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Comprehensive Comparison of Tomographic Reconstruction Techniques

This study delves into various methods like FBP, Laminography, ART, SART, and SIRT for tomographic imaging, assessing their merits and demerits. Emphasis on ART, SART, and SIRT comparison.

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Comprehensive Comparison of Tomographic Reconstruction Techniques

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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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