Imaging%20from%20Projections

1
Imaging from Projections

• Eric Miller
• With minor modifications by
• Dana Brooks

2
Outline

• Problem formulation
• Whats 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

3
A Projection
The total amount of f(x,y) along the line defined
by t and q
4
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

5
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 for all q and t
• Limited view tomography a topic for advanced
  course

6
The Radon Transform
Polar equation for line
So the line exists only where this equation is
true
Function oft and q
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What does it do?
Simplest case f(x,y) a d function only exists
at a single point

• Proof only by limiting argument as products of
  ds 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

8
In Pictures
Kind of 2D impulse response (PSF)
y
t
q
x
The Image
Called the Radon Transform (a.k.a.the sinogram)
9
More Examples
t
q
t
q
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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.

11
Recall 2D Fourier Transform
Analysis
Synthesis

• Space variable x goes with frequency
  variable u
• Space variable y goes with frequency
  variable v
• (u,v) called spatial frequency domain

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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
• 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

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Fourier-Slice
t
1D Fourier Transform
y
x
Variables w and q are the polar form of u and v
So FST is
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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 space on a polar
  grid
• Interpolate onto rectangular grid
• Inverse 2D FT and we are done!!

u
15
An Alternate ApproachFiltered Backprojection

• This requires lots of Fourier Transforms
• This means we cant begin processing until we
  have all slices
• Turns out theres 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 differentcomputer
implementation by simply re-writing the
fundamentalexpressions for the underlying
theory - Kak and Slaley, CTI
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FBP Motivation in Pictures
v
v
w
u
u

• 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

By linearity, could in theory break up
reconstruction intocontribution from
independentwedges in 2D Fourier space

• For K projections, the width of the wedge at w
  is just

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FBP Theory
Now, change right side from polar to rectangular
To get rectangular coordinates in space, polar in
frequency
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FBP Theory II
Make use of two facts
To arrive at
Backproject
Filter (in space)
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FBP Interpretation

• Recall from linear systems
• 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
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FBP Interpretation
Backprojection Note that Qq(t) needs only one
(filtered) projection
Think of this as Qq(t) evaluated at the point
t xcosq y sinq
Sum up over allangles
t

• Along this line in image space set the
  value to
• Qq(t0)
• All points get a value
• Do for all angles
• Add up

y
Region we arereconstructing
x
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FBP Example
Orig.
Recon
Zoom
22
Limited data IAngle decimation
23
Limited data IILimited Angle
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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
• 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)

25
Other Fourier Imaging Applications
v
v
u
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

• Standard SAR
• Collects data on wedge shaped regions of
  Fourier space
• Very limited view
• Similar math to X-ray

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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)