x-ray computed tomography
DESCRIPTION
X-Ray Computed Tomography. Radon Transform Fourier Slice Theorem Backprojection Operator Filtered Backprojection (FBP) Algorithm Implementation Issues Total Variation Reconstruction. X-Ray Tomography. “Tomography” comes from Greek and means cross-sectional representations - PowerPoint PPT PresentationTRANSCRIPT
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
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CT images (from wikipedia)
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X-Ray Tomography
Radon transform
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Example of Radon Transform: sinogram
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6IPIM, IST, José Bioucas, 2007
Example of Radon Transform: sin0gram
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Fourier Slice Theorem
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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 Radom transform of implies the knowledge of the Fourier transform of
is spacelimited
can be computed from samplestaken apart
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Fourier Slice Theorem: Consequences
Assumnig that is bandlimited to then it is possible to restore with a resolution
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Backprojection Operator
is the sum of all line integrals that crosses the point
o
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Backprojection Operator
is the adjoint operator of when the usual inner product is considered for both the data functions and the objects
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Backprojection Operator
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Filtered Backprojection Algorithm
periodic function of
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Filtered Backprojection Algorithm
From the Fourier slice theorem
Defining the filtered backprojections as
Then
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Filtered Backprojection
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Hann filter
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Ram-Lak filter
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