Math of Optical Tomography
An Overview of the Mathematics of Optical
Tomography
Joe Eichholz
Rose-Hulman Institute of Technology
January 18, 2012
Math of Optical Tomography
Outline
1
Motivation
2
Intuition
3
Mathematics
4
Recent Work
Math of Optical Tomography
Motivation
tomography
(n) - a method of producing a three-dimensional
image of the internal structures of a solid object (such as the
human body or the earth) by the observation and recording of the
dierences in the eects on the passage of waves of energy
impinging on those structures.
Math of Optical Tomography
Motivation
Tomography techniques play a large role in current diagnostic
medicine, and the development of new techniques continues to be
an area of active research.
type of energy wave
name of scan
X-rays
CT
gamma rays
SPECT
radio-frequency waves
MRI
Math of Optical Tomography
Motivation
We seek to develop a new imaging modality in which optical light is
the energy source. Compared to other technologies this technique
has several potential advantages:
Use of non-ionizing radiation
Potential for high resolution even in soft tissue
Relatively inexpensive
Math of Optical Tomography
Motivation
Several optical tomography imaging techniques exist in various
forms, with varying degrees of maturity. They can be loosely
categorized in to two camps.
Bioluminescense Tomography
The structures of interest are emitting light, we measure the
amount of light escaping the object of interest and reconstruct
what the source looks like.
Diuse Optical Tomography
We expose the object of interest to a known light source,
measure the resulting light emitted from the object, and try to
recover how much/where the light was scattered/absorbed.
This will give us information about the internal structure of
the object.
Math of Optical Tomography
Intuition
Intuition: A source reconstruction problem
Math of Optical Tomography
Intuition
A source reconstruction problem
?
Pinhole Sensors
Math of Optical Tomography
Intuition
A source reconstruction problem
?
Pinhole Sensors
Math of Optical Tomography
Intuition
A source reconstruction problem
Pinhole Sensors
Math of Optical Tomography
Intuition
A source reconstruction problem
Pinhole Sensors
Math of Optical Tomography
Intuition
A source reconstruction problem
Pinhole Sensors
Math of Optical Tomography
Intuition
A source reconstruction problem
Pinhole Sensors
Math of Optical Tomography
Intuition
A source reconstruction problem
Pinhole Sensors
Math of Optical Tomography
Intuition
A source reconstruction problem
Pinhole Sensors
Math of Optical Tomography
Intuition
What information did we use to come to our conclusion?
Measurements from an experiment
Our notion that light moves in straight lines.
Used our idea of how light moves to predict the outcome of
the experiment if the light source is in various places
Chose the source position that gave results most consistent
with the actual outcome
Math of Optical Tomography
Intuition
Intution: An absorption reconstruction
problem
Math of Optical Tomography
Intuition
Absorption reconstruction problem
Math of Optical Tomography
Intuition
Absorption reconstruction problem
OW
Math of Optical Tomography
Intuition
Absorption reconstruction problem
X-ray source
X-ray detector
Math of Optical Tomography
Intuition
Assumptions we will use to determine where break is:
X-rays move in straight lines through the body
X-rays are mostly absorbed by bone
X-rays mostly pass through soft tissue
Math of Optical Tomography
Intuition
Absorption reconstruction problem
X-ray source
Detector reading
X-ray detector
position
Math of Optical Tomography
Intuition
Absorption reconstruction problem
X-ray source
Detector reading
X-ray detector
position
Math of Optical Tomography
Intuition
Absorption reconstruction problem
X-ray source
Detector reading
X-ray detector
position
Math of Optical Tomography
Intuition
Absorption reconstruction problem
X-ray source
Detector reading
X-ray detector
position
Math of Optical Tomography
Intuition
Absorption reconstruction problem
We used the same kind of procedure as the last problem:
Take measurements
Develop a model that tells us what the expected results would
be if the break were in a certain place in the bone
Break must be in a place that is consistent with the
measurements we have
Math of Optical Tomography
Mathematics
Mathematics
Math of Optical Tomography
Mathematics
Bioluminescense Tomography (BLT)
The general procedure for BLT is
1
Genetically modify some cells of interest to emit light
2
Measure amount of light coming out of animal
3
Reconstruct light source distribution, which gives the position
of the modied cells
Math of Optical Tomography
Mathematics
BLT
We will need a model of light propagation. i.e., if we are given a
source distribution, how do we predict what our measurements will
be?
Math of Optical Tomography
Mathematics
Given a source distribution, we are going to end up nding the
intensity of light, u , at every point in the object of interest
moving in every possible direction.
Let X be the spatial domain corresponding to the object we
are imaging, and x a typical point in X .
Let
Γ
Let S
be the boundary of X
2
be the unit sphere, and
ω
a typical direction in S
Γ+ be the outow boundary of X × S 2 i.e.
(x , ω) such that x ∈ Γ and ω points away from
Let
2
all the points
X at x .
Γ− be the inow boundary of X × S 2 , i.e. all points
(x , ω), such that x ∈ Γ and ω points in to X at the point
Let
x.
Math of Optical Tomography
Mathematics
Light source distributions are represented as functions
f
:X →R
Possible experimental outcomes are functions on
Γ+
to
R.
A model of light propagation is a function that takes a source
distribution f to the resulting boundary measurements. We
call this function M (f ).
Reconstructing the light source means nding the light source
whose resulting boundary measurements will most closely
agree with the actual measurements taken.
argmin
f ||M (f ) − umeas ||
Math of Optical Tomography
Mathematics
What's the problem?
If we decide to look for f in some nite dimensional function space
we can represent f as an n dimensional vector, ~
f.
Dene
E (~
f)
= ||M (f ) − umeas ||
The problem
min E (~
f)
~f
is now a simple nite dimensional optimization problem (possibly
constrained), which is a well understood area of applied
mathematics.
Math of Optical Tomography
Mathematics
The problem
ω · ∇u (x , ω) + µt (x )u (x , ω) = µs (x )
Z
S2
0
0
g (ω, ω )u (x , ω )d ω
µs ( x )
gives the amount of scattering at each point x
µa ( x )
gives the amount of absorption at each point x
µt (x )
is the total attenuation coecient (µ
0
+ f (x )
a ( x ) + µs ( x ) )
g (ω, ω 0 ) gives the probability of a photon traveling in direction
ω 0 redirecting to direction ω
u (x , ω) gives the intensity of light at each position x
moving in direction
∈X
ω
M (f ) is the solution u to the
radiative transport equation
Math of Optical Tomography
Mathematics
The problem, again
In order to evaluate our model M (f
)
ω · ∇u (x , ω) + µt (x )u (x , ω) = µs (x )
u |Γ−
This will pose a serious challenge.
we need to solve the RTE
Z
S
=0
0
2
0
g (ω, ω )u (x , ω )d ω
0
+ f (x )
Math of Optical Tomography
Mathematics
A restatement
Another way to think of this problem is: If we are given the source
function f , the resulting light distribution u will be the solution to
ω · ∇u (x , ω) + µt (x )u (x , ω) = µs (x )
Z
S
0
2
0
g (ω, ω )u (x , ω )d ω
0
+ f (x )
However, if we are given information about u (from measurement),
can we gure out what the function f was?
This problem falls in the category of
large and well studied eld.
inverse problems,
which is a
Math of Optical Tomography
Mathematics
The RTE
ω · ∇u (x , ω) + µt (x )u (x , ω) = µs (x )
u |Γ−
Z
S
0
2
0
g (ω, ω )u (x , ω )d ω
0
+ f (x )
=0
Is classied as an integro-dierential equation in 5 independent
variables. Except in the simplest of scenarios, there is no hope of a
closed form solution.
Numerical solution of this equation will be dicult because:
High dimensionality leads to a huge number of unknowns
Presence of integral terms leads to non-local eects
Math of Optical Tomography
Mathematics
Recap
In the BLT problem we wish to recover the distribution of light
inside a biological tissue from some measurements of light escaping
the tissue.
Steps
meas
Measure light escaping tissue, call it u
Use knowledge of the domain to create model of light escaping
tissue for any source function, call it M (f ).
meas
Recreate the light source that corresponds to u
min E (~
f)
f ∈F
where
E (~
f)
= ||umeas − M (f )||
by
Math of Optical Tomography
Mathematics
Problems
min E (fˆ)
f ∈F
Is it clear that there is a unique f that minimizes E ?
Typical minimization algorithms require many evaluations of
the objective function, E . Worse still, they will require
information about
∇E .
Evaluation of E requires the numerical
solution of RTE with high accuracy. Evaluation of
signicantly more expensive.
∇E
is
Math of Optical Tomography
Mathematics
One solution
min E (fˆ)
f ∈F
The ill-posedness is usually xed via regularization
min E (fˆ)
f ∈F
+ ε||f ||
We just pick out the function f that minimizes E and has
minimal norm (in some appropriate norm).
You may have seen this idea in linear algebra when solving
under-determined systems of equations.
Math of Optical Tomography
Mathematics
Current Research Areas
Math of Optical Tomography
Mathematics
Current Research Topics
The more dicult problem is to evaluate E and
∇E
quickly. That
is, we want to be able to solve the RTE for a given source function
f very quickly.
Find ways to avoid calculating
∇E
Replace the RTE with some other model of light propagation
when appropriate
Find ways to speed up the numerical solution of the RTE
Math of Optical Tomography
Mathematics
Approximating the RTE
Under the hypothesis that attenuation is scattering dominated
and that scattering is nearly isotropic it is appropriate to
replace the RTE with a diusion approximation
−div(D (x )∇u ) + µt u = f
where D is a term that involves both absorption and scattering.
While these are usually reasonable assumptions, in many
biological tissues scattering is highly anisotropic. Examples
include synovial uid, cerebrospinal uid, etc.
Math of Optical Tomography
Mathematics
What's the harm?
Pinhole Sensors
Math of Optical Tomography
Mathematics
What's the harm?
Pinhole Sensors
Math of Optical Tomography
Mathematics
What's the harm?
Interface
Pinhole Sensors
Math of Optical Tomography
Mathematics
The moral of the story is that replacing the RTE with an
approximation is a great idea when it works, but it will give terrible
results in many important cases.
Math of Optical Tomography
Mathematics
Speeding up solution to the RTE
We can replace the RTE with other approximations of varying
accuracy.
We can try to accelerate the numerical solution of the RTE
A wide variety of numerical schemes have been designed,
including nite dierence, nite element, collocation,
discontinuous Galerkin, etc.
Any reasonable numerical method will boil down to solving a
huge linear system of equations
Math of Optical Tomography
Recent Work
Solve
Tu
= Su + f
T comes from the transport part of the equation, and can be
made lower triangular (ish), S comes from the scattering part of
the equation and is usually dense.
There are many well studied methods in the literature for this kind
of system, all are iterative and most are accelerated using
preconditioners or Krylov subspace methods.
Math of Optical Tomography
Recent Work
Solve
Tu
= Su + f
The most classic technique in the eld is the source iteration
method
1
2
u0
=0
Tu n = Su n−1 + f
This actually works pretty well when the attenuation is not
scattering dominated.
Math of Optical Tomography
Recent Work
Solve
Tu
= Su + f
Some recent work during a summer REU is applying classic
methods in numerical linear algebra to the above matrix equation.
Results were encouraging.
S
L
where S , S
D
and S
U
= SL + SD + SU
are block lower triangular, diagonal, and
upper triangular resp.
The rst method is basically block Gauss-Seidel iteration:
(T + SL + SD )u n = SU u n−1 + f
Math of Optical Tomography
Recent Work
Solve
Tu
= Su + f
where
S
= SL + SD + SU
and
T
= TD + TL
the second method is a block form of the successive-over-relaxation
method
(TD + SD )u n = (1 − ω)(TD + SD )u n−1
+ ω −(TL + SL )u n+1 − (TU + SU )u n + f
Here,
ω
is an acceleration parameter that needs to be chosen
appropriately.
Math of Optical Tomography
Recent Work
Solve
Tu
= Su + f
The development of the Gauss-Seidel and SOR acceleration
techniques resulted in a huge reduction in the number of iterations
required to solve the linear system without even the need for
preconditioning.
Math of Optical Tomography
Recent Work
In addition to reducing the number of iterations of a linear solver
required to solve the RTE, we may attempt to speed up each
iteration using parallel computations.
Multicore programming
GPU computing
Distributed memory (MPI) computing
Math of Optical Tomography
Recent Work
Final Recap
Solution of the BLT (and OT problems in general) is theoretically
possible, but currently extremely slow. Research needs to go in to
Reducing number of RTE evaluations required in the
optimization routines
Accelerating the solution of the RTE when required
Designing better numerical schemes tailored for use in OT
(better convergence order, multigrid, adaptive)
Designing faster methods to solve the resulting systems of
linear equations
Using parallel computing to speed up the solution of the RTE
Math of Optical Tomography
Recent Work
Thanks for listening!