1. 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

2. 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

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
X-rays t
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),

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

13. 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:

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 Applied Mathematics 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),
A. Fardani, E. L. Ritman, and K. T. Smith, Local tomography, SIAM J. Appl. Math., 52(1992),
A. G. Ramm, A. I. Katsevich, The Radon Transform and Local 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 called a candidate image.
can be written as
where
is called an ambiguity 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, : Article ID:
H. Kudo, M. Courdurier, F. Noo and M. Defrise, Tiny a priori knowledge solves the interior problem in computed tomography. Phys. Med. Biol., (9): p
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,

22. 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, : Article ID:
H. Kudo, M. Courdurier, F. Noo and M. Defrise, Tiny a priori knowledge solves the interior problem in computed tomography. Phys. Med. Biol., (9): p

23. 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, 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,

24. Piecewise Constant ROI
The object image is
piecewise constant in ROI
that is
can be
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. 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:

27. 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, (9): p
H. Y. Yu, J. S. Yang, M. Jiang, G. Wang, Supplemental analysis on compressed sensing based interior tomography. Physics In Medicine And Biology, Vol. 54, No. 18, pp. N425 - N432, 2009.

28. 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

29. 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

30. Counter Example 1 2

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,

32. 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,

33. 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,

34. Main point

35. 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

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): , 2012.

38. Interior CT (Sheep Lung)

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

40. (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.

41. Thanks for your attention!