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Image Reconstruction

Image Reconstruction. Atam P Dhawan. y. b. Radiating Object f( a,b,g ). Image g(x,y,z). Image Formation System h. g. z. Image Domain. Object Domain. x. a. Image Formation. b. y. Radiating Object. Image. Image Formation System h. Selected Cross-Section. g. z.

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Image Reconstruction

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  1. Image Reconstruction Atam P Dhawan

  2. y b Radiating Object f(a,b,g) Image g(x,y,z) Image Formation System h g z Image Domain Object Domain x a Image Formation

  3. b y Radiating Object Image Image Formation System h Selected Cross-Section g z Image Domain Object Domain x a Radiation Source Reconstructed Cross-Sectional Image Image Formation: External Source

  4. b y Image Radiating Object Image Formation System h Selected Cross-Section g z Image Domain Object Domain x a Reconstructed Cross-Sectional Image Image Formation: Internal Source

  5. Fourier Transform

  6. y q p f(x,y) q x p P(p,q) q Line integral projection P(p,q) of the two-dimensional Radon transform. Radon Transform

  7. Projection p1 A Reconstruction Space B Projection p3 Projection p2 Radon Transform

  8. Fourier Slice Theorem • X-y coordinate system rotated to p-q u = w cos q v= w sin q

  9. v Sqk(w) Sq2(w) F(u,v) Sq1(w) qk q2 q1 u Fourier Slice Theorem…

  10. Inverse Radon Transform

  11. Filtered Backprojection The integration over the spatial frequency variable w should be carried out from But in practice, the projections are considered to be bandlimited. This means that any spectral energy beyond a spatial frequency, say W, must be ignored. can be computed as in the spatial domain and is bandlimited. is the Fourier transform of the filter kernel function

  12. in the spatial domain and is bandlimited. is the Fourier transform of the filter kernel function

  13. H(w) 1/2t -1/2t 1/2t w

  14. If the projections are sampled with a time interval of t, the projections can be represented as Using the Sampling theorem and the bandlimited constraint, all spatial frequency components beyond W are ignored such that For the bandlimited projections with a sampling interval of t

  15. h(t) hR-L(p) t HHamming(p) H(w) w -1/2t 1/2t Filter Function

  16. The Final Algorithm: FBP

  17. f1 f2 f3 Overlapping area for defining wi,j Ray with ray sum pi fN Iterative ART

  18. PET ML Image Reconstruction

  19. ML-EM Algorithm

  20. Multi-Grid EM Algorithm

  21. MGEM Reconstruction

  22. Reconstruction in MRI Fourier Transform Reconstruction Method

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