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IPIM, IST, José Bioucas, 2007 1 X-Ray Computed Tomography Radon Transform Fourier Slice Theorem Backprojection Operator Filtered Backprojection (FBP) Algorithm.

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Presentation on theme: "IPIM, IST, José Bioucas, 2007 1 X-Ray Computed Tomography Radon Transform Fourier Slice Theorem Backprojection Operator Filtered Backprojection (FBP) Algorithm."— Presentation transcript:

1 IPIM, IST, José Bioucas, X-Ray Computed Tomography Radon Transform Fourier Slice Theorem Backprojection Operator Filtered Backprojection (FBP) Algorithm Implementation Issues Total Variation Reconstruction

2 IPIM, IST, José Bioucas, 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 R body L dl

3 IPIM, IST, José Bioucas, CT images (from wikipedia)

4 IPIM, IST, José Bioucas, X-Ray Tomography Radon transform

5 IPIM, IST, José Bioucas, Example of Radon Transform: sinogram

6 IPIM, IST, José Bioucas, Example of Radon Transform: sin0gram

7 IPIM, IST, José Bioucas, Fourier Slice Theorem

8 IPIM, IST, José Bioucas, Fourier Slice Theorem: Illustration

9 IPIM, IST, José Bioucas, 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 IPIM, IST, José Bioucas, Fourier Slice Theorem: Consequences Assumnig that is bandlimited to then it is possible to restore with a resolution

11 IPIM, IST, José Bioucas, Backprojection Operator is the sum of all line integrals that crosses the point o

12 IPIM, IST, José Bioucas, Backprojection Operator is the adjoint operator of when the usual inner product is considered for both the data functions and the objects

13 IPIM, IST, José Bioucas, Backprojection Operator

14 IPIM, IST, José Bioucas, Filtered Backprojection Algorithm periodic function of 

15 IPIM, IST, José Bioucas, Filtered Backprojection Algorithm From the Fourier slice theorem Defining the filtered backprojections as Then

16 IPIM, IST, José Bioucas,

17 IPIM, IST, José Bioucas, Filtered Backprojection Hann filter Ram-Lak filter

18 IPIM, IST, José Bioucas,


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