High Order Total Variation Minimization For Interior Computerized Tomography

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

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Physical Principle of CT Beers Law
Monochromic X-ray radiation
4
Projection DataLine 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
Cant 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
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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 Applied Mathematics 2001,
Philadelphia Society for Industrial and Applied
Mathematics.
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Interior Problem (IP)
An image
is compactly supported in a disc

Seek to reconstruct
in a region of interest (ROI)

only from projection data corresponding to the
lines which go through the ROI
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Non-uniqueness of IP
Theorem 1
(Non-uniqueness of IP)
There exists
satisfying
an image
(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)

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Truncated FBP
,
.
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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.
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Lambda CT
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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

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Candidate Images
Any solution of IP
satisfies
(1)
(2)
and is called a candidate image.
can be written as
is called an ambiguity image and satisfies
where
(1)
(2)
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Property of Ambiguity Image
Theorem 2
is an arbitrary ambiguity image,
If
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.
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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
be an arbitrary ambiguity image.
Let
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.
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Piecewise Constant ROI
The object image
is
piecewise constant in ROI
can be
that is
partitioned into finite sub-
regions
such that
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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.
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TV of Candidate Images
Assuming that the object
Theorem 4
is piecewise constant in
image
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.
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TV-based iCT
Assume that the object image
is piecewise
Theorem 5
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.
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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
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How to Define High Order TV?
For any distribution
on
order TV
if
of
is trivially defined by
where
for a smooth function
on
But for a piecewise smooth function
on
It is most likely
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Counter Example
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High Order TV (HOT)
Definition 1
For any distribution
on
the
order
TV of
is defined by
where
is an arbitrary partition
is the diameter of
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
HOT of Candidate Images
If the object image
Theorem 6
is
polynomial in the ROI.
piecewise
For any candidate image
we have
where
Poly-
is
is the boundary between subregions
and
nomial 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..
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HOT-based iCT
Assume that the object image
is piecewise
Theorem 7
polynomial in 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.
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Main point
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HOT Minimization MethodAn unified Approach
Assume that the object image
is piecewise
Theorem 8
Let be a Linear function space on
polynomial in .
. If satisfies
(Null space)
(1) Every is analytic
(2) Any cant be polynomial unless
.
Then
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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.
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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.
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Interior CT (Sheep Lung)
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Interior CT (Human Heart)
Raw data from GE Medical Systems, 2011
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Yang JS, Yu HY, Jiang M, Wang G High order total
variation minimization for interior tomography.
Inverse Problems 261-29, 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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Thanks for your attention!