pupil control systems
DESCRIPTION
Pupil Control Systems. By: Darja Kalajdzievska, PhD-University of Manitoba & Parul Laul, PhD-University of Toronto Supervisor: Alex Potapov. Pupil Light Reflex of the Human Eye. Why does the pupil radius expand/contract? - PowerPoint PPT PresentationTRANSCRIPT
Pupil Control SystemsPupil Control Systems
By: Darja Kalajdzievska, PhD-University of Manitoba
&
Parul Laul, PhD-University of Toronto
Supervisor:
Alex Potapov
Pupil Light Reflex of the Human Eye
Why does the pupil radius expand/contract?
• The size of the pupil controls the amount of light let in to the eye
• As intensity of light increases, pupil contracts
•As intensity of light decreases, pupil dilates
•The pupil cannot react instantaneously to light disturbances, there is a delay in reaction time
First we must determine the optimal light intensity for the eye, We consider intensity as a function of area (radius R):
We assume that there is an equilibrium amount of light (which corresponds to an optimal area for that light intensity) that the eye
prefers-A* and that the pupil will expand/contract until it allows this amount of light in:
nn RI
RI
RI
0
202
101
*)(~ AAdt
dR
Here, A represents the area of light on the pupil, which is also the intensity amount,We determine area to be:
aRaRA 222
Introducing a Model
We know that the change in radius is proportional to light intensity:
)(1 RIdt
dR
)()( 2 RARI
*)(223 AaRaR
dt
dR
If A(R) is the region of intersection of light and the eye:
So we have the ODE:
Letting and
then
haR 22 cAha *)2(3
chRdt
dR 23
Formulating the Model
Solving this ODE with initial condition R(0)=Ro, we get an expression for R as a function of time:
h
Ahae
h
AhaRtR ht
2
*)2(
2
*)2()( 2
03
The Final Model – Instantaneous Reaction
Instantaneous Reaction
Radius of Pupil versus Time with Fixed Light Intensity
Time (sec)
Pup
il ra
dius
(m
m)
Parameters
Delay – Intuitive Idea
Assumptions
•Intensity - normalized between 0 (low intensity) and 1 (high intensity)
•Radius size fluctuates between 2 and 4 mm, R_o =4mm
•Time Delay – 0.18 ms
•Instantaneous change in radius after delay
We can change our previous ODE to incorporate a delay in reaction time:
Now we use the Taylor Series Expansion to turn our 1st order ODE to a 2nd order ODE:
cthRdt
dR )(23
)(23 thrdt
dR
dt
dr
To get rid of the inhomogeneous term-c, we let r = R-c
...)(''2
)(')()(2
tohtrtrtrtr
)(''
2)(')(2)('
2
3 trtrtrhtr
Refining the Model – Introducing Delay
If we let
and
21
)21(
h
h
22
2
We get an ODE which is simple to solve:
0)()(')('' 21 trtrtr
We look for solutions of the form:
Since we need a solution that exhibits oscillations, we are looking for the case of complex roots:
ptpt
pt
eptrpetr
etr2)(''&)('
)(
2
4,
0
22
1121
212
pp
ppe pt
2
)4(,04
2121
2122
1
ipp
Plugging this back into the ODE:
This gives a solution:
tt
ectt
ectr
2
4sin2
1
2
4cos2
1
)(2
`122
2`12
1
Delay Reaction...Almost There
To determine
We use the initial condition
21 ,cc
0)0( rr
and the fact that the initial velocity =0 due to the delay 0)0(' r
Our solution is now:
terrtertt
2
4sin
2
4
2
4cos
2
4 2122
212
00
2122
212
0
11
The Final Model – Delay Reaction
Delay – Growth of Pupil Radius
Time (sec)Pup
il ra
dius
(m
m)
Parameters
terrtertrt
o
t
o )2
4sin(4
2
1)
2
4cos(4
2
1)( 122
1
120122
1
12
11
sec5.0
75.0
)(2
5.1
)2(
2*
22
mmA
acrh
mmcro
Radius of Pupil versus Time with Fixed Light Intensity
22
21
2
21
h
h
Delay – Decay of Pupil Radius
Time (sec)Pup
il ra
dius
(m
m)
Parameters
terrtertrt
o
t
o )2
4sin(4
2
1)
2
4cos(4
2
1)( 122
1
120122
1
12
11
sec3.0
75.0
)(2
5.1
)2(
2*
22
mmA
acrh
mmcro
Radius of Pupil versus Time with Fixed Light Intensity
22
21
2
21
h
h
In this case, we want
To obtain equal amplitude oscillations, we fix values for all other parameters and try to determine this value of
0
1
1
21
t
e
hh
h
h
2
1120
2121
In our experiment: 3779644729.075.12
1*
*
Oscillations grow
Oscillations decay
Equal Amplitude Oscillations
*
Delay – Equilibrium Oscillations
Time (sec)Pup
il ra
dius
(m
m)
Parameters
terrtertrt
o
t
o )2
4sin(4
2
1)
2
4cos(4
2
1)( 122
1
120122
1
12
11
sec9.3779644720
75.0
)(2
5.1
)2(
2*
22
mmA
acrh
mmcro
Radius of Pupil versus Time with Fixed Light Intensity
22
21
2
21
h
h
Numerical Approximations – Delay Model
cMiRtRR
ctMihRt
RR
AhatMtihRt
R
cthR
AhathRdt
dR
ii
ii
~)(~
))((2
)2()(2
)(2
)2()(2
1
1
*
*
Recall:
Converting to Discrete Time, we obtain:
Numerical Approximations – Delay Model
t0 t2 t3{Measured data from first M steps
constant is that assume toreasonable isit
so vanishes, of influence the time,someAfter
value
constant aby timesMfirst theeapproximat we
times,Mpast on depends ofsolution Since 1
past
past
past
i
R
R
R
R
Numerical Simulations – Growth
Time (sec)
Pup
il ra
dius
(m
m)
Radius of Pupil versus Time with Fixed Light Intensity
cMiRtRR ii~)(~
1
Numerical Simulations – Decay
cMiRtRR ii~)(~
1
Time (sec)
Pup
il ra
dius
(m
m)
Radius of Pupil versus Time with Fixed Light Intensity
Numerical Simulations – Equilibrium Oscillations
cMiRtRR ii~)(~
1
Time (sec)
Pup
il ra
dius
(m
m)
Radius of Pupil versus Time with Fixed Light Intensity
Plot illustrates approximate equilibrium oscillations
3905.0
Numerical Simulations – Incorporating Non-Linearity
)( size, radius vary withlet now wemodel, theimprove To R
To account for the fact that the radius of the eye is bounded, we define:
where, ~)(~ cRRdt
dR ~
otherwise ,0
R , maxmin RR {
Converting to the Discrete Model, we obtain:
cMiRtRRR ii~)()(~
1
Numerical Simulations – Non-linear
cMiRtRRR ii~)()(~
1
Time (sec)
Pup
il ra
dius
(m
m)
Radius of Pupil versus Time with Fixed Light Intensity
Further Questions/Model Flaws
•Human error / human disturbances•Assume that the area of the light on the eye changes to head movements, fatigue of eye muscles, etc.•These disturbances will increase with time, so let distance )(taa
Questions
Flaws
•In reality, the diameter of the pupil can neither increase or decrease unboundedly, so a nonlinear model should be introduced to regulate oscillations•Height of the area of intersection of light and pupil was assumed to be constant (h), in reality, this height and indeed the entire area of intersection would be a function of light intensity•Experimentation should be done to find more accurate parameter values
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