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

X-Ray Computed Tomography

  • Radon Transform

  • Fourier Slice Theorem

  • Backprojection Operator

  • Filtered Backprojection (FBP) Algorithm

  • Implementation Issues

  • Total Variation Reconstruction


X ray computed tomography

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


X ray computed tomography

CT images (from wikipedia)


X ray computed tomography

X-Ray Tomography

Radon transform


X ray computed tomography

Example of Radon Transform: sinogram


X ray computed tomography

Example of Radon Transform: sin0gram


X ray computed tomography

Fourier Slice Theorem


X ray computed tomography

Fourier Slice Theorem: Illustration


X ray computed tomography

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


X ray computed tomography

Fourier Slice Theorem: Consequences

Assumnig that is bandlimited to then it is possible to restore

with a resolution


X ray computed tomography

is the sum of all line integrals that crosses the point

o

Backprojection Operator


X ray computed tomography

Backprojection Operator

is the adjoint operator of when the usual inner product is

considered for both the data functions and the objects


X ray computed tomography

Backprojection Operator


X ray computed tomography

Filtered Backprojection Algorithm

periodic function of 


X ray computed tomography

Filtered Backprojection Algorithm

From the Fourier slice theorem

Defining the filtered backprojections as

Then


X ray computed tomography

Filtered Backprojection

Hann filter

Ram-Lak filter


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