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3. Unknown 3

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10. 10 Backprojection usually produce a blurred version of the image.

11. 11 3 1 3 1 3 05330

12. 12 0.6 0.2 0.6 0.2 0.6 3 1 3 1 3

13. 13 010.6 0 01 0 01 0 01 0 01 0 05330

14. 14 0.6 0.2 0.6 0.2 0.6 01 0 01 0 01 0 01 0 01 0 1.61.2 0.6 0.21.20.8 0.2 0.61.61.2 0.6 0.21.20.8 0.2 0.61.61.2 0.6 BP: Numerical Example

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17. 2D-FT(I) 1D-FT(Radon(I)) 0°10° 90° 120° 17

18. 18 Fundamentals of Medical Imaging Paul Suentes

19. 19 2D Inverse Fourier Transform Function we want to reconstruct Let’s change F from cartesian coordinates to polar coordinates

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21. 21 Form half lines to full lines :

22. 22 Now the Central Slice Theorem become simply : And therefore:

23. 23 And its 1D inverse Fourier transform from k to r. In the Radon domain it’s a convolution over r :

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25. 25 Note that it’s a backprojection ! This is called Filtered Backprojection

26. 26 Fundamentals of Medical Imaging Paul Suentes

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31. 31 Ram-Lak Filter (or smoothed version of it) Projections Backproject Reconstructed image

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33. 33 3 1 3 1 3 05330

34. 34 0.6 0.2 0.6 0.2 0.6 3 1 3 1 3 05330

35. 35 0.6 0.2 0.6 0.2 0.6 3 1 3 1 3 05330 2.2

36. 36 0.6 0.2 0.6 0.2 0.6 3 1 3 1 3 05330 -2.22.80.8 -2.2 Rectification

37. 37 0.161.160.76 0.16 -0.220.760.36 -0.22 0.161.160.76 0.16 -0.220.760.36 -0.22 0.161.160.76 0.16 3 1 3 1 3 05330 -2.22.80.8 -2.2 Rectification

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