1. Extrapolation and iteration for the problem of LFOV Dr. Shuangren Zhao Research Associate Radiation Physics Department Princess Margaret Hospital

2. What is LFOV and ROI LFOV is Limited field of view ROI is region of interest Crop is the image inside the ROI Crop outside ROI is the image outside the ROI Projections we study are truncated

3. The Influence of truncated projections

4. Phantoms 1. Shepp-Logan head phantom 2. Body phantom 3. Modified Shepp-Logan head phantom 4. Strong Modified Shepp-Logan head phantom 5. further Modified Shepp-Logan head phantom 6. crops for the ROI

5. Truncated projections and their direct reconstruction

6. Extrapolations Zero extrapolation *0 (a) Constant extrapolation *c (b) Linear extrapolation *(bx+c) Exponential extrapolation *exp(-x/αL) Exponential extrapolation *exp(-(x/αL)^2) (c) Cos extrapolation *cos(x) Quadratic extrapolation *(ax^2+bx+c) (d) Mixed extrapolation *(ax^2+bx+c)exp(-x/α) (e) Mixed extrapolation *(ax^2+bx+c)exp(-(x/α)^2) (f) Original projection without extrapolation (g)

7. Extrapolations for phantom 3

8. Extrapolations for phantom 3 and 4

9. Quadratic extrapolation (ax^2+bx+c) (d) Projection should positive:

10. Update from quadratic extrapolation to mixed extrapolation {exp(-x/aL)(ax^2+bx+c)}

11. Different fits for the boundary values: 1. The values of projections 2. The differential values of the projections

12. Update for fitting boundary values

13. Update for the mixed extrapolation of (ax 2 +bx+c)exp(-x/αL)

15. The distances of reconstructed images to the image of phantom ideal distance: reconstruction with non-truncated projections.

16. Reconstructions with different extrapolations using phantom 1

17. Reconstructions with different extrapolations using phantom 2

18. Reconstructions with different extrapolations using phantom 3

19. Reconstructions with different extrapolations using phantom 4

20. Reconstructions with different extrapolations using phantom 5

21. Iterative reconstruction algorithm:

22. Projections filter (for phantom 2)

23. Iterative reconstruction results for the phantom 1 with exp(-(x/αL)^2) extrapolation α=0.5

24. Iterative reconstruction results for the phantom 2 with exp(-(x/αL)^2) extrapolation α =0.5

25. Iterative reconstruction results for the phantom 3 with exp(-(x/αL)^2) extrapolation α =0.5

26. Iteration results for phantom 5 with exp(-(x/αL)^2)(ax^2+bx+c) and exp(-(x/αL)^2) extrapolation α=0.5

27. Further find the optimal parameters for for phantom 5

28. The stability of the parameters

29. Further find the optimal parameters for for phantom 4

30. Further find the optimal parameters for for phantom 5

31. The stability of the parameters

32. Find the optimal parameters for for phantom 4

33. Reconstruction with …menthod Errors of iterative reconstruction without truncation Phantom 5 Iterative reconstruction without truncation Crop of Phantom iterative Reconstruction with truncation Reconstruction without truncation Errors reconstruction without truncation Errors of iterative reconstruction with truncation Number of Projections=180 Distance=0.0253Distance=0.0221 Distance=0.0348Distance=0

34. Reconstruction with …menthod Phantom 5 Crop of Phantom Reconstruction without truncation Iterative reconstruction without truncation Errors reconstruction without truncation Errors of iterative reconstruction without truncation iterative Reconstruction with truncation Errors of iterative reconstruction with truncation Projections:360, 1st=mix 2, 2ed=exp 2, α1=0.65, α2=0.068,k=-1.04 Distance=0Distance=0.0167 Distance=0.0145Distance=0.0191

35. Contradiction Our shield (extrapolation) is the best shield, it can resist all spears in the world. Our spear (iteration) is the best spear, it can destroy all shields in the world. Which one would you like to buy? The extrapolation or the iteration?