1. Computed Tomography

2. Introduction
Tomography is a method to reconstruct the cross-section of an object.
FBP, Laminograpgy, ART, SART, and SIRT are the most popular ways in tomography with different principles.
We will also try to find out the advantages and disadvantages of ART, SART, and SIRT.

3. Outline Introduction Laminography
Algebraic Reconstruction Technique (ART)
Variations on ART
Simultaneous Iterative Reconstruction Technique (SIRT)

4. Laminography Translational Laminography Rotational Laminography
Focal slice 焦平面

5. Laminography

6. Outline Introduction Laminography
Algebraic Reconstruction Technique (ART)
Variations on ART
Simultaneous Iterative Reconstruction Technique (SIRT)

7. Algebraic Reconstruction Technique (ART)

8. ART

9. ART
Additive ART process: 12 8

10. ART 5 7 6 2

11. ART 5 7 6 2 10

12. ART 5 7 6 2

13. ART
Final: 5 7 6 2

14. 𝑏 𝑗 𝑥,𝑦 = 1, in 𝑗th pixel 0, otherwise
ART
To reconstruct a 2D continuous function 𝑓(𝑥,𝑦), we define an image basic function:
𝑏 𝑗 (𝑥,𝑦)
𝑏 𝑗 𝑥,𝑦 = 1, in 𝑗th pixel 0, otherwise

15. ℜ 𝑖 𝑓 𝑥,𝑦 ≈ 𝑗=1 𝑁 𝑓 𝑗 ℜ 𝑖 𝑏 𝑗 𝑥,𝑦 , 𝑖=1,2,…,𝑀
ART
Let 𝑓 𝑥,𝑦 be discretized by N = n*n, and 𝑓 𝑗 is the mean of 𝑗th grid:
𝑓 𝑥,𝑦 ≈ 𝑗=1 𝑁 𝑓 𝑗 ∙ 𝑏 𝑗 (𝑥,𝑦)
After Radon transform, 𝑓 𝑥,𝑦 can be represented as:
ℜ 𝑖 𝑓 𝑥,𝑦 ≈ 𝑗=1 𝑁 𝑓 𝑗 ℜ 𝑖 𝑏 𝑗 𝑥,𝑦 , 𝑖=1,2,…,𝑀
𝑀: Total number of rays
ℜ: An operator of Radon transform
ℜ 𝑖 : the Radon transform of 𝑖th ray
將f(x,y,z)經過N=n*n*n 離散化
Fi 為每個欲重建voxel的衰減率
R是拉登轉換運算子 Rj是第j條射線的拉登轉換

16. ℜ 𝑖 𝑓 𝑥,𝑦 ≈ 𝑗=1 𝑁 𝑓 𝑗 ℜ 𝑖 𝑏 𝑖 𝑥,𝑦 , 𝑖=1,2,…,𝑀
ART
Let 𝑤 𝑖𝑗 = ℜ 𝑖 𝑏 𝑗 𝑥,𝑦 , and 𝑝 𝑖 is projection value of 𝑖th ray.
ℜ 𝑖 𝑓 𝑥,𝑦 ≈ 𝑗=1 𝑁 𝑓 𝑗 ℜ 𝑖 𝑏 𝑖 𝑥,𝑦 , 𝑖=1,2,…,𝑀
𝑗=1 𝑁 𝑤 𝑖𝑗 𝑓 𝑗 = 𝑝 𝑖 , 𝑖=1,2,…,𝑀

17. ART 𝑗=1 𝑁 𝑤 𝑖𝑗 𝑓 𝑗 = 𝑝 𝑖 , 𝑖=1,2,…,𝑀 𝑀: Total number of rays
𝑗=1 𝑁 𝑤 𝑖𝑗 𝑓 𝑗 = 𝑝 𝑖 , 𝑖=1,2,…,𝑀
𝑀: Total number of rays
𝑤 𝑖𝑗 : weighted coefficient

18. ART

19. ART
Each equation can be regarded as a hyperplane of N-dimensional space, if unique solution exists, these M hyperplanes must intersect at one point.
If M and N are small enough, we can use inverse matrix to find the solution, however, there are several factors that make these difficult to achieve:
To reconstruct a 256*256 pixel image, N = and the weighted coefficient matrix size = 65536*65536
M ≠ N, M < N, M > N
Error and noise make the matrix contradiction or no solution
矩陣大部分為0 為一個龐大的稀疏矩陣 耗時
M不等n使 無法使用逆矩陣去解
M<N無限解
M>N無解

20. ART

21. ART
M>N

22. Outline Introduction Laminography
Algebraic Reconstruction Technique (ART)
Variations on ART
Simultaneous Iterative Reconstruction Technique (SIRT)

23. Variations on ART

24. Variations on ART

25. Variations on ART Multiplicative ART:
In MART (Multiplicative ART), each reconstructed element is changed in proportion to its magnitude.
This is in sharp contrast to additive ART, where each element in the ray is changed a fixed amount, independent of its magnitude.

26. Variations on ART
Multiplicative ART process: 7 11 9 13 12 8

27. Variations on ART Vertical rays: 𝑓 1 1 = 𝑓 3 1 = 11 2 ∗1= 5.5
𝑓 1 1 = 𝑓 3 1 = 11 2 ∗1= 5.5
𝑓 2 1 = 𝑓 4 1 = ∗1=4.5 5 7 6 2 2 2 1

28. Variations on ART Horizontal rays:
𝑓 1 2 = ∗5.5= 𝑓 2 2 = ∗4.5=5.4
𝑓 3 2 = 8 10 ∗5.5= 𝑓 4 2 = 8 10 ∗4.5=3.6 5 7 6 2
5.5
4.5 10 10

29. Variations on ART Diagonal rays:
𝑓 1 3 = ∗6.6= 𝑓 2 3 = ∗5.4=7.16
𝑓 3 2 = ∗4.4= 𝑓 4 3 = ∗3.6=2.47 5 7 6 2
10.2
9.8
6.6
5.4
4.4
3.6

30. Variations on ART
Result:
4.53
7.16
5.84
2.47

31. Variations on ART Vertical rays:5 7 6 2
Vertical rays:
𝑓 1 1 = ∗4.53= 𝑓 2 1 = ∗7.16=6.69
𝑓 3 1 = ∗5.84= 𝑓 4 1 = ∗2.47=2.30
10.37
9.63
4.53
7.16
5.84
2.47

32. Variations on ART Horizontal rays:5 7 6 2
Horizontal rays:
𝑓 1 2 = ∗4.8= 𝑓 2 2 = ∗6.69=6.99
𝑓 3 2 = ∗6.19= 𝑓 4 2 = ∗2.3=2.16
11.49
4.8
6.69
6.19
2.3
8.49

33. Variations on ART Diagonal rays:5 7 6 2
Diagonal rays:
𝑓 1 3 = ∗5.01= 𝑓 2 3 = ∗6.99=7.09
𝑓 3 2 = ∗5.83= 𝑓 4 3 = ∗2.16=2.10
7.17
12.82
5.01
6.99
5.83
2.16

34. Variations on ART 5 7 6 2
Final:
4.89
7.09
5.91
2.10

35. Outline Introduction Laminography
Algebraic Reconstruction Technique (ART)
Variations on ART
Simultaneous Iterative Reconstruction Technique (SIRT)

36. Simultaneous Iterative Reconstruction Technique (SIRT)
Iterative algorithm:
: total number of elements in a projection
: iteration
: a projection that passes
: an element along the th projection
: the sum of measured data for a projection
: total number of projections that pass

37. SIRT 12 8

38. SIRT 5 7 6 2

39. SIRT 5 7 6 2
10.67
9.33 5
5.67
5.33 4

40. ART DEMO

43. 你只要了解：
ART的步驟
ART與SIRT的差別