Deblurring in ct

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Presentation about basics of image deblurring, two popular approaches, and two computed tomography applications.

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Deblurring in ct

  1. 1. Deblurring & Applications inComputed Tomography Kriti Sen Sharma Graduate Research Assistant
  2. 2. Outline 1st Half (15min)Deblurring Basics 2nd Half (15min)Deblurring in CT
  3. 3. Blurring Imaging DefectsWhat if we cannot improve imaging process anymore!!!
  4. 4. SolutionInvert the imaging defects MATHEMATICALLY = DEBLURRING (Deconvolution)
  5. 5. ExamplesAcquired Image After Deconvolution Real life photography
  6. 6. ExamplesAcquired Image After Deconvolution Astronomical Imaging
  7. 7. ExamplesAcquired Image After Deconvolution Microscopic Imaging
  8. 8. Mathematical Model-1 Imaging Defects Ax b b = Ax + n xd = A-1 b
  9. 9. Mathematical Model-2 Imaging Defects "(x) p(x) g(x) g(x) = p(x) " #(x) + n(x), x $ R 2 l(x) inverse filter of p(x) "(x) = l(x) # g(x)! ! !
  10. 10. Mathematical Model-2 PSF: Point Spread Function
  11. 11. Deblurring Example-1 Noise = 10-10
  12. 12. Deblurring Example-2 ill-posedness of the Inverse problem Noise = 10-5
  13. 13. Deblurring Example-3 Noise unknown
  14. 14. Solutions-1 Imaging Defects A x b b = Ax + nRecap xd = A-1 b
  15. 15. Solutions-1 Truncated Singular Value Decomposition•  A = U Σ VT = [u1 … uN] diag(s1… sN) [v1 … vN]T•  Truncation Ak* = [v1 … vN] diag(1/s1… 1/sN) [u1 … uN] T•  xk = Ak* b
  16. 16. Solutions-1 Using k = 53 i.e. 53 major singular values used
  17. 17. Visible now?
  18. 18. Solutions-2 Recap Imaging Defects "(x) p(x) g(x) g(x) = p(x) " #(x) + n(x), x $ R 2 l(x) inverse filter of p(x) "(x) = l(x) # g(x)! ! !
  19. 19. Solutions-2 Wiener Filter L( u) : PSD of inverse filter l(x) Sn ( u) : PSD of noise P ( u) : PSD of blurring filter p(x) S" ( u) : PSD of object P(u*) P(u)P(u*) = P ( u) 2 L(u) = 2 S ( u) ! P ( u) + n! "1 P(u*) S" ( u) L(u) = P(u) = 2 P ( u) P(u*) = 2 1 P ( u) + SNR(u) ! !
  20. 20. End of Deblurring Basics!!Now to discuss some real applications of Deblurring in CT
  21. 21. Jing Wang, Ge Wang, Ming Jiang Blind deblurring of spiral CT images Based on ENR and Wiener filterJournal of X-Ray Science and Technology – 2005[previous: IEEE Trans. on Medical Imaging 2003]
  22. 22. Blind Deconvolution P(u*) 1st Problem: L(u) = 2 S ( u) Finding P(u) P ( u) + n S" ( u) PSD of p(x) = ? P(u*) = 2 1 P ( u) + 2nd Problem: SNR(u) Finding SNR(u) SNR at different u = ?!
  23. 23. Solution to 1st ProblemAssume p(x) → Gaussian with σ = ? Deblur at multiple σ Find σ that gives best deblurring How to find best σ: Use ENR
  24. 24. Solution to 2nd Problem Assume SNR(σ) = k Find k by phantom studies
  25. 25. ENR•  Edge to Noise Ration•  in terms of I-divergence (Information Theoretic approach)•  Noise effect•  Edge effect•  ENR = Edge effect / Noise effect
  26. 26. ENR Maximization Principle maximize ENR(σ,k) to get optimal σ
  27. 27. Rollano Hijarrubia et. al.Selective Deblurring for Improved Calcification Visualization and Quantification in Carotid CT Angiography: Validation Using Micro-CT IEEE Transactions on Medical Imaging 2009
  28. 28. Wiener Filter•  Two problems to be solved: – 1. Point Spread Function (PSF) = ? – 2. Signal to Noise Ratio (SNR) = ?
  29. 29. Solution to 2nd Problem•  Phantom designed•  Scanned•  Reconstructed•  Deblurred at various SNR•  Optimum SNR value chosen
  30. 30. Solution to 1st Problem•  By measuring PSF of a bead image1•  Resolution of scanner: 0.3 - 0.4mm Bead size: 0.28mm [1]. Meinel JF, Wang G, Jiang M, et al. Spatial variation of resolution and noise in multi-detector row spiral CT. Acad Radiol. 2003;10:607– 613.
  31. 31. Selective Deblurring Axial MIPs (Maximum Intensity Projection) ofthe original, deconvolved, and restored images of the phantom.
  32. 32. MicroCT Reference
  33. 33. Thanks!Questions?
  34. 34. Summary of J. Wang et.al. Given g(x) Wiener Filter at various σ to get λ(x,σ) Wiener filter SNR parameter chosen from phantom studies ENR(σ) calculated Max ENR(σ): σOPT λ(x,σOPT)
  35. 35. Edge to Noise Ratio u(x)•  I-divergence I(u,v) = # u(x)log " # [ u(x) " v(x)] x v(x) x•  Noise effect ! N(" ,k) = I ( g,G" # $k ( g,k," ))•  Edge effect E(" ,k) = I ( #k ( g,k," ),G" $ #k ( g,k," )) ! E(" ,k)•  ENR ENR(" ,k) = ! N(" ,k) !

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