Tomography
• The Radon transform is the key technology in
CAT scanning, now used in every hospital since
1972. Nowadays the research frontier has shifted
to MRI= magnetic resonance imaging. I will
discuss both.
• An X-ray moves through an object of density f(x,y)
at the point (x,y). It is absorbed or deflected with
probability f(x,y) ds where ds is the element of
length along a line, L, through (x,y). The chance
it goes all the way along L is exp(-Pf(L) ) where
Why line integrals?
• Beer’s law says that the log of the ratio of input to
detected X-ray photons is proportional to the line
integral of the density along the straight line path
of the X-ray beam.
• If an X-ray passes through an object of density f(s)
at the point s, then the probability that it gets to
s+ds given that it gets to s is 1-f(s) ds +o(ds).
• Multiplying all these probabilities proves Beer’s
law. So the Radon transform, Pf(L), the line
integral of a object with density (gm/cc), f(x,y)
can be measured. Radon’s theorem does the rest.
Rando phantom
• He has no neck. He is
used to calibrate
scanners. Note that Xray images are better
for finding cavities
than to study brain
tumors. Why is this?
Interior tissue density
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•
•
•
•
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Fat = .9
Bone = 2
Water = 1
Blood = 1.05
Tumor = 1.03
Gray matter = 1.02
First commercial CAT scanner
EMI 1972 Godfrey Hounsfield
• It measured one line
integral at a time. The
X-ray source is visible
at the bottom, there is
a detector at the top. It
measured 100 line
integrals and then
rotated 1 degree and
went back.
Anatomical “phantom” model
Hounsfield invented
tomography but didn’t
think of using an
anatomical model. This
idea turned out useful. The
line integrals can be
calculated exactly. Errors
in algorithm can be
separated from errors due
to noise in data.
The density values are chosen
• Note the “skull”,
“ventricles”, “tumors”
Seems pretty silly, but
I got very lucky with
this idea as we’ll see.
The line integrals of the phantom
• If the line misses the
head the integral is
zero. The small
“tumors” contribute
only to the 4th decimal
place. Need many
projections.
How to invert the Radon
transform, ie “reconstruct”
• The Fourier transform of the projection
is equal to the two-dimensional Fourier
transform of the object.
Thus we know the Fourier transform of f.
Now the Fourier inversion formula gives f
Derivative of the Hilbert
transform operator
In polar coordinates
The Jacobian is |r|, and
the product of f ^ with
• |r| = (ir)( -i sgn(r))
• ir = derivative
• -isgn(r) = Hilbert
transform
What is the contribution from
each line integral to the final
reconstruction?
• It’s a linear operator,
f(x,y) = sum over L of c(L,(x,y) ) Pf(L), and
we need only to know the coefficients
c(L,x,y) of the inverse. These depend only
on the distance from (x,y) to L. The filter
function is the function of the distance.
• It’s known as the Shepp-Logan filter. Many
other filters would be as good.
The first 60 back-projected
convolutions
• These are the first 60
• Projections +
convolutions
Convoluted backprojections
60-120
• These are the
accumulated next 60
convoluted
backprojections.
Convoluted backprojections 120180
• After 180
backprojected
convolution the
reconstruction (upper
left image) is complete.
Fourier reconstruction
• Note streak artifacts
outside the skull. Why
are they present?
• If God made us with
the skull at the center
of our brain and the
brain on the outside
CAT scanning would
be much less useful.
Artifact due to an error in one
line integral
• Shows the filter’s
contribution. But each
line integral
contributes to the
whole reconstruction.
Old tomography
• Simple backprojection
with no filtering.
Dates back to 1932 but
never caught on. Not
quite good enough to
be useful.
Filtering allows cancellation
• Old tomography gives
not f(x,y) but f * (1/r).
• Filtering removes the
1/r and gives back
f(x,y) after
accumulating the
backprojections.
Note the accuracy
• Except for some
averaging this gives
back the actual values
chosen in the phantom
1.02
• The value in the tumor
was 1.03, gray matter
was 1.02.
Hounsfield’s reconstruction
• Note the white just
inside the skull. Is it
real? It must be an
artifact. It wasn’t in
the original phantom.
Lucky me.
Hounsfield’s
algorithm was iterative
like Gauss-Seidel.
Later reconstruction by EMI
• Much better, but still
artifacted. This one was
due to another engineer at
EMI , Christopher Lemay.
• Lemay could not convince
Hounsfield to use a
formula. However he did
not use the Fourier
approach either but a
different one where there
was no choice of filter.
Can use to set thresholds
• CAT measurements of
line integrals are
accurate to .1%. f(x,y)
is reconstructed to .5%
• Radon inversion is a
singular integral
operator but it can be
done practically as we
see here.
CAT is sensitive to consistent
errors in the 80th line integral
• Some later CAT scanner
designs allowed the
detectors to rotate with the
tube. These were subject
to circle artifacts.
• The 4th generation design
avoided this problem but
ASE lost to GE.
Amer Sci Eng’g 1974 600
detectors
• The 4th generation
design. Stationary
detectors.
Some bad news
• For every finite n, it is
not enough to know n
projections. There are
invisible functions. In
fact for every 0 < f < 1
there is a g = 0,1 with
the same line integrals
as f in the n directions.
• Can CAT scanners be?
Coronal view
• Can see ventricles
• Not ellipses, alas.
Can use for the rest of the body
too, but less useful
• The fact that interior
head tissue is nearly
all the same becomes
an advantage.
What is this body part?
• Keep your guesses
clean.
Lungs and chest
• Note the rings in the
board. This was an
early test case at ASE.
Industrial application
• Delamination in exit
cone of rocket engine
NASA
Simulation of the NASA situation
• Even small
delaminations can be
found thanks to the
streak artifacts we saw
outside the skull.
Limited angle tomography
• Can one do
tomography with only
160 degrees of
projections?
Best we could do
• Judged not good
enough for the
application to fast
CAT scanning
• Probably not a good
research problem.
Analytic continuation
is involved.
New topics
• Emission tomography; PET, SPECT
• The subject ingests a radio-pharmaceutical which
moves under metabolic action to the place where
the body’s chemistry needs it. It emits radiation
which is measured. One can use a Poisson model
of radiation which has no errors and attempt to
find the maximum likelihood distribution that
makes the observed photon counts in the detectors
most likely. The problem with this technology is
that it is too slow to be used to study fast mental
processes.
Emission scanner PET
• Gets lower resolution
than CAT but it is
more effective than
CAT for metabolism
studies. CAT cannot
do metabolism at all.
• CAT measures
electron density.
Functional Magnetic Resonance
Imaging
• A hydrogen atom acts like a compass needle in a
magnetic field and oscillates (spins) with a
frequency proportional to field strength. Its
spectrum changes with the local surrounding
atoms and so magnetic resonance can be used for
spectroscopy. In particular, oxy and deoxy
hemoglobin can be distinguished by measuring
their resonances due to the fact that the nearby
oxygen atom changes slightly the rate of spin also
due to the iron atom nearby. The possibility of this
was pointed out by Pauling but Seiji Ogawa did it.
Magnetic Resonance Imaging
• Paul Lauterbur made the magnetic field
have a gradient so that the spins at different
points would be separated. The spins induce
an electrical current in a pick-up coil
surrounding the subject. In this way if the
local spin density at (x,y,z) is f(x,y,z) then
• The current induced in the coil is called the
free induction decay signal and is
Simple Fourier inversion
• The current induced in the coil is thus the Fourier
transform of the hydrogen spin density. Choosing
different gradients (a,b,c) allows
the Fourier transform to be measured at many
points in k-space = Fourier space and the spin
density f(x,y,z) can be obtained by direct Fourier
inversion. This is standard MRI. I want to discuss
a sub-topic, functional MRI.
Functional MRI
• When you are thinking about lunch, which part of
your brain is active? When you are
instantaneously recognizing Monica Lewinsky,
how is this done?
• In fMRI, the difference of the spin density pre and
post task is taken. This allows one to distinguish
oxy and deoxy hemoglobin. But this has to be
done in real time or we will never be able to see
where the image of Monica is stored etc. How to
sample the Fourier transform of the difference in
real time. It costs 1 ms to sample f^(k) at one k.
Space-time trade off
• We need good time resolution and are
willing to give up spatial resolution if
necessary. This can be done using the
uncertainly principle analog. Suppose we
measure the Fourier transform of the spin
density on a small subset of Fourier or
k-space. Then we can use the Parseval
identity
Prolate spheroidal wave functions
• We want to choose a phi^(k) that vanishes except
on a small set A of k’s.
• Then the right side is known if we only measure
f ^ on A . This takes only 50 ms if A is small.
• We also want to choose phi^ so that in brain space
phi is compactly supported. The uncertainty
principle says that if phi^ is compactly supported
then phi cannot be. There is a MOST compactly
supported phi though say for a sphere A in an L2
sense, maximizing the L2 integral over a region in
brain space given that the L2 norm of phi is one.
The trajectory of the k-space
measurements of f^(k)
• The best region A in k-space is a sphere of
low spatial frequencies, which is at first
surprising. One might think one should try
to sample k-space sparsely. We take A to be
a small sphere. We then have to loop
through A with a space-filling path along
which we take our measurements of f^(k).
• Which path to choose?
Ball of yarn 1
• One ball of yarn trajectory
Ball of yarn 2
• Second ball of yarn trajectory
How to choose the best trajectory?
• The first trajectory seems to be more spacefilling but it is also more complicated. A
still loosely formulated problem is how to
choose a curve which is most uniformly
dense in a sphere. In 2D people use an
Archimedean spiral but there are several
natural generalizations to 3D. Best? Hector?