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

IPIM, IST, José Bioucas, 2007 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% S R body L dl

IPIM, IST, José Bioucas, 2007 3 CT images (from wikipedia)

IPIM, IST, José Bioucas, 2007 4 X-Ray Tomography Radon transform

IPIM, IST, José Bioucas, 2007 5 Example of Radon Transform: sinogram

IPIM, IST, José Bioucas, 2007 6 Example of Radon Transform: sin0gram

IPIM, IST, José Bioucas, 2007 7 Fourier Slice Theorem

IPIM, IST, José Bioucas, 2007 8 Fourier Slice Theorem: Illustration

IPIM, IST, José Bioucas, 2007 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

IPIM, IST, José Bioucas, 2007 10 Fourier Slice Theorem: Consequences Assumnig that is bandlimited to then it is possible to restore with a resolution

IPIM, IST, José Bioucas, 2007 11 Backprojection Operator is the sum of all line integrals that crosses the point o

IPIM, IST, José Bioucas, 2007 12 Backprojection Operator is the adjoint operator of when the usual inner product is considered for both the data functions and the objects

IPIM, IST, José Bioucas, 2007 13 Backprojection Operator

IPIM, IST, José Bioucas, 2007 14 Filtered Backprojection Algorithm periodic function of 

IPIM, IST, José Bioucas, 2007 15 Filtered Backprojection Algorithm From the Fourier slice theorem Defining the filtered backprojections as Then

IPIM, IST, José Bioucas, 2007 16

IPIM, IST, José Bioucas, 2007 17 Filtered Backprojection Hann filter Ram-Lak filter

IPIM, IST, José Bioucas, 2007 18

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