Back Projection Reconstruction for CT, MRI and Nuclear Medicine

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Back Projection Reconstructionfor CT, MRI and
Nuclear Medicine

• F33AB5

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CT collects Projections
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• Introduction
• Coordinate systems
• Crude BPR
• Iterative reconstruction
• Fourier Transforms
• Central Section Theorem
• Direct Fourier Reconstruction
• Filtered Reconstruction

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To produce an image the projections are back
projected
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Crude back projection

• Add up the effect of spreading each projection
  back across the image space.
• This assumes equal probability that the object
  contributing to a point on the projection lay at
  any point along the ray producing that point.
• This results in a blurred image.

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Crude v filtered BPR
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Crude BPR
Filtered BPR
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Sinograms
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r
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Stack up projections
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Solutions

• Two competitive techniques
• Iterative reconstruction
• better where signal to noise ratio is poor
• Filtered BPR
• faster
• Explained by Brooks and di Chiro in Phys. Med.
  Biol. 21(5) 689-732 1976.

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Coordinate system

• Data collected as series of
• parallel rays, at position r,
• across projection at angle ?.
• This is repeated for various angles of ?.

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Attenuation of ray along a projection

• Attenuation occurs exponentially in tissue.
• ?(x) is the attenuation coefficient at position x
  along the ray path.

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Definition of a projection

• Attenuation of a ray at position r, on the
  projection at angle ?, is given by a line
  integral.
• s is distance along the ray, at position r across
  the projection at angle ?.

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Coordinate systems

• ?(x,y) and ?(r,s) describe the distribution of
  attenuation coefficients in 2 coordinate systems
  related by ?.
• where i 1..M for M different projection
  orientations
• angular increment is ?? ?/M.

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Crude back projection

• Simply sum effects of back-projected rays from
  each projection, at each point in the image.

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Crude back projection

• After crude back projection, the resulting image,
  ?(x,y), is convolution of the object (?(x,y))
  with a 1/r function.

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Convolution

• Mathematical description of smearing.
• Imagine moving a camera during an exposure. Every
  point on the object would now be represented by a
  series of points on the film the image has been
  convolved with a function related to the motion
  of the camera

?
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Iterative Technique

• Guess at a simulated object on a PxQ grid (?j,
  where j1?PxQ),
• Use this to produce simulated projections
• Compare simulated projections to measured
  projections
• Systematically vary simulated object until new
  simulated projections look like the measured
  ones.

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• For your scanner calculate ?jj(r,?i), the path
  length through the jth voxel for the ray at
  (r,?i)
• ?j need only be estimated once at the start of
  the reconstruction,
• ?j is zero for most pixels for a given ray in a
  projection

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?j0 ?20.1 ?71.2
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• The simulated projections are given by
• ?j is mean simulated attenuation coefficient in
  the jth voxel.

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Object and projections
First guess
From Physics of Medical Imaging by Webb
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To solve

• Analytically, construct P x Q simultaneous
  equations putting ?(r,?i) equal to the measured
  projections, p(r,?i)
• this produces a huge number of equations
• image noise means that the solution is not exact
  and the problem is 'ill posed
• Instead iterate modify ?j until ?(r,?i) looks
  like the real projection p(r,?i).

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Iterating

• Initially estimate ?j by projecting data in
  projection at ? 0 into rows, or even simply by
  making whole image grey.
• Calculate ?(r,?i) for each ?i in turn.
• For each value of r and ?, calculate the
  difference between ?(r,?) and p(r,?).
• Modify ?i by sharing difference equally between
  all pixels contributing to ray.

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122/3
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Next iteration
First guess
Object
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Fourier Transforms

• Imagine a note played by a flute.
• It contains a mixture of many frequency sound
  waves (different pitched sounds)
• Record the sound (to get a signal that varies in
  time)
• Fourier Transforming this signal will give the
  frequencies contained in the sound (spectrum)

Time
Frequency
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Fourier transforms of images

• A diffraction pattern is the Fourier transform of
  the slit giving rise to it

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Central Section theorem

• The 1D Fourier transform of a projection through
  an object is the same as a particular line
  through 2DFT of the object.
• This particular line lies along the conjugate of
  the r axis of the relevant projection.

Projection
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Direct Fourier Reconstruction

• Fourier Transform of each projection can be used
  to fill Fourier space description of object.

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Direct Fourier Reconstruction

• BUT this fills in Fourier space with more data
  near the centre.
• Must interpolate data in Fourier space back to
  rectangular grid before inverse Fourier
  transform, which is slow.

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Relationship between object and crude BPR results

• Crude back projection from above
• Defining inverse transform of projection as
• then

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• The right hand side has been multiplied and
  divided by k so that it has the form of a 2DFT in
  polar coordinates
• k conjugate to r
• ?k conjugate to ?r
• the integrating factor is kdrd? ? dxdy

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• Crude back projected image is same as the true
  image, except Fourier amplitudes have been
  multiplied by (magnitude of spatial frequency)-1.
• Physically because of spherical sampling.
• Mathematically because of changes in coordimates.

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Filtered BPR

• Multiplying 2 functions together is equivalent to
  convolving the Fourier Transforms of the
  functions.
• Fourier transform of (1/k) is (1/r)
• Multiplying FT of image with 1/k is same as
  convolving real image with 1/r
• ie BPR has effect we supposed.

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Filtered BPR

• Therefore there are two possible approaches to
  deblurring the crude BPR images
• Deconvolve multiplying by f (1/f x f 1) in
  Fourier domain.
• Convolve with Radon filter in the image domain,
  to overcome effect of being filtered with 1/r by
  crude BPR.