```Visual and Auditory Systems
Tutorial 7-8
The Technion - Israel Institute of Technology
Electrical engineering faculty
Tutorials 7-8: MTF, Retina Illuminance and Lenses
Questions:
Question 1:
A sinusoidal lattice is projected on a screen. The luminance varies along the x axis
according to: B  B0  B1 sin 0 x  , B1  B0 .
cycles
.
cm
a. Calculate the spatial frequency (in cycles per degree-cpd) seen in a distance R
from the screen, assuming it's approximately a part of a spherical dome around
the viewing point.
0  4 rad cm , thereofre the lattice appears in a spatial frequency of 2


An animal is looking at the screen. The animal has a retina containing
receptors
10, 000
in uniform density, and its eye lens is located 10 mm from the
mm 2
retina (figure1). The animal has proper vision for objects placed 10 cm from her eyes
and above. Assume the luminance aforementioned matches the animal's photopic
curve.
The animal's MTF curve is displayed in figure 2. To answer this question treat
MTF   as the frequency response of a filter affected from the spatial frequency,
without being affected from its direction. The frequency response is real.
b. Find the range of R values that allow the animal to notice the
periodic structure, when the upper limit, H , is determined by the Nyquist rate
of the retina sampling.
c. Calculate the effective contrast the animal feels (meaning the one received
based on the contrast filtered in the MTF) with dependence in the distance R
between her and the screen (can be written as an expression of MTF   ).
What is the distance from which the lattice will appear the most clear
(maximum contrast)?
d. Assuming the MTF in figure 2 refers to a linear shift-invariant filter (LSI)
derived from processing in the retina, how can you calculate the receptive
field of the animal's eye?
1
Visual and Auditory Systems
Tutorial 7-8
The Technion - Israel Institute of Technology
Electrical engineering faculty
Question 2:
Given the impulse response of the visual system of a certain animal is:


h  x, y   2 exp   ax2  by 2   exp   bx2  ay 2  ,
ab
a. Calculate and sketch the animal's frequency response H  u, v  .
b. Explain the difference between this animal's MTF curves for an experiment
where the illuminance lattice is horizontal, vertical or diagonal.
c. Explain how the outcome of section b will change, dependent on the constants
a, b .
Question 3:
The receptors density of the retina in a certain animal is displayed below:
The animal's distance between retina and eye lens is 10  mm , the refractive index of
the vitreous (transmission medium inside the eye) is n  1 , and the internal
transmission coefficient is    0.3 .
a. The animal is capable of clearly identifying distant objects, starting from
10 cm . What is the focal length range of the animal's eye lens?
The animal is placed in front of a screen, looking at point A:
b. Gratings in varying spatial frequencies are projected on the screen. What are
the maximal frequencies (in cpd) that can be noticed by the animal in points
A, B, C? For the sake of this solution use a spherical dome approximation for
the screen. Note the center of the animal's retina is pointing to point A in all
cases.
2
Visual and Auditory Systems
Tutorial 7-8
The Technion - Israel Institute of Technology
Electrical engineering faculty
 cd 
c. The screen is lightened uniformly with luminance B  10  2  .
m 
What is the animal's retina illuminance in Lux, with and without taking in to
consideration the C-S phenomenon (Stiles-crawford)? The answer should be
given in Troland and Eff. Troland. The relation between the pupil's diameter
and the luminance is according to the human model (using the tanh formula).

2
The C-S phenomenon effect is:   r   exp r
10

,
 r   mm
Question 4:
A person with myopia, who can see clearly up to 0.5 m , is wearing glasses with
2  D .
a. What is the maximal distance to which this person can now see clearly?
b. The same person accidentally put on a 2  D glasses. What are his vision
limitations now and what is the maximal distance to which he can now see
clearly?
Question 5:
An approximation of the photopic curve is given in figure 1, to be used to calculate
the lumen value in this question, and an additional curve of another kind of flux, in
the UV range, called uven, is given in figure 2.
This question involves the human visual system, so the uven units express flux, but do
not describe a light sensation, because they are not part of the visible light range.
 in the range
A source with A  1  m 2  is producing uniform radiation of 8 mW
nm

100  800  nm . The luminous emmitance is uniform and lambertian all over the
source's area.
a. Calculate how much lumen and uven are produced from the source and the
luminances B and BUV based on the lumen and uven definitions respectively.
b. The source covers the viewer's visual field fully. What will be the retina's
visible light illuminance ( E ' ) according to the lumen, and the retina's UV
illuminance ( EUV ' ) according to the uven definition? The pupil's diameter is
identical in both cases and determined by the visible light based on the lumen.
3
Visual and Auditory Systems
Tutorial 7-8
The Technion - Israel Institute of Technology
Electrical engineering faculty
Assume identical transmission and refraction for both visible light and UV in
the medium inside the eye - n  4 ,    0.5 . The C-S effect can be
3
neglected.
The information on the pupil's diameter, the transmission and refraction in
section b are all valid for section c as well.
c. UV radiation is known to be harmful for the eye. Figure 3 describes two
options providing protection for the eyes from strong radiation, visible and
non-visible ("sunglasses") of the aforementioned source.
1) Which sunglasses will create a more substantial sensation of reduced
light?
2) What is the retina's UV illuminance ( EUV ' ) in each of the following
cases? In UV damages terms, is it preferable to not wear sunglasses at
all in compare to one of the possibilities? The C-S effect can be
neglected.
d. Calculate the amount of Troland and Eff. Troland that arrive to the retina
without sunglasses.
4
Visual and Auditory Systems
Tutorial 7-8
The Technion - Israel Institute of Technology
Electrical engineering faculty
Solutions:
Question 1:
a. A view from above:
The period length is x0  1
f0
1
2
cm
The angular period length is:
x0
1 180
90

deg   deg 
R
2R 
R
1 R
  
cpd 
 0 90
 0  tan 0  
b. First we shall calculate the longitudinal receptors density:
10, 000  100  mm 1 

x 
1
 mm  sample distance
100
For small angles   n ' (Snell's law).
RR - Distance between the lens and retina.
'
x 0.01 mm 
180 3

 103  rad  
10  deg 
RR
1 cm 

Assuming    '  n  1 :
According to Nyquist law s 
1

H 
 2 H (sampling frequency in the retina)
1


 8.36  cpd 
2 0.36
5
Visual and Auditory Systems
Tutorial 7-8
The Technion - Israel Institute of Technology
Electrical engineering faculty
There are two limitations that need to be taken in to account in the R calculation:
1) The given vision range R  10 cm
H
R
 H
10 90
Using (2) we get 25  cm  R  250  cm  , which also fits limitation (1).
2) MTF:

c. The contrast for a sinusoidal lattice is:
B
C 1
B0
Ceff  C  MTF   
B1
R 
 MTF 

B0
 90 
C eff is maximal in the frequency where the MTF is maximal, meaning for
  1 2 H , therefore:
Ropt 
90 H

 125  cm
 2
d. The retina response assuming a LSI system is R  x   E  x  * g   x  .
The retina's impulse response: h  x   g   x  .
Since the MTF is the retina's frequency response MTF    F h  x  .
Therefore:
g   x   F 1 MTF  
6
Visual and Auditory Systems
Tutorial 7-8
The Technion - Israel Institute of Technology
Electrical engineering faculty
Question 2:
a.
H  u, v   F 2 D h  x, y 
2

F2D
exp  r 2  
   exp   

4




F
exp   ax 2  by 2  

H  u, v  
2D
r
;
 x2  y 2 ;  2  u 2  v2 
  u 2 v2  
 exp      
ab
  4b 4a  


 u 2  v2  
ab
2
  u 2 v2  
  u 2 v2  
exp



exp

 
     

4
a
4
b


 
  4b 4a   
7
Visual and Auditory Systems
Tutorial 7-8
The Technion - Israel Institute of Technology
Electrical engineering faculty
b. A comparison between MTF curves received for an experiment where the lattice is
horizontal/vertical (continuous line) to an experiment where the lattice is diagonal
(Dashed line):
c. The closer the ratio a is to 1, we receive a bigger radial symmetry, and of course the
b
MTF for an experiment with a diagonal lattice approaches the MTF for an experiment
with a horizontal/vertical lattice. In the graph presented in section the ratio is
In the following graph the ratio is
a
 1.33 :
b
8
a
 3.
b
Visual and Auditory Systems
Tutorial 7-8
The Technion - Israel Institute of Technology
Electrical engineering faculty
Question 3:
a. For objects placed in  :
1 1 1 1 1
   

f max  1 cm
f u v  1
For a larger focal length, the eye lens won't be converging enough.
For objects placed in a distance of 10 cm :
1 1 1 1 1
   
f u v 10 1
Therefore the focal length range is:

f min 
10
cm
11
0.909 cm  f  1cm
b. For point A:
The point is right in front of the center of the retina, where the receptors density is
105  mm1  . The distance between 2 receptors is therefore 105  mm , and the
angular sampling distance is:
105

 0.16 104  rad   0.018 deg 
10
This is the angular sampling distance outside the eye, found using the connection for
small angles    n  .
The maximal frequency, according to the Nyquist rate on the retina sampling is:
H 
1
1
1
1
 H 


 27.8  cpd 
2n 2 1  0.018 0.036
2
For points B, C:
The calculation is done in the same way, only the receptors density is according to the
point location.
0.15 180

 2.9  deg 
3

0.35 180
C 

 6.7  deg 
3

B 
B 

receptors density  5 10 4  mm 2 

receptors density  3 10 4  mm 2 
2 105
 4.47 104  rad   0.026  deg 
10
3.3 105
C 
 5.77 104  rad   0.033 deg 
10

H 
1
 19.2  cpd 
2 1  0.026

H 
1
 15.2  cpd 
2 1  0.033
Meaning, the sensitivity to the spatial frequency decreases as the distance from the center
of the vision field rises.
c. Calculation of the pupil's diameter:
d  5  3tanh  0.4log10 10  3.68mm
Calculation of  :
n 2 12
  2  2  1  1 2 
 cm 
v
1
9
Visual and Auditory Systems
Tutorial 7-8
The Technion - Israel Institute of Technology
Electrical engineering faculty
Calculation of the pupil's area:
S
d2
 11.7  mm 2   0.117 cm 2 
4
Calculation of the retina's illuminance in Lux:
E    BS  0.3 110  0.117  0.35Lux 
Calculation of the retina's illuminance in Troland:
B  S  10 11.7  117 Troland 
Calculation of the effective area:
d
2
d
Seff  2    r  rdr  2
0
2

re
r2
10
0
d
dr  10 1  e

2
40
  9.77  mm 2 
 


 d 3.86mm
Calculation of the retina's illuminance in Eff.Troland:
B  Se  10  9.77  97.7  Eff. Troland
Question 4:
a.
1
 2 
f
f -1
2
m
We have to find the focal distance from the lens u which will result in v   1
(the distance in which the person sees clearly)
1 1 1
1
 

2  2 

u
f v u
u
The conclusion – now this person can see clearly to any distance!
b. The eye has a f max which focuses to the retina starting from u  1  m .
2
10
2
Visual and Auditory Systems
Tutorial 7-8
The Technion - Israel Institute of Technology
Electrical engineering faculty
1
f max

1
1

2.2 50
f max  2.1 cm

For 2 lenses in a row (glasses lens + eye lens):
f f
2.1 50
f eq  1 2 
 2  cm
f1  f 2 2.1  50
f eq Was calculated for f max of glasses lens + eye lens, and it will give us the
maximal distance:
1
1 1


2 2.2 u
umax  22  cm 

Question 5:
a.

f     const  8  mW
nm 


F   f    K    d   8 103  300 100  700 100  300 100  1040 lumen 
0

1
FUV   f    KUV    d   8 103   200  700  560 uven 
2
0
Since the surface is lambertian:
L
B

L
F 1040

 1040 lux 
A
1
LUV 
FUV 560
 uven 

 560  2 
A
1
 m 

B

BUV
 cd 
 331  2 

m 
560
UV  cd 

 178.75 
2


 m
1040
b. First we shall calculate the pupil's diameter:
d  5  3 tanh  0.4 log B   5  3 tanh  0.4 log 331  2.71 mm
2
d 
 S      5.73  mm 2   5.73 102 cm 2 
2
2
4 
n  3 
   
 0.36 cm 2 
 v   2.2 


'
E       B  S  0.5  0.36  331 5.73 102  3.41lux 
2
 uven 
'
EUV
      BUV  S  0.5  0.36 178.25  5.73 102  1.87  2 
 m 
c.
(1) Fa  0.5F  520  Lumen
11
Visual and Auditory Systems
Tutorial 7-8
The Technion - Israel Institute of Technology
Electrical engineering faculty

Fb   f    b    K    d  
0


 8 103 0.6  300 100  0.15  700 100  0.15  300 100   264 lumen 
500  600 nm
600  700 nm
 400500 nm

L
Fa
520
 cd 
 Ba  a 

 162.52  2 
   A  1
m 
 Bb 
Lb


Fb
264
 cd 

 84  2 
  A  1
m 
d a  5  3 tanh  0.4 log165.52   2.87  mm 
db  5  3 tanh  0.4 log 84   3.06  mm 

2
d 
Sa    a   6.47  mm 2   6.47 102 cm 2 
 2 
Sb  7.35 102 cm 2 

E    Ba Sa  0.5  0.36 165.52  6.47 10 2  1.93 lux 
'
a
Eb'    Bb Sb  0.5  0.36  84  7.35 102  1.11lux 
Conclusion: Ea'  Eb' , so in terms of visible light, glasses b are preferable!
(2)
FUV ,a  0.5 F  280 uven 

FUV ,b
  f    b    KUV    d  
0




 8 103 1  0.5 100  700  0.6  0.5  700 100   448 uven 
400 500 nm
500  600 nm


F
280
UV  cd 
BUV ,a  UV ,a 
 89.13 
2

  A  1
 m
F
448
UV  cd 
BUV ,b  UV ,b 
 142 
2

  A  1
 m

'
2
EUV
 1.04 lux 
, a    BUV , a S a  0.5  0.36  98.13  6.47 10
'
2
EUV
 1.89 lux 
,b    BUV ,b Sb  0.5  0.36 142.6  7.35 10
'
'
'
Conclusion: EUV
, a  EUV  EUV ,b , meaning glasses a improve (pass more visible
light and less UV) and glasses b spoil.
12
Visual and Auditory Systems
Tutorial 7-8
The Technion - Israel Institute of Technology
Electrical engineering faculty
d. The formula for the retina illuminance:
 cd 
B  2   S  mm 2 
m 
troland
BS  331 5.73  1895 troland 
BSe  B
d2 
d 2 0.002d 4 
1

0.085

 331 5.2988  1753 eff .Troland 
4 
8
48 
13
Visual and Auditory Systems
Tutorial 7-8
The Technion - Israel Institute of Technology
Electrical engineering faculty
Appendix – A Visual Representation of the MTF Curve
Last update-may 2011
14
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