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- 1. Deblurring & Applications inComputed Tomography Kriti Sen Sharma Graduate Research Assistant
- 2. Outline 1st Half (15min)Deblurring Basics 2nd Half (15min)Deblurring in CT
- 3. Blurring Imaging DefectsWhat if we cannot improve imaging process anymore!!!
- 4. SolutionInvert the imaging defects MATHEMATICALLY = DEBLURRING (Deconvolution)
- 5. ExamplesAcquired Image After Deconvolution Real life photography
- 6. ExamplesAcquired Image After Deconvolution Astronomical Imaging
- 7. ExamplesAcquired Image After Deconvolution Microscopic Imaging
- 8. Mathematical Model-1 Imaging Defects Ax b b = Ax + n xd = A-1 b
- 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. Mathematical Model-2 PSF: Point Spread Function
- 11. Deblurring Example-1 Noise = 10-10
- 12. Deblurring Example-2 ill-posedness of the Inverse problem Noise = 10-5
- 13. Deblurring Example-3 Noise unknown
- 14. Solutions-1 Imaging Defects A x b b = Ax + nRecap xd = A-1 b
- 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. Solutions-1 Using k = 53 i.e. 53 major singular values used
- 17. Visible now?
- 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. 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. End of Deblurring Basics!!Now to discuss some real applications of Deblurring in CT
- 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. 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. 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. Solution to 2nd Problem Assume SNR(σ) = k Find k by phantom studies
- 25. ENR• Edge to Noise Ration• in terms of I-divergence (Information Theoretic approach)• Noise effect• Edge effect• ENR = Edge effect / Noise effect
- 26. ENR Maximization Principle maximize ENR(σ,k) to get optimal σ
- 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. Wiener Filter• Two problems to be solved: – 1. Point Spread Function (PSF) = ? – 2. Signal to Noise Ratio (SNR) = ?
- 29. Solution to 2nd Problem• Phantom designed• Scanned• Reconstructed• Deblurred at various SNR• Optimum SNR value chosen
- 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. Selective Deblurring Axial MIPs (Maximum Intensity Projection) ofthe original, deconvolved, and restored images of the phantom.
- 32. MicroCT Reference
- 33. Thanks!Questions?
- 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. 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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