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Computed Tomography

Computed Tomography. Slide made by 呂英弘 , 張 易 冰 指導教授 : 傅 楸 善 E-mail: b04902086@ntu.edu.tw. Outline. Introduction (Filtered Back Projection) FBP Laminography Algebraic Reconstruction Technique (ART) Variations on ART The Factors Affecting the Algebraic Reconstruction Technique

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Computed Tomography

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  1. Computed Tomography Slide madeby 呂英弘, 張易冰 指導教授:傅楸善 E-mail: b04902086@ntu.edu.tw

  2. Outline • Introduction • (Filtered Back Projection) FBP • Laminography • Algebraic Reconstruction Technique (ART) • Variations on ART • The Factors Affecting the Algebraic Reconstruction Technique • Simultaneous Iterative Reconstruction Technique (SIRT) • Simultaneous Algebraic Reconstruction Technique (SART) • Comparison • References

  3. Outline • Introduction • (Filtered Back Projection) FBP • Laminography • Algebraic Reconstruction Technique (ART) • Variations on ART • The Factors Affecting the Algebraic Reconstruction Technique • Simultaneous Iterative Reconstruction Technique (SIRT) • Simultaneous Algebraic Reconstruction Technique (SART) • Comparison • References

  4. 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.

  5. Outline • Introduction • (Filtered Back Projection) FBP • Laminography • Algebraic Reconstruction Technique (ART) • Variations on ART • The Factors Affecting the Algebraic Reconstruction Technique • Simultaneous Iterative Reconstruction Technique (SIRT) • Simultaneous Algebraic Reconstruction Technique (SART) • Comparison • References

  6. (Filtered Back Projection) FBP

  7. FBP • Original image => 2D Fourier Transform => Filter => Inverse Fourier Transform => Result • Run the projections back through the image (hence the name ``back projection'') to obtain a rough approximation to the original. • Blurring (star-like artifacts) that occur in other parts of the reconstructed image. • The optimal way to eliminate these patterns in the noiseless case is through a ramp filter.

  8. FBP • Problem

  9. FBP

  10. FBP

  11. FBP • The combination of back projection and ramp filtering is known as filtered back projection.(to eliminate the low frequency component around 0 Hz in frequency domain after Fast Fourier Transform)

  12. Outline • Introduction • (Filtered Back Projection) FBP • Laminography • Algebraic Reconstruction Technique (ART) • Variations on ART • The Factors Affecting the Algebraic Reconstruction Technique • Simultaneous Iterative Reconstruction Technique (SIRT) • Simultaneous Algebraic Reconstruction Technique (SART) • Comparison • References

  13. Laminography Translational LaminographyRotational Laminography

  14. Laminography

  15. Outline • Introduction • (Filtered Back Projection) FBP • Laminography • Algebraic Reconstruction Technique (ART) • Variations on ART • The Factors Affecting the Algebraic Reconstruction Technique • Simultaneous Iterative Reconstruction Technique (SIRT) • Simultaneous Algebraic Reconstruction Technique (SART) • Comparison • References

  16. Algebraic Reconstruction Technique (ART)

  17. Algebraic Reconstruction Technique (ART)

  18. ART : Total number of rays : weighted coefficient

  19. ART

  20. ART • Additive ART process: 7 11 9 13 12 8

  21. ART 0 0

  22. ART 10 10

  23. ART 10 10

  24. ART • Final:

  25. ART : Total number of rays : weighted coefficient

  26. ART

  27. ART • Each equation can be regarded as a hyperplane of N-dimensional space, if unique solution exists, these Mhyperplanesmust 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 = 65536 and the weighted coefficient matrix size = 65536*65536 • MN, M < N, M > N • Error and noise make the matrix contradiction or no solution

  28. ART

  29. ART M>N

  30. ART • 1937 S.KaczmarzKaczmarz method • i: iteration times

  31. ART • In the reconstruction process, it is easy to generate salt-and-pepper noise. • : Relaxation factor (in order to improve the image's quality)

  32. Outline • Introduction • (Filtered Back Projection) FBP • Laminography • Algebraic Reconstruction Technique (ART) • Variations on ART • The Factors Affecting the Algebraic Reconstruction Technique • Simultaneous Iterative Reconstruction Technique (SIRT) • Simultaneous Algebraic Reconstruction Technique (SART) • Comparison • References

  33. Variations on ART

  34. Variations on ART

  35. 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.

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

  37. Variations on ART • Vertical rays: • 5.5 2 2

  38. Variations on ART • Horizontal rays: 10 10

  39. Variations on ART • Diagonal rays: 9.8 10.2

  40. Variations on ART • Result:

  41. Variations on ART • Vertical rays: • 4.8 • 6.19 9.63 10.37

  42. Variations on ART • Horizontal rays: 11.49 8.49

  43. Variations on ART • Diagonal rays: 12.82 7.17

  44. Variations on ART • Final:

  45. Outline • Introduction • (Filtered Back Projection) FBP • Laminography • Algebraic Reconstruction Technique (ART) • Variations on ART • The Factors Affecting the Algebraic Reconstruction Technique • Simultaneous Iterative Reconstruction Technique (SIRT) • Simultaneous Algebraic Reconstruction Technique (SART) • Comparison • References

  46. The Factors Affecting the Algebraic Reconstruction Technique • Relaxation factor : • The smaller the relaxation factor, the better the reconstruction quality, and the artifacts caused by noise less likely to appear, but its convergence speed slow. • The larger the relaxation factor, the more blurred the image, but its convergence speed fast. • In some paper, relaxation factor is associated with the number of iterations.

  47. The Factors Affecting the Algebraic Reconstruction Technique

  48. The Factors Affecting the Algebraic Reconstruction Technique If <0, moves further from . If =0, no movement. If 0< <1, moves towards but can not reach . If =1, reach . If 1< <2, moves to the other side of . If >=2, moves to the place further than its mirror in .

  49. The Factors Affecting the Algebraic Reconstruction Technique • Based on the discussion above, it is recommended that if we want to converge at the solution.

  50. The Factors Affecting the Algebraic Reconstruction Technique • Projection order: • The projection order decides the weighted matrix order, also affects the convergence speed. • EX:

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