If F{f(x,y)} = F(u,v) or F(?,?)

1
Central Section or Projection Slice Theorem
If Ff(x,y) F(u,v) or F(?,?) then F(?,?)
F g? (R) The Fourier Transform of a
projection at angle ? is a line in the Fourier
transform of the image at the same angle. The
Central Section Theorem is also referred to as
the projection-slice theorem.
2
CT Reconstruction Methods - Filtered Back
Projection
p 8 ? d? ? F-1Fg? (R) p ? ( x
cos ? y sin ? - R) dR 0 -8
Each projection is filtered to account for
oversampling of lower spatial frequencies before
back projection. Here filtering is done in the
frequency domain
3
Convolution Back Projection
p 8 ? d? ? g? (R) c (R) ? ( x
cos ? y sin ? - R) dR 0 -8 Same idea as
filtered back projection, but filtering is done
in projection domain. Here each projection is
convolved with c (R) and then back
projected. Describe c (R) C (p) p c (R)
lim 2 (?2 - 4p2R2) / (?2 4p2R2)2 ? ?0
4
CT Artifacts Beam Hardening
I0
I I0 e - ? ? dl ln (I0 / I) ? ?
dl Assumptions - zero width pencil beam
- monoenergetic Look at horizontal and
vertical projection measure of attenuation at
point P. How will bone affect the vertical
projection?
?(x,y)
Bone

P
5
Should get same answer for each projection.
Normal emitted energy spectrum How does beam look
after moving through bone tissue?
Relative Intensity
E
Bone attenuates
Cupping artifact Soft tissue cannot demonstrate
its attenuation ? no photons to show it
6
First Generation CT Scanner
From Webb, Physics of Medical Imaging
7
Physics of Medical Physics, EditorWebb
8
Interpolation during Back Projection
9
Second Generation CT Scanner
From Webb, Physics of Medical Imaging
10
3rd and 4th Generation Scanners
From Webb, Physics of Medical Imaging
11
Electron Beam CT
12
Krestel- Imaging Systems for Medical Diagnosis
13
Krestel- Imaging Systems for Medical Diagnosis
14
Krestel- Imaging Systems for Medical Diagnosis
15
Sampling Requirements in CT

• How many angles must we acquire?
• Samples acquired by each projection are shown
  below as they fill the F(u,v) space.

v
u
F(u,v)
16
Impulse Response
Lets assume that the CT scanner acquires exact
projections. Then the impulse response is the
inverse 2D FT of the sampled pattern.
v
u
Inverse 2D FT
F(u,v)
Impulse response h(x,y)
17
Impulse response

• Bright white line points to correctly imaged
  impulse.
• Inside of White circle shows region that is
  correctly imaged
• Region outside of white circle suffers streak
  artifacts.
• How big is the white circle?
• How does it relate to the number of angles?
• Easier to think of in frequency domain

Impulse response h(x,y)
18
Recall Band limiting and Aliasing

• Fg(x) G(u)
• G(u) is band limited to uc, (cutoff frequency)
• Thus G(u) 0 for u gt uc.

uc
To avoid overlap (aliasing) with a sampling
interval X,
uc
Nyquist Condition Sampling rate must be greater
than twice the frequency component.
19
Apply to Imaging
Fg(x,y) G(u,v)
G(u,v)
g(x,y)
If we sample g(x,y) at intervals of X in x and y,
then G(u,v) replicates
v
1/X

G(u,v)
u
1/X
20
Sampling in Frequency Domain
If we sample in the frequency domain, then the
image will replicate at 1/frequency sampling
interval.
G(u,v)
DF
Intersections depict sampling points. Let
sampling interval in frequency domain be equal in
each direction and be DF. Then we expect the
images to replicate every 1/ DF
21
1/ DF FOV
FOV
22
How many angles do we need?
Three angles acquired from a CT exam are shown,
each acquired Dq apart. Data acquisition is
shown in the Fourier space using the Central
Section Theorem. The radial spacing, DFr , is
the separation between the vertical hash marks on
the horizontal projection. If we consider what
the horizontal projection will look like as it is
back projected, we can appreciate that 1/ DFr
Field of View.
23
How many angles do we need?
To avoid aliasing of any spatial frequencies, the
spacing of the points in all directions must be
no larger than DFr . The largest spacing between
points will come along the circumference. If we
scale the radius of the circle so it has radius
1, the circumference will be 2p. However, we
only need projections to sample a distance of p
since one projection will provide two points on
the circle.
24
How many angles do we need?
To avoid aliasing of any spatial frequencies, the
spacing of the points in all directions must be
no larger than DFr . The largest spacing between
points will come along the circumference. If we
scale the radius of the circle so it has radius
1, the circumference will be 2p. However, we
only need projections to sample a distance of p
since one projection will provide two points on
the circle.
25
How many angles do we need?
For a radius of 1 or diameter of 2, DFr 2/N
where N is the number of detectors. That is,
there are N hash marks ( samples) on the
projection above. The spacing of points on the
circumference will then be p/M where M is the
number of projections. So the azimuthal spacing
on the circumference, p/M,must be equal or
smaller to the radial spacing 2/N. Solving for
M gives M p/2 N
26
Azimuthal Sampling requires M angles p/2
Ndetector elements

• Azimuthal Undersampling
• Sampling Pattern - frequency space or projections
  acquired
• Plot of one line of 2D impulse response
• 2D impulse response with undersampling
• Image of a Square
• What causes artifacts? Where do artifacts
  appear?