Tomography

1
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

2
Why line integrals?

• Beers 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 sds given that it gets to s is 1-f(s) ds
  o(ds).
• Multiplying all these probabilities proves Beers
  law. So the Radon transform, Pf(L), the line
  integral of a object with density (gm/cc), f(x,y)
  can be measured. Radons theorem does the rest.

3
Rando phantom

• He has no neck. He is used to calibrate scanners.
  Note that X-ray images are better for finding
  cavities than to study brain tumors. Why is this?

4
Interior tissue density

• Fat .9
• Bone 2
• Water 1
• Blood 1.05
• Tumor 1.03
• Gray matter 1.02

5
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.

6
Anatomical phantom model

• Hounsfield invented tomography but didnt
  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.

7
The density values are chosen

• Note the skull,
• ventricles, tumors
• Seems pretty silly, but
• I got very lucky with this idea as well see.

8
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.

9
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

10
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

11
What is the contribution from each line integral
to the final reconstruction?

• Its 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.
• Its known as the Shepp-Logan filter. Many
• other filters would be as good.

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The first 60 back-projected convolutions

• These are the first 60
• Projections convolutions

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Convoluted backprojections 60-120

• These are the accumulated next 60 convoluted
  backprojections.

14
Convoluted backprojections 120-180

• After 180 backprojected convolution the
  reconstruction (upper left image) is complete.

15
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.

16
Artifact due to an error in one line integral

• Shows the filters contribution. But each line
  integral contributes to the whole reconstruction.

17
Old tomography

• Simple backprojection with no filtering. Dates
  back to 1932 but never caught on. Not quite good
  enough to be useful.

18
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.

19
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.

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Hounsfields reconstruction

• Note the white just inside the skull. Is it real?
  It must be an artifact. It wasnt in the original
  phantom. Lucky me. Hounsfields algorithm was
  iterative like Gauss-Seidel.

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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.

22
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.

23
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.

24
Amer Sci Engg 1974 600 detectors

• The 4th generation design. Stationary detectors.

25
Some bad news

• For every finite n, it is not enough to know n
  projections. There are invisible functions. In
  fact for every 0 lt f lt 1 there is a g 0,1 with
  the same line integrals as f in the n directions.
• Can CAT scanners be?

26
Coronal view

• Can see ventricles
• Not ellipses, alas.

27
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.

28
What is this body part?

• Keep your guesses clean.

29
Lungs and chest

• Note the rings in the board. This was an early
  test case at ASE.

30
Industrial application

• Delamination in exit cone of rocket engine NASA

31
Simulation of the NASA situation

• Even small delaminations can be found thanks to
  the streak artifacts we saw outside the skull.

32
Limited angle tomography

• Can one do tomography with only 160 degrees of
  projections?

33
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.

34
New topics

• Emission tomography PET, SPECT
• The subject ingests a radio-pharmaceutical which
  moves under metabolic action to the place where
  the bodys 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.

35
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.

36
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.

37
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

38
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.

39
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.

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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

41
Prolate spheroidal wave functions

• We want to choose a phi(k) that vanishes except
  on a small set A of ks.
• 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.

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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?

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Ball of yarn 1

• One ball of yarn trajectory

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Ball of yarn 2

• Second ball of yarn trajectory

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How to choose the best trajectory?

• The first trajectory seems to be more
  space-filling 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?

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