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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 Tomography. “Tomography” comes from Greek and means cross-sectional representations

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X-Ray Computed Tomography

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  1. X-Ray Computed Tomography • Radon Transform • Fourier Slice Theorem • Backprojection Operator • Filtered Backprojection (FBP) Algorithm • Implementation Issues • Total Variation Reconstruction

  2. 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

  3. CT images (from wikipedia)

  4. X-Ray Tomography Radon transform

  5. Example of Radon Transform: sinogram

  6. Example of Radon Transform: sin0gram

  7. Fourier Slice Theorem

  8. Fourier Slice Theorem: Illustration

  9. 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

  10. Fourier Slice Theorem: Consequences Assumnig that is bandlimited to then it is possible to restore with a resolution

  11. is the sum of all line integrals that crosses the point o Backprojection Operator

  12. Backprojection Operator is the adjoint operator of when the usual inner product is considered for both the data functions and the objects

  13. Backprojection Operator

  14. Filtered Backprojection Algorithm periodic function of 

  15. Filtered Backprojection Algorithm From the Fourier slice theorem Defining the filtered backprojections as Then

  16. Filtered Backprojection Hann filter Ram-Lak filter

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