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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

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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

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

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

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

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

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

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

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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

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

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

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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

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IPIM, IST, José Bioucas, 2007 13 Backprojection Operator

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

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

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IPIM, IST, José Bioucas, 2007 16

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

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