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High Order Total Variation Minimization For Interior Computerized Tomography

High Order Total Variation Minimization For Interior Computerized Tomography. Jiansheng Yang School of Mathematical Sciences Peking University, P. R. China July 9, 2012. This is a joint work with Prof. Hengyong Yu, Prof. Ming Jiang,

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High Order Total Variation Minimization For Interior Computerized Tomography

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  1. High Order Total Variation Minimization For Interior Computerized Tomography Jiansheng Yang School of Mathematical Sciences Peking University, P. R. China July 9, 2012 This is a joint work with Prof. Hengyong Yu, Prof. Ming Jiang, Prof. Ge Wang

  2. Outline • Background • Computerized Tomography (CT) • Interior Problem • High Order TV (HOT) • TV-based Interior CT (iCT) • HOT Formulation • HOT-based iCT

  3. Physical Principle of CT: Beer’s Law Monochromic X-ray radiation:

  4. Projection Data:Line Integral of Image

  5. CT: Reconstructing Image from Projection Data Measurement Projection data: Sinogram p Image t X-rays Reconstruction

  6. Projection data corresponding to all line which pass through any given point Projection data associated with :

  7. Backprojection Can’t be reconstructed only from projection data associated with

  8. Complete Projection Data and Radon Inversion Formula Radon transform (complete projection data) Radon inversion formula Filtered-Backprojection (FBP)

  9. Incomplete Projection Data and Imaging Region of Interest(ROI) Interior problem Truncated ROI Exterior problem

  10. Truncated ROI F. Noo, R. Clackdoyle and J. D. Pack, “A two-step Hilbert transform method for 2D image reconstruction”, Phys. Med. Biol., 49 (2004), 3903-3923.

  11. Truncated ROI: Backprojected Filtration (BPF) Differentiated Backprojection (DBP) Filtering (Tricomi)

  12. Exterior Problem Ill-posed Uniqueness Non-stability F. Natterer, The mathematics of computerized tomography. Classics in AppliedMathematics 2001, Philadelphia: Society for Industrial and Applied Mathematics.

  13. Interior Problem (IP) An image is compactly supported in a disc : Seek to reconstruct in a region ofinterest (ROI) ROI : only from projection data corresponding to the lines which go through the ROI:

  14. Non-uniqueness of IP Theorem 1 (Non-uniqueness of IP) There exists an image satisfying (1) (2) (3) Both and are solutions of IP. F. Natterer, The mathematics of computerized tomography. Classics in AppliedMathematics 2001, Philadelphia: Society for Industrial and Applied Mathematics.

  15. How to Handle Non-uniqueness of IP Truncated FBP Lambda CT Interior CT (iCT)

  16. Truncated FBP , .

  17. Lambda CT Lambda operator: Sharpened image Inverse Lambda operator: Blurred image Combination of both: More similar to the object image than either is a constant determined by trial and error E. I. Vainberg, I. A. Kazak, and V. P. Kurozaev, Reconstruction of the internal three dimensional structure of objects based on real-time internal projections , Soviet J. Nondestructive testing, 17(1981), 415-423 A. Fardani, E. L. Ritman, and K. T. Smith, Local tomography, SIAM J. Appl. Math., 52(1992), 459-484. A. G. Ramm, A. I. Katsevich, The Radon Transform andLocal Tomography, CRC Press, 1996.

  18. Lambda CT

  19. Interior CT (iCT) • Landmark-based iCT The object image is known in a small sub-region of the ROI • Sparsity-based iCT The object image in the ROI is piecewise constant or polynomial

  20. Candidate Images Any solution of IP satisfies (1) (2) and is calledacandidate image. can be written as where is calledanambiguity image and satisfies (1) (2)

  21. Property of Ambiguity Image Theorem 2 If is an arbitrary ambiguity image, then is analytic, that is, can be written as Y.B. Ye, H.Y. Yu, Y.C. Wei and G. Wang, A general local reconstruction approach based on a truncated Hilbert transform. International Journal of Biomedical Imaging, 2007. 2007: Article ID: 63634. H. Kudo, M. Courdurier, F. Noo and M. Defrise, Tiny a priori knowledge solves the interior problem in computed tomography. Phys. Med. Biol., 2008. 53(9): p. 2207-2231. J. S. Yang, H. Y. Yu, M. Jiang and G. Wang, High Order Total Variation Minimization for Interior Tomography. Inverse Problems 26(3): 1-29, 2010.

  22. Sub-region ROI Landmark-based iCT If a candidate image satisfies we have Therefore, and Method: Analytic Continuation Y.B. Ye, H.Y. Yu, Y.C. Wei and G. Wang, A general local reconstruction approach based on a truncated Hilbert transform. International Journal of Biomedical Imaging, 2007. 2007: Article ID: 63634. H. Kudo, M. Courdurier, F. Noo and M. Defrise, Tiny a priori knowledge solves the interior problem in computed tomography. Phys. Med. Biol., 2008. 53(9): p. 2207-2231.

  23. Further Property of Ambiguity Image Theorem 3 Let be an arbitrary ambiguity image. If then cannot be polynomial unless That is, H. Y. Yu, J. S. Yang, M. Jiang, G. Wang, Supplemental analysis on compressed sensing based interior tomography. Physics In Medicine And Biology, 2009. Vol. 54, No. 18, pp. N425 - N432, 2009. J. S. Yang, H. Y. Yu, M. Jiang and G. Wang, High Order Total Variation Minimization for Interior Tomography. Inverse Problems 26(3): 1-29, 2010.

  24. ROI Piecewise Constant ROI The object image is piecewise constant in ROI can be that is partitioned into finite sub- regions such that

  25. Total Variation (TV) For a smooth function on In general,for any distribution on where W. P. Ziemer, Weakly differential function , Graduate Texts in Mathematics, Springer-Verlag, 1989.

  26. ROI TV of Candidate Images Theorem 4 Assuming that the object is piecewise constantin image the ROI. For any candidate image: we have where is the boundary between neighboringsub-regions and W. M. Han, H. Y. Yu, and G. Wang, A total variation minimization theorem for compressed sensing based tomography. Phys Med Biol.,2009.Article ID: 125871.

  27. TV-based iCT Theorem 5 Assume that the object image ispiecewise constantin the ROI. For any candidate image: if and then That is H. Y. Yu and G. Wang, Compressed sensing based Interior tomography. Phys Med Biol, 2009. 54(9): p. 2791-2805. H. Y. Yu, J. S. Yang, M. Jiang, G. Wang, Supplemental analysis on compressed sensing based interior tomography. Physics In Medicine And Biology, 2009. Vol. 54, No. 18, pp. N425 - N432, 2009.

  28. ROI Piecewise Polynomial ROI The object image is piecewise order polynomial can be that is, in the ROI partitioned into finite subregions such that Where any could be

  29. How to Define High Order TV? For any distribution on order TV if of is trivially defined by where on for a smooth function But for a piecewise smooth function on It is most likely

  30. 1 2 1 Counter Example

  31. High Order TV (HOT) Definition 1 For any distribution on the order TV of is defined by where is an arbitrary partition of is the diameter of and J. S. Yang, H. Y. Yu, M. Jiang and G. Wang, High Order Total Variation Minimization for Interior Tomography. Inverse Problems 26(3): 1-29, 2010.

  32. ROI HOT of Candidate Images Theorem 6 If the object image is polynomial in the ROI. piecewise For any candidate image we have where Poly- is nomial and is the boundary betweensubregions and J. S. Yang, H. Y. Yu, M. Jiang and G. Wang, High Order Total Variation Minimization for Interior SPECT. Inverse Problems 28(1): 1-24, 2012..

  33. HOT-based iCT Theorem 7 Assume that the object image ispiecewise polynomialin the ROI. For any candidate image if and then That is, J. S. Yang, H. Y. Yu, M. Jiang and G. Wang, High Order Total Variation Minimization for Interior Tomography. Inverse Problems 26(3): 1-29, 2010.

  34. Main point

  35. HOT Minimization Method:An unified Approach Theorem 8 Assume that the object image ispiecewise Let be a Linear function space on polynomialin . . If satisfies (Null space) (1) Every is analytic; (2) Any can’t be polynomial unless . Then

  36. HOT-based Interior SPECT J. S. Yang, H. Y. Yu, M. Jiang and G. Wang, High Order Total Variation Minimization for Interior SPECT. Inverse Problems 28(1): 1-24, 2012.

  37. HOT-based Differential Phase-contrast Interior Tomography Wenxiang Cong, Jiangsheng Yang and Ge Wang, Differential Phase-contrast Interior Tomography, Physics in Medicine and Biology 57(10):2905-2914, 2012.

  38. Interior CT (Sheep Lung)

  39. Interior CT (Human Heart) Raw data from GE Medical Systems, 2011

  40. (a) (b) (d) (c) cm cm (f) (e) • Yang JS, Yu HY, Jiang M, Wang G: High order total variation minimization for interior tomography. Inverse Problems 26:1-29, 2010 • Yang JS, Yu HY, Jiang M, Wang G: High order total variation minimization for interior SPECT. Inverse Problems 28(1):1-24, 2012.

  41. Thanks for your attention!

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