Imaging from Projections
Eric Miller
With minor modifications by
Dana Brooks
These slides based almost entirely on
a set provided by Prof. Eric Miller
Outline
• Problem formulation
– What’s a projection?
– Application examples
– Why is this interesting?
• The forward problem
– The Radon transform
– The Fourier Slice Theorem
• The Inverse Problem
– Undoing the Radon transform with the help of Fourier
– Filtered Backprojection Algorithm
• Complications and Extensions
These slides based almost entirely on
a set provided by Prof. Eric Miller
A Projection
The total amount of f(x,y) along the line defined by t and q
Pq  t 
t
y
q
f ( x, y )
x
t0
These slides based almost entirely on
a set provided by Prof. Eric Miller
Application Examples
• CAT scans:
– X ray source moves around the body
– f(x,y) is the density of the tissue
• MRI
– Not as clear cut what the “projection” is, but in a peculiar way, the
math is the same (remind me to talk about this when we get to the
MRI Imaging equation …)
– f(x,y) is the spin density of molecules in the tissue
• Synthetic Aperture Radar
– Satellite moves down a linear track collecting radar echoes of the
ground
– Used for remote sensing, surveillance, …
– Again: math is the same (after much pain and anguish)
– f(x,y) is the reflectivity of the earth surface
These slides based almost entirely on
a set provided by Prof. Eric Miller
Motivation
• In all cases, one observes a bunch of sum or
integrals of a quantity over a region of
space: these are “projections”
• The goal is to use a collection of these
projections to recover f(x,y).
• Here we will talk about the full data case
– Assume we see Pq  t  for all q and t
• Limited view tomography a topic for
advanced course
These slides based almost entirely on
a set provided by Prof. Eric Miller
The Radon Transform
Pq  t 
Polar equation for line:
t
x cosq  y sin q  t
ds
So the line exists only where
this equation is true
y
Function of
t and q
Pq  t  
f ( x, y )
x

q

f  x, y  ds
,t  line
 

q
  f  x, y    x cosq  y sin q  t  dxdy
 
These slides based almost entirely on
a set provided by Prof. Eric Miller
What does it do?
Simplest case: f(x,y) a  function: only exists at a single point
Pq  t  
 
    x  x , y  y    x cosq  y sin q  t  dxdy
0
0
 
   x0 cosq  y0 sin q  t 
• Proof only by limiting argument as products of ’s not well defined
• Interpretation:
• A “function” in (t,q) space which “is” 1 along a sinusoidal
curve and zero elsewhere: note that a point in 2D  a curve
• Say y0 = 0 and x0 = 1 then this is an “image” which “is” 1 when
t = cos q
These slides based almost entirely on
a set provided by Prof. Eric Miller
In Pictures
Kind of 2D impulse response (PSF)
y
t
q
x
Pq  t 
The Image
Called the Radon Transform
(a.k.a.the sinogram)
Note that we draw Pq  t  as a rectangular “image” in t and q
These slides based almost entirely on
a set provided by Prof. Eric Miller
More Examples
t
q
t
These slides based almost entirely on
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q
Fourier Slice Theorem
• Key idea here and for a large number of
other problems
• Analytically relate the 1D Fourier
transform of P to the 2D Fourier
transform of f.
• Why?
– If we can do this, then a simple inverse 2D
Fourier gives us back f from the “data” P.
These slides based almost entirely on
a set provided by Prof. Eric Miller
Recall 2D Fourier Transform
Analysis
 
F  u, v  
  f  x, y  exp  j 2 ux  vy dxdy
 
Synthesis
f  x, y  
 
  F  u, v  exp  j 2 ux  vy dudv
 
• “Space” variable x goes with “frequency” variable u
• “Space” variable y goes with “frequency” variable v
• (u,v) called “spatial frequency domain”
These slides based almost entirely on
a set provided by Prof. Eric Miller
Fourier – Slice Theorem (FST)
• Let F(u,v) be defined as on last slide
• Define Sq(w) as the 1D Fourier transform of P along t
for some frequency variable w
Sq  w  

 Pq t  exp  j 2 wt dt

• FST says that Sq is equal to F(u,v) along a line
tilted at an angle q with respect to the (u,v)
coordinate system
•To make this more precise …
These slides based almost entirely on
a set provided by Prof. Eric Miller
Fourier-Slice
Pq  t 
v
F(u,v) along line
t
1D Fourier Transform
w
y
f ( x, y )
q
q
x
Variables w and q are the polar form of u and v
u  w cosq
v  w sin q
So FST is:
Sq  w  F  w cosq , w sin q 
These slides based almost entirely on
a set provided by Prof. Eric Miller
u
Reconstruction Implications
v
• Collect data from lots and
lots of projections.
•Take 1D FT of each to get
one line in 2D frequency
space
• Fill up 2D spatial frequency
u space on a polar grid
• Interpolate onto rectangular
grid
• Inverse 2D FT and we are
done!!
These slides based almost entirely on
a set provided by Prof. Eric Miller
An Alternate Approach
Filtered Backprojection
• This requires lots of Fourier Transforms
• This means we can’t begin processing until we have all slices
• Turns out there’s a more efficient way to organize things
• This requires “ugly” interpolation, worse at high frequencies
The derivation of this algorithm is perhaps one of the most
illustrative examples of how we can obtain a radically different
computer implementation by simply re-writing the fundamental
expressions for the underlying theory
- Kak and Slaley, CTI
These slides based almost entirely on
a set provided by Prof. Eric Miller
FBP Motivation in Pictures
v
v
u
w
u
w
By linearity, could in theory
break up reconstruction into
contribution from independent
“wedges” in 2D Fourier space
• In practice, we measure over
lines.
• Idea: build a 2D filter
which covers the line, but has
the same “weight” as the
wedge at that frequency, w
• In other words “mush”
triangle to a rectangle
• Then “sum up” filtered
projections
These slides based almost entirely on
a set provided by Prof. Eric Miller
• For K projections, the width
of the wedge at w is just
width 
2 w
K
FBP Theory
f  x, y    F  u , v  exp  j 2  ux  vy  dudv
Now, change right side from polar to rectangular
u  w cosq
v  w sin q
dudv  wdwdq
To get rectangular coordinates in space, polar in frequency:
f  x, y  
2 
  F  w,q  w exp  j 2 w  x cosq  y sin q  dwdq
0 0
These slides based almost entirely on
a set provided by Prof. Eric Miller
FBP Theory II
Make use of two facts:
F  w,q     F   w,q 
t  x cos q  y sin q



To arrive at f  x, y      Sq  w  w exp  j 2 wt dw dq
0  


  Qq  x cos q  y sin q dq
Qq  t  
 0

Backproject
Sq  w  w exp  j 2 wt dw Filter (in space)

These slides based almost entirely on
a set provided by Prof. Eric Miller
FBP Interpretation
• Recall from linear systems
Fourier
transform
d
f  t   jF   
dt
• So |w| filter is more or less a differentiator. Accentuated high
frequency information leads to problems with noise amplification
• In practice, roll off response.
w
w
These slides based almost entirely on
a set provided by Prof. Eric Miller
FBP Interpretation
Backprojection: Note that Qq(t) needs only one (filtered) projection

 Qq  x cosq  y sin q dq
0
Sum up over all
angles
Think of this as Qq(t) evaluated
at the point t = xcosq + y sinq
Qq  t0 
t
y
Region we are
reconstructing
q
These slides based almost entirely on
a set provided by Prof. Eric Miller
• Along this line
in “image space”
set the value to
Qq(t0)
x
• All points get a value
• Do for all angles
• Add up
FBP Example
Orig.
Recon
Zoom
These slides based almost entirely on
a set provided by Prof. Eric Miller
Limited data I:
Angle decimation
These slides based almost entirely on
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Limited data II:
Limited Angle
These slides based almost entirely on
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Artifact Mitigation
• Take a more matrix-based “inverse problems” perspective
• Discretized Radon transform, data, and object to arrive at a
forward model
y  Cx
• Where C has many fewer rows than columns
• Use SVD, TSVD, Tikhonov, or other favorite
regularization scheme to improve reconstruction results
• Note: significant move from analytical to numerical
inversion means a basic shift in how we are approaching
the problem. No more FBP (at least not easily)
These slides based almost entirely on
a set provided by Prof. Eric Miller
Other Fourier Imaging Applications
v
v
u
• Standard SAR
• Collects data on
wedge shaped regions
of Fourier space
• Very limited view
• Similar math to X-ray
u
• Diffraction tomography
• Collects data on petal shaped regions
of Fourier space
• Very limited view
• More sophisticated math than X ray
• Arises in geophysical and medical
imaging problems
These slides based almost entirely on
a set provided by Prof. Eric Miller
Generalized Radon Transforms
• Radon transform = integral of object over straight
lines
• Many extensions
– Integration over planes in 3D
– Over circles in 2D (different type of SAR)
– Over much more arbitrary mathematical structures
(asymptotic case of some acoustics problems with
space varying background).
– Of weighted object function (attenuated Radon
transform)
These slides based almost entirely on
a set provided by Prof. Eric Miller