School of Mathematical Sciences Peking University, P. R. China 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

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

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

Projection Data:Line Integral of Image

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

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

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

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

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

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.

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

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

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

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 Applied Mathematics 2001, Philadelphia: Society for Industrial and Applied Mathematics.

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

, Truncated FBP .

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 and Local Tomography, CRC Press, 1996.

Lambda CT

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

Candidate Images Any solution of IP satisfies (1) (2) and is called a candidate image. can be written as where is called an ambiguity image and satisfies (1) (2)

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.

Landmark-based iCT Sub-region If a candidate image satisfies we have ROI 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.

Further Property of Ambiguity Image Theorem 3 Let be an arbitrary ambiguity image. If then That is, cannot be polynomial unless 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.

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

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.

TV of Candidate Images ROI Theorem 4 Assuming that the object image is piecewise constant in the ROI. For any candidate image: we have where is the boundary between neighboring sub-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.

TV-based iCT Theorem 5 Assume that the object image is piecewise constant in 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.

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

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

Counter Example 1 2

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.

HOT of Candidate Images ROI Theorem 6 If the object image is piecewise polynomial in the ROI. For any candidate image we have where is Poly- nomial and is the boundary between subregions 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..

HOT-based iCT Theorem 7 Assume that the object image is piecewise polynomial in the ROI. For any candidate image if then and 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.

Main point

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

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.

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.

Interior CT (Sheep Lung)

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

(a) (b) (c) (d) (e) (f) cm 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.

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