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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

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Presentation Transcript
slide1

X-Ray Computed Tomography

  • Radon Transform
  • Fourier Slice Theorem
  • Backprojection Operator
  • Filtered Backprojection (FBP) Algorithm
  • Implementation Issues
  • Total Variation Reconstruction
slide2

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

slide4

X-Ray Tomography

Radon transform

slide9

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

slide10

Fourier Slice Theorem: Consequences

Assumnig that is bandlimited to then it is possible to restore

with a resolution

slide12

Backprojection Operator

is the adjoint operator of when the usual inner product is

considered for both the data functions and the objects

slide14

Filtered Backprojection Algorithm

periodic function of 

slide15

Filtered Backprojection Algorithm

From the Fourier slice theorem

Defining the filtered backprojections as

Then

slide17

Filtered Backprojection

Hann filter

Ram-Lak filter

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