### slides for Chapter 3

```Intensity Discrimination
Intensity discrimination is the process of
distinguishing one stimulus intensity
from another
Two types:
Difference thresholds – the two stimuli are
physically separate
Increment thresholds – the two stimuli are
Fig. 1.1
Increment threshold
stimuli (edges of stimuli
touch each other)
Difference threshold
(separated stimuli)
A
B
L
T
= L + L
L
T
= L - L
Fig. 1.2
L
T
L
L
T
L = L + L
T
L u m in a n c e
+ L
L
- L
L T = L-  L
0
L
Theory and Practice
Theory:
Quantum fluctuations provide a theoretical lower limit
for intensity discrimination by an “ideal” observer
Fig. 2.7
Mean = 16
From Dr. Kraft’s
course – Hecht,
Shlaer & Pirenne
Mean = 8
Photon emission
follows a Poisson
distribution
Mean = 4
Probability that the
numbers of quanta
on the x-axis will
occur in any given flash
Mean = 2
0.4
Mean = 1
0.0
0
5
10
15
20
25
30
Number of Quanta in a Flash
To distinguish a flash
with a mean of 8 from
a flash with a mean
of 9 quanta is
impossible! The
distributions overlap
almost completely
Mean of 8, vs. mean of 9
0.15
0.1
Series1
0.05
36
31
26
21
16
11
0
6
Series2
1
If present the flash with
a mean of 8 photons
many times and a flash
with a mean of 9
photons presented
many times, there will
be many times that the
9 photon flash will have
more photons than the
8 photon flash
S1
Mean of 8, vs. mean of 12
0.15
0.1
Series1
0.05
36
31
26
21
16
11
6
0
1
Series2
S1
Mean of 8, vs. mean of 16
0.15
0.1
Series1
0.05
36
31
26
21
16
11
6
0
1
Series2
S1
Mean of 8, vs. mean of 20
0.15
0.1
Series1
0.05
36
31
26
21
16
11
6
0
1
Series2
S1
In a Poisson distribution, the variance is equal to the
mean.
The standard deviation (SD) is the square root of the
mean.
In a two-alternative forced-choice task, to reach
threshold (75% correct), LT must differ from L by 0.95
SD. (e.g., L = 0.95 SD)
0.15
0.1
Series1
0.05
36
31
26
21
16
11
6
0
1
Series2
S1
Moreover, as L increases, the minimum threshold L also
increases with the L
because the variance in a Poisson distribution
equals the mean, so the SD changes with the
square root of the mean
0.15
0.1
Series1
0.05
36
31
26
21
16
11
6
0
1
Series2
S1
As a result, an “ideal” observer would follow the
deVries-Rose Law (deVries, 1943;Rose, 1948):
L
L=K
where L is the threshold luminance difference, L is the
background, or reference, luminance and K is a constant.
Theory and Practice
In practice:
at low background intensities, human observers behave
as an ideal detector (follow the deVries-Rose Law)
L (milliLamberts)
10000
1000
Fig. 3.1
100
10
1
0.1
0.01
0.001
0.0001
0.00001
0 1 0 00 1 .00 1 0 .0 1
0
0
0
0.
0.0
0.1
1
10
L (milliLamberts)
10 0 1 00 0 10 00 0
At higher intensity levels, the intensity discrimination
threshold is higher than expected from an ideal detector
(e.g., Weber’s Law holds)
The constant proportional relationship between the increment
threshold and the reference or background level is called
Weber’s law, which is mathematically expressed as:
L L = K
where L is the threshold luminance difference, L is the
background, or reference, luminance and K is a constant.
L (milliLamberts)
10000
1000
Fig. 3.1
100
10
1
0.1
0.01
0.001
0.0001
0.00001
00 1 .0 00 1 0.00 1 0 .0 1
0
0
0
0.
0.1
1
10
L (milliLamberts)
10 0 1 00 0 10 00 0
the fraction: threshold L divided by the reference luminance,
L is called the “Weber fraction” (threshold L / L)
You always can determine the Weber fraction,
even when Weber’s Law does not hold
L/L
Fig. 3.2
1.0
0.8
Weber’s Law does NOT hold
(L/ L rises as L decreases)
0.6
Weber’s Law holds
0.4
0.2
0.0
0.00001 0.0001 0.001
0.01
0.1
1
10
L (milliLamberts)
100
1000
10000 100000
Both the deVries-Rose and Weber’s laws fail to account
for thresholds at high light intensities
Fig. 3.3
Threshold, L
6
4
2
0
-2
-4
Predicted by DeVries Rose Law
Predicted by Weber's Law
-6
-6
-4
-2
0
2
4
6
Log Background Intensity, L (cd/m2)
The increment threshold data of a rod monochromat (circles) plotted along
with the theoretical lower limit (deVries-Rose, dotted line) and the predictions of Weber’s
Law (solid line). Luminance values are in cd/m 2. (Redrawn from Hess et al. (1990)
More practical issues:
How changes in other stimulus dimensions
affect the Weber fraction
The Weber Fraction is affected by stimulus size, duration,
wavelength, and retinal location (eccentricity from the fovea)
#1 Stimulus size: the Weber fraction is lower (smaller) for
larger test stimuli
Log Weber Fraction, L/L
Fig. 3.4
2
Test Field Diameter
4'
10'
18'
55'
121'
1
0
-1
-2
-3
-7
-6
-5
-4
-3
-2
-1
0
Log Background Intensity, L (cd/m2)
1
2
3
More practical issues: Is a target visible under certain conditions?
Log Weber Fraction, L/L
2
This is the target’s Weber
fraction. It is NOT a threshold
Test Field Diameter
4'
1
121'
0
Is a spot with a particular
-1 relative to
luminance,
background, visible? It
depends on its size.
-2
If the target is 121’, it is visible
If 4’, it is not visible
-3
-7
-6
-5
-4
-3
-2
-1
0
1
2
Log Background Intensity, L (cd/m)
2
3
Need to distinguish between the Weber fraction of
a target vs. the threshold of a viewer.
For a subject or patient viewing a target, if the
subject’s Weber fraction is below a line, then the
subject’s threshold is better (smaller).
If the Weber fraction of a target is below the line,
the target is NOT visible to someone whose
threshold is on the line.
The smaller the threshold L,
the smaller is the value of the Weber fraction for a
given background L, (only the numerator changes)
and the more sensitive the visual system is to
differences in light intensity.
A
B
L
T
= L + L
L
T
= L - L
Fig. 1.2
L
T
L
L
T
L = L + L
T
L u m in a n c e
+ L
L
- L
L T = L-  L
0
L
The “dinner plate” example:
Plate with luminance of 0.0102
footlamberts. Background is 0.01
footlamberts
L is thus 0.0102– 0.01 = 0.0002.
L/L = 0.0002/0.01 = 0.02 (plate is 2%
more intense)
From Figure 3-4, can learn that this is
not visible.
Log Weber Fraction, L/L
2
Test Field Diameter
4'
10'
18'
55'
121'
1
0
-1
.
-2
-3
-7
-6
-5
-4
-3
-2
-1
0
Log Background Intensity, L (cd/m2)
1
2
3
More practical issues: Is a target visible under certain conditions?
Log Weber Fraction, L/L
2
Test Field Diameter
4'
1
Is a spot with a particular
luminance, relative to
0
background, visible? It
depends on its size.
121'
-1
-2
Plate’s Weber fraction
-3
-7
-6
-5
-4
-3
-2
-1
0
1
2
Log Background Intensity, L (cd/m)
2
3
Continuing: How changes in other stimulus
dimensions affect the Weber fraction
#2 Short-duration flashes are harder to see
(are less discriminable) than long-duration
flashes
That is, the threshold L increases as flash
duration becomes shorter.
Sensitivity = 1/threshold
Continuing: How changes in other stimulus
dimensions affect the Weber fraction
#3 Threshold L varies with eccentricity from the fovea
At low luminance levels, threshold is lowest (sensitivity is
highest) about 15-20 degrees from fovea and the fovea is
“blind”
At high luminance levels, threshold is lowest at the fovea
Threshold, L
(Apostilbs)
This is the basis for visual field tests
-4
-3
Background
luminance, L
0
0.001
0.01
0.1
1
10
100
1000
-2
Fig. 3.5
-1
0
1
2
3
-60
-45
-30
-15
0
15
30
45
60
75
90
Angular Distance From Fixation (deg)
Increment threshold as a function of eccentricity from the fovea for several luminance
levels. The top line shows the threshold when the background luminance (L) is very low
(0 apostilbs). The bottom line shows the threshold for a background L of 1000 apostilbs.
Note, on the Y-axis, that lower thresholds (higher sensitivities) are upwards on the
graph. (Modified from (Lynn, Felman & Starita, 1996).)
Intensity discrimination can be limited at many places
within the visual system
Sensory Magnitude Scales Revisited
Using the “just noticeable difference” (jnd) to create a
scale for sensory magnitude vs. stimulus magnitude
L + threshold L = LT
LT is one jnd more intense than L.
LT + threshold L = LT2
LT2 is one jnd more intense than LT
And so on…
Sensory Magnitude
12
10
8
6
4
L
2
0
0
50
100
Stimulus Luminance, L (cd/m2)
150
200
Sensory Magnitude
12
10
L + threshold L = LT
8
LT is one “just noticeable difference”
(jnd) more intense than L.
6
LT
4
L
2
0
0
50
100
Stimulus Luminance, L (cd/m2)
150
200
Sensory Magnitude
12
LT + threshold L = LT2
10
LT2 is one jnd more intense than LT
8
LT2
6
and 2 jnd’s larger than L
LT
4
L
2
0
0
50
100
Stimulus Luminance, L (cd/m2)
150
200
Sensory Magnitude
12
10
LTn+1
LTn
8
When Weber’s Law holds, the
threshold Ls keep getting
larger, so 1 jnd is a larger
increase in stimulus luminance
LT2
6
LT
4
L
2
0
0
50
100
Stimulus Luminance, L (cd/m2)
150
200
Fechner’s Law
Sensory Magnitude
12
10
8
6
4
2
Fechner's Law: Log(L)
0
0
50
100
Stimulus Luminance, L (cd/m2)
150
200
Fechner’s Law relates the magnitude of sensation to the
increment threshold
Fechner’s law:
 = k log(  ) 
where  is sensory magnitude,  is an arbitrary constant
determining the scale unit, and  is the stimulus magnitude

Comparing Fechner’s Law with Stevens’ Power Law
Sensory Magnitude
12
Fig. 3.6
10
8
Stevens’ Power Law
resembles Fechner’s Law
when the exponent is <1
6
4
2
Stevens' Law: L0.15
Fechner's Law: Log(L)
0
0
50
100
Stimulus Luminance, L (cd/m2)
150
200
```