determined as follows:
During a test run, a systematic variation of the stimulus intensity (contrast change) occurrs in many intermediate steps. Therefore the contrast levels are randomized.
The participant’s answers are recorded by pressing a button on a input device (yes/
Method Presentation Task Order Criteria Method of
average error
Objects should be
indistinguishable
Participant adjusts equal lu-minance
- Contrast
sensitiv-ity difference is expressed by the adjustment error Limits
method Objects have variable con-trast
Threshold is adjusted with
up/ down
method
From seen
to not
seen and conversely
Mean value of all measurements
Method of contstant stimuli
Objects with different dis-crete contrast values
Decision, if seen
or not randomized frequency distribu-tion by 50% detec-tion probability
Table 2.2: Test methods for threshold contrast determination for interval scaled and nor-mally distributed variables according to [34].
Minimum
determination Yes
threshold Maximum threshold
1
intensityStimulus
No threshold
2 n Repetition
Detection
Figure 2.13: Determination of the detection threshold. Repeated target presentation at dif-ferent intensities; Individual minimum and maximum thresholds are set for two luminance levels. Thus each subject receives the same boundary conditions.
no answer), whether the stimulus was seen or not. If the participant is not sure of his answer, he does not press any button. In this case the perceived threshold is below the threshold value. This does not mean that the difference is invisible or imperceptible, but the detection probability decreases (< 50%). Depending on how the participant reacts to the stimulus, the intensity of the subsequent stimuli is automatically adjusted. The stimulus is thus adapted individually to the sensitivity of the subject and set to a fixed value at the beginning. The individual threshold values were determined at the end of the adaptation time.
In order to determine the perception thresholds from the respective contrast stimuli, the responses given by the subject were fed into a psychometric function. From this function the absolute threshold contrast is calculated. In case of yes /no experi-ments, the threshold is usually set to a probability of 50% (of the “yes” responses).
In this investigation the 99% detection probability was determined and is also
de-2.4. PSYCHOPHYSICS 45 fined as threshold value in the further course of this work.
2.4.3 Automatic Staircase
Classical psychophysical methods have the disadvantage that, depending on the design, they lose a considerable amount of time that is used to examine stimuli intensities away from the threshold. Adaptive Staircase processes stand out due to their simplicity and flexibility. The stimuli are varied by an algorithm depending on the previous response of the test person. This results in a temporal advantage in determination of the threshold as the detection object intensities are dynamically adapted. The Staircase method approaches the suspected threshold very quickly and thus uses the maximum amount of information from each stimulus presenta-tion [67]. Therefore, most of the presented stimuli are very close to the threshold.
The required time for a threshold determination can thus be greatly shortened.
In Figure 2.14 an example for the course of the threshold determination by means of one test iteration is presented. On the left hand side the initial values of the upper (SO, solid line) and lower (SU, dashed line) Staircase response are indicated: If the object is not perceived the intensity is increased (+), if the object is perceived the intensity is reduced (-).
Stim uliintensit
y
Number of steps
0 2 4 6 8 10
Upper Staircase Lower Staircase 0.5
0 0.25
Figure 2.14: Schematic representation of the double Staircase according to [6] [7]. Initial values of the upper (SO, solid line) and lower (SU, dashed line) Staircase response: if the object is not perceived the intensity is increased (+), if the object is perceived, the intensity is reduced (-).SO: start value.
The initial values SO and SU of the two staircase procedures are determined individually for each participant in a rough determination of the threshold [7] [6].
For this purpose, the intensity of the detection object is increased by the participant
based on the maximum representable contrast of the object from 0% in steps of 10%, until the object can be safely detected for the first time. Based on this value, the initial values can be set flexibly. According to preliminary tests SO was set to 99%
and SU to 5% of this contrast.
A detailed description of the adaptive methods is omitted here. A more detailed description of the Automatic Staircase method used in this work can be found in [6].