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1IPIM, IST, José Bioucas, 2007
X-Ray computed tomography
� Radon Transform
� Fourier Slice Theorem
� Backprojection Operator
� Filtered Backprojection (FBP) Algorithm
� Implementation Issues
� Total Variation Reconstruction
2IPIM, IST, José Bioucas, 2007
X-Ray tomography
� “Tomography” comes from Greek and means cross-sectional representations
of a 3D object
� X-ray tomography, introduced by Hounsfield in 1971, is usually called
computed tomography (CT)
� CT produces images of human anatonomy with a resolution of about 1mm
and an attenuation coefficient attenuation of about 1%
S
R
bodyL
dl
3IPIM, IST, José Bioucas, 2007
CT images (from wikipedia)
4IPIM, IST, José Bioucas, 2007
X-Ray tomography
Radon transform
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Example of Radon transform: sinogram
θ (degrees)
x'
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6IPIM, IST, José Bioucas, 2007
Example of Radon transform: sinogram
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7IPIM, IST, José Bioucas, 2007
Fourier slice theorem
8IPIM, IST, José Bioucas, 2007
Fourier slice theorem: illustration
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9IPIM, IST, José Bioucas, 2007
Fourier slice theorem: consequences
1- The solution of the inverse problem , when exists,
is unique
2- The knowledge of the the Radon transform of implies the
knowledge of the Fourier transform of
is spacelimited
can be computed from samples
taken apart
10IPIM, IST, José Bioucas, 2007
Fourier slice theorem: consequences
Assumnig that is bandlimited to then it is possible to restore
with a resolution
11IPIM, IST, José Bioucas, 2007
Backprojection operator
is the sum of all line integrals that crosses the point
o
12IPIM, IST, José Bioucas, 2007
Backprojection operator
is the adjoint operator of when the usual inner product is
considered for both the data functions and the objects
13IPIM, IST, José Bioucas, 2007
Backprojection operator
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14IPIM, IST, José Bioucas, 2007
Filtered backprojection algorithm
periodic function of φ
15IPIM, IST, José Bioucas, 2007
Filtered backprojection algorithm
From the Fourier slice theorem
Defining the filtered backprojections as
Then
16IPIM, IST, José Bioucas, 2007
Summary of the filtered backprojection algorithm
17IPIM, IST, José Bioucas, 2007
Filtered backprojection
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Hann filter
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Ram-Lak filterTV - Reconstruction
