1. Medical Imaging Mohammad Dawood Department of Computer Science University of Münster Germany

2. 2 Medical Imaging, SS-2010 Mohammad Dawood Image Reconstruction

3. 3 Medical Imaging, SS-2010 Mohammad Dawood Reconstruction Law of Attenuation

4. 4 Medical Imaging, SS-2010 Mohammad Dawood Reconstruction Parallel projections of a plane

5. 5 Medical Imaging, SS-2010 Mohammad Dawood x y r s θ n Reconstruction Radon Transformation f

6. 6 Medical Imaging, SS-2010 Mohammad Dawood Reconstruction Radon Transformation (Line Integrals at different angles)

7. 7 Medical Imaging, SS-2010 Mohammad Dawood Reconstruction Radon Transformation OriginalSinogram (Radon Transform)

8. 8 Medical Imaging, SS-2010 Mohammad Dawood Reconstruction Inverse Radon Transformation H: Hilbert transform

9. 9 Medical Imaging, SS-2010 Mohammad Dawood Reconstruction Filtered Back Projection

10. 10 Medical Imaging, SS-2010 Mohammad Dawood Reconstruction Filtered Back Projection

11. 11 Medical Imaging, SS-2010 Mohammad Dawood Projections Backproject Filter 1D Filter 2D Backproject Image Reconstruction Filtered Back Projection 2D/3D filtering is costly

12. 12 Medical Imaging, SS-2010 Mohammad Dawood Reconstruction Fourier Slice Theorem

13. 13 Medical Imaging, SS-2010 Mohammad Dawood Reconstruction Fourier slice theorem Take a two-dimensional function f(r), project it onto a line, and do a Fourier transform of that projection Take that same function, but do a two-dimensional Fourier transform first, and then slice it through its origin parallel to the projection line

14. 14 Medical Imaging, SS-2010 Mohammad Dawood Reconstruction Fourier Slice Theorem

15. 15 Medical Imaging, SS-2010 Mohammad Dawood

16. 16 Medical Imaging, SS-2010 Mohammad Dawood

17. 17 Medical Imaging, SS-2010 Mohammad Dawood 1=Ram-Lak (ramp), 2=Shepp-Logan, 3=Cosine, and 4=Hamming Reconstruction FBP: Commonly used filters

18. 18 Medical Imaging, SS-2010 Mohammad Dawood Reconstruction Iterative Reconstruction b: measured values x: unknown attenuation coefficients a ij : weights f1f1 f2f2 …fnfn LOR 1 LOR 2 … LOR n

19. 19 Medical Imaging, SS-2010 Mohammad Dawood Reconstruction Iterative Reconstruction Kaczmarz Method (=ART: Algebraic Reconstruction Technique)

20. 20 Medical Imaging, SS-2010 Mohammad Dawood Reconstruction Iterative Reconstruction Kaczmarz Method (=ART: Algebraic Reconstruction Technique) 1. Start by setting x(0) = 0 2. Compute the forward projection from the n-th estimate, i.e. b(n) = A x(n) 3. Choose i and correct the current estimate x(n) 4. Iterate steps 2,3 until the difference between new forward projection b(n), computed in 2, and the old one is below tolerance

21. 21 Medical Imaging, SS-2010 Mohammad Dawood

22. 22 Medical Imaging, SS-2010 Mohammad Dawood Reconstruction Iterative Reconstruction EM (Expectation Maximization)

23. 23 Medical Imaging, SS-2010 Mohammad Dawood Reconstruction Iterative Reconstruction OSEM (Ordered Subset Expectation Maximization)