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Mapping Internal Variations in Translucency within a Translucent
Object Using Beams of Light
(c) Copyright 1995, 1997 Ian Clarke
Index
Abstract
Background
Introduction
The Mechanics of Translucency
A First Step
Improving the Primary Map
An Iterative Procedure
Conclusion
This document describes an iterative mathematical method which can be applied
to creating 2 or 3 dimensional maps of variations of translucency within
an object, using data obtained by passing beams of light through that object.
This problem is interesting because while a large quantity of data can
be obtained using this method, there will be parts of the object for which
the only information available will have passed through other parts of
the object. The interference from the surrounding data must therefore
be be removed, yet in order to remove the interference from the surrounding
areas we must know what their translucency is, which of course we don't.
I first began considering this problem during the Summer of 1994, in September
of that year I decided that I would enter this project in a national science
contest which gave me until January to solve the problem, and write a 50
page report. It was not until November that I hit on the idea that
would eventually allow me to solve the problem.
When I entered the project in the competion I obtained first place in my
section and an award from the Irish Institute of Physists, I was also invited
to a further exhibition for projects with industrial potential.
For thousands of years man has been curious about the world around him.
He would look at the objects that made up his world, and try to find out
how and why they worked. His ability to disassemble things into their
component parts, to look inside them, has been essential to mans scientific
development. Now man has the ability to peer inside objects without
damaging them, he can see pictures of a human heart beating or of blood
flowing around a body, but thereare still limitations to this ability,
and it is some of these limitations that I intend to tackle.
Consider the problem of a team of scientists who want to find out what
our planet consists of. The most direct method would be simply to
drill into it, however this is simply not possible. Our current knowledge
comes from what we can infer from the mass of our planet, and what we see
on the surface. One possibility is to send a suitible electromagnetic
through the earth and attempt to receive it at the other side, and to measure
how much of the wave makes it through. We could repeat this hundreds
of times from different positions on the Earth and collect a large amount
of data, but there is a problem.
Take any given location within the Earth, we want to know the translucenly
of that specific location, so we arrange for all of our probe beams to
intersect at this point, in the hope that we can gather as much information
as possible, but no matter how many beams we pass through this point, they
will always have to travel through other material before reaching the location
we are interested in, and then yet more material afterwards, and all we
get from that beam is a single value, how do we separate the data we are
looking for from the other data which we are not interested in? This
is the question I set out to answer.
The first thing we must look at is what happens when a beam of electromagnetic
energy travels through a translucent object.
In the image shown, l is the length of the segment of the beam which
passes through the object, T is the 'translucency' of that object,
a is the initial strength of the beam, and b is the final
strength. We will define translucency of an object as the intensity
of a beam of light after travelling through 1 unit of that material if
its initial intensity was 1. Now we get our first equation:
This equation tells us several very useful things about translucency, one
of the most important is that if we have two translucent objects, it does
not matter which order the beam encounters the objects, the combined translucency
will remain the same.
It is time to begin tackling the problem head on, before we can do this
we shall need a toy example.
Here we have an object consisting of 4 areas of equal size, these are named
A, B, C and D. There are also 4 beams travelling through this object
called 1,2,3 and 4. Now we are going to need some notation:
| Initial Intensity
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I(yy)
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yy is a given beam of light
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| Final Intensity
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F(yy)
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yy is a given beam of light
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| I(yy) / F(yy)
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N(yy)
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Length of segment of
beam in a part of
an object
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L(yy,xx)
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yy is a given beam of light,
xx is a part of an object
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Total Length of beam
inside object
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L(yy)
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yy is a given beam of light
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| Translucency
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T(xx)
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xx is a part of an object
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From the diagram we will know the following values:
I(1...4), F(1...4) (and therefore N(1..4)), and L(1..4,A..D).
Our task will be to calculate T(A..D).
Now let us work backwards somewhat, using our first equation we know the
following:
N(yy) = T(A)L(yy,A) x T(B)L(yy,B)
x T(C)L(yy,C) x T(D)L(yy,D)
The whole problem is that we don't know what T(xx) is, but what
if, in for each value of N(yy) we assumed that T(xx) is constant?
We could then calculate this constant, and we shall call it P(xx).
P(xx)=(N(yy)1/L(yy))L(yy,xx)
We can now use this equation to assign values to A, B, C
and D, but note that each of A,B,C, and D have two beams of light going
through them, meaning that we will get two values, what do we do?
Simple, we take their multiplicative mean, ie. we use the square root of
their product.
The resulting imprefect map which we produce shall be called
the Primary Map.
So how can we improve our primary map? Well, let's first examine
what is wrong with it. Basically, for each part of the map the value
we have is a combination of the values not only of the area we are interested
in, but of the surrounding areas too. If we could somehow remove
this influence then we would obtain the actual value for that part of the
map. To examine this problem we will use an even simpler model, probably
the simplest model possible that remains interesting.
We know that:
We can also express this in terms of the actual T(xx) values rather
than in terms of N:
Phew! Now, let us assume that we know what T(B) is, and we
want to get rid of it from this equation. We find that if we multiply
P(A)by the following:
We can remove T(B) from P(A) and get an improved approximation
to T(A), so what would we end up with if we removed T(C)
in the same manner too? We would obtain:
All we now have to do is to remove the power, simply by taking this to
the power of
and we have finally obtained T(A)!! But we can't break out
the bubbly just yet. Don't forget that rather bold assumption we
made two equations ago, we assumed that we knew the values of T(B)
and T(C), now if we knew these values we would probably know T(A)
too and this would all be a pointless exercise, so why bother? Well,
the thing is that we do have an approximation to T(B) and
T(C) in the guise of P(B) and P(C), so what if we
use these instead? Well, we are obviously not going to get a perfect
answer, but we do get a better answer, which we will call B(A),
and this is very useful.
Finally comes the trickery, we have seen how given information about the
lengths and paths of the light beams (what we refer to L(yy,xx))
and their initial (I(yy)) and final (F(yy)) intensities,
we can create a primary map of the object in question (calling it P(xx)),
and then using an approximation to the final map (which happens to be the
primary map itself) produce an improved approximation to the actual map,
which we call B(xx). But why stop there, we can now do the
same thing again, but this time using B(xx) for our approximation,
creating a still better approximation, and so on until eventually B(xx)
will converge to T(xx), and we have done what we have set out to
do!
I have set out to do what I originally intended to do, we now have an iterative
procedure which can produce an accurate map of the inside of an object
from data obtained by passing beams of electromagnetic radiation through
that body.
Further work needs to be done to determine the situations where the algorithm
will and will not work correctly, and the time which it will require to
converge to appropriate values.
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