X-Ray Computed Tomography
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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

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X-Ray Computed Tomography

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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

    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%

L

S

body

dl

R


CT images (from wikipedia)


X-Ray Tomography

Radon transform


Example of Radon Transform: sinogram


Example of Radon Transform: sin0gram


Fourier Slice Theorem


Fourier Slice Theorem: Illustration


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 samples

taken apart


Fourier Slice Theorem: Consequences

Assumnig that is bandlimited to then it is possible to restore

with a resolution


is the sum of all line integrals that crosses the point

o

Backprojection Operator


Backprojection Operator

is the adjoint operator of when the usual inner product is

considered for both the data functions and the objects


Backprojection Operator


Filtered Backprojection Algorithm

periodic function of 


Filtered Backprojection Algorithm

From the Fourier slice theorem

Defining the filtered backprojections as

Then


Filtered Backprojection

Hann filter

Ram-Lak filter


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