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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
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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: Beer’s Law
Monochromic X-ray radiation:
0 1, : radiation powerI I: absorption densityc
0 : path lengthl
01 0 (Beer's Law)clI I e
1
01 0( ) ln( )
x
xf x dl I I
0I 1I
0l
c
0I1I( )f x0x 1x
x
( )I x
( )dI I f x dl
(differential form)dI I cdl( ) : the distribution of absorption density of a cross-section of the objectf x
Il
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Projection Data:Line Integral of Image
( , )( , ) ( ) ( )
L tRf t f x dl f t s ds
2x
1x
(cos ,sin )
( , )L t
x t s
( sin ,cos ) t
( , ) { : }L t t s s
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CT: Reconstructing Image from Projection Data
t
Sinogram
X-rays
Proj
ectio
n
data
:
Measurement
Reconstruction
Image ( )f x
1x
2xt
( , )Rf t
s
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Projection data corresponding to all line which pass through any given point x
2x
1x
(cos ,sin )
( , )L t
x t s
( sin ,cos ) t
t x x
( , )L x
Projection data associated with :
x ( , ),Rf x 0 .
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Backprojection
x
( , )L x
0( , ) ,c Rf x d
( )f x Can’t be reconstructed only from
projection data associated with
.x
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Complete Projection Data and Radon Inversion
Formula
( , ) ( ) ,Rf t f t s ds
( sin ,cos ), (cos ,sin ),
2 0
( , )1( ) .
2t Rf t
f x dtdx t
R
Radon transform (complete projection data)
Radon inversion formula
0 , .t
1 2( , )x x x
( , ) ( ) ,Rf t f t s ds
Filtered-Backprojection (FBP)
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Incomplete Projection Data and Imaging Region of Interest(ROI)
ROI
ROI
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.
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Truncated ROI: Backprojected Filtration (BPF)
1 ( )H ( ) PV d
f x sf x s
s
0
0
0
1Hg(s)=H ( ) ( , )d
2 tf x Rf x
( )( )H ( )d1 1( )( ) ( ) ( ) PV
b b
a a
b s s a g s sb s s a g s g s ds
s s
(Tricomi)
Differentiated Backprojection (DBP)
supp [ , ]g a b
( sin ,cos )
Filtering
(cos ,sin ),
( , )s a b
0
0 0( ) ( )g s f s t
a
b
( )a s b 0
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Exterior Problem
UniquenessNon-stability
Ill-posed
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)
0 ( )f x
A
0 ( )f x
An image
is compactly supported in a disc
Seek to reconstruct in a region of interest (ROI)
a
only from projection data corresponding to the lines which go through the ROI:
0 ( , ), 0 , Rf t a t a
: | ,|x A
| |x aROI
0suppf
:
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Non-uniqueness of IP
Supp ;Au
20 ( )u C
( , ) 0, 0 , ;Ru t a t a
Theorem 1
satisfying
(1)
(2)
(3) ( ) 0, .au x x
F. Natterer, The mathematics of computerized tomography. Classics in Applied Mathematics 2001, Philadelphia: Society for Industrial and Applied Mathematics.
(Non-uniqueness of IP)
an image
There exists
Both
0f and 0f u are solutions of IP.
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How to Handle Non-uniqueness of IP
Truncated FBPLambda CT Interior CT (iCT)
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Truncated FBP
2 0 | |
1 ( , )( ) ,
2t
a at a
Rf tT f x dtd x
x t
,
.
( ) : Shepp-Logan
Phantom
f x ( )aT f x
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Lambda CT
2 21 0
( )( ) ( , )sf x c Rf x d
ˆ( )( ) | | ( )f f
212 0
( )( ) ( , )f x c Rf x d
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.
1 1 ˆ( )( ) | | ( )f f
1(1 )Lf f f
Lambda operator: Sharpened image
Inverse Lambda operator: Blurred image
Combination of both:
More similar to the object image than eitheris a constant determined by trial and
error
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Lambda CT
( ) : Shepp-Logan
Phantom
f x ( )( )f x 1( )( )f x1
( ) 0.15 ( )
0.85( )( )
Lf x f x
f x
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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
0 ( )f x
0 ( )f x
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Candidate Images
( )f x
Supp ;Af
0( , ) ( , ), 0 , .Rf t Rf t a t a
Any solution of IP
satisfies
(1)
(2)
and is called a candidate image.
( )f x
0( ) ( ) ( )f x f x u x
( )u x
Suppu ;A
( , ) 0, 0 , .Ru t a t a
can be written as
is called an ambiguity image and satisfies
(1)
(2)
where
Null Spaceu
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Property of Ambiguity Image
( )u x
|a
u
1 2
1 2
1 2
, 1 2,
( ) , .n nn n a
n n
u x b x x x
Theorem 2
is an arbitrary ambiguity image,
If
then
is analytic, that is,
|a
can be written as
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.
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.
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Landmark-based iCT
Sub-regionsmall ROI
0suppfIf a candidate image
0f f u satisfies
0| ,|small small
f f
we have | 0.small
u Therefore,
ROI| 0u
and ROI 0 ROI .| |f f
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.
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Further Property of Ambiguity Image
( )u xTheorem 3
be an arbitrary ambiguity image.
Let
|a
u | 0.a
u cannot be polynomial unlessH. 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.
If1 2
1 2
1 2
, 1 2( ) , ,n nn n a
n n n
u x b x x x
then
( ) 0, .au x x
That is,
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Piecewise Constant ROI
4 5
1
23
ROI0suppf
The object image
0 ( )f x
piecewise constant in ROI
is
, that is
can be
partitioned into finite sub-regions
1
m
ii
such that
0 | , 1 .i if c i m
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Total Variation (TV)
2 2
1 21 2
TV( ) .f f
f dx dxx x
For a smooth function
20TV( ) sup div : ( ) ,| | 1 ,f f dx C
1 2( , ) , 2 21 2| | .
W. P. Ziemer, Weakly differential function , Graduate Texts in Mathematics, Springer-Verlag, 1989.
In general, for any distribution
where
f on
f on
1 2
1 2
div ,x x
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,1
2 2
1 2
TV( ) | | | |
( ) ( )
i ji ji j m
f c c
u udx
x x
Assuming that the object
0 ( )f x4 5
1
23
ROI0suppf
1,4
2,3
1,5
is piecewise constant in the ROI. For any candidate
image:0 ,f f u we have
where ,i j is the boundary betweenneighboring sub-
regionsi and
.jW. 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.
Theorem 4image
TV of Candidate Images
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TV-based iCT
Assume that the object image
0 ( )f x is piecewiseconstant in the
ROI.For any candidate image:
0 ,h f u
if
Theorem 5
0
TV( ) min TV( ),f f u
h f
then
| 0 and 0| | .h f
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.
H. Y. Yu and G. Wang, Compressed sensing based Interior tomography. Phys Med Biol, 2009. 54(9): p.
2791-2805.
0
0 arg min TV( ).f f u
f f
That is
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Piecewise Polynomial ROI
4 5
1
23
ROI0suppf
The object image
0 ( )f x is piecewise; that
is,can
bepartitioned into finite subregions
1
m
ii
such that 1 2
1 2
1 2
0 , 1 20
| ( ) ( ) ,
1 .
i
nk ki
k k ik k
f x b x x P x
i m
in the ROI
n-th order polynomial
Where any
could be
1 2,ik kb 0.
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How to Define High Order TV? ,For any
distribution f on if n-th (n 2) order TV
of f is trivially defined by
where
1n n 0 0HTV ( ) sup div : ( ) ( ) , | | 1 in n n
r rf f dx C
2
0
| | | | ,n
ll
n0 1 2
div ,nn
rr n r
r x x
for a smooth function
f on ,2
n 1 20 1 2
HTV ( ) .nn
l n ll
ff dx dx
x x
But for a piecewise smooth function
f on
,nHTV ( ) .f
It is most likely
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Counter Example
(0,2) 1, (0,1]
( )2, (1,2)
xf x
x
TV( ) 1f
2 0
0
HTV ( ) sup : ( ), | | 1 in
sup (1) : ( ), | | 1 in
f f dx C
C
O
x
( )f x
1
2
2
1
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High Order TV (HOT)
,For any distribution
Definition 1
f on the n-th orderTV of f is defined
by
1
n1max { } 0
HOT ( ) limsup ( )k
k M
Mnk
kdiam Q
f I f
n( ) min{TV( | ),HTV ( | )} .k k
nk Q QI f f f
1{ }Mk kQ
( )kdiam Q ,
where
is an arbitrary partition ,kQis the diameter of and
ofkQ
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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HOT of Candidate Images
,
n+11
2 2 211
10 1 2 1 2
HOT ( )
min ,
i j
i ji j m
nn
l n ll
f P P ds
f f fdx
x x x x
If the object image
0 ( )f x is
4 5
1
23
ROI0suppf
1,4
2,3
1,5
polynomial in the ROI. For any candidate
image0 ,f f u
we have
where
,i j is the boundary between subregions
i and
.j
Theorem 6
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..
| (1 )i if P i m i
s nomial and
n-th
n-th Poly-
piecewise
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HOT-based iCTAssume that the object image
0 ( )f x is piecewisepolynomial in the
ROI.For any candidate image
0 ,h f u
if
Theorem 7
0n+1 n+1HOT ( ) min HOT ( ),
f f uh f
then | 0u and 0| | .h f
n-th
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.
That is,0
0 n+1arg min HOT ( ).f f u
f f
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Main point
0,
n+1 n+1 01
min HOT ( ) HOT ( )i j
i jf fi j m
f f P P ds
2 2 211
10 1 2 1 2
min , 0;nn
l n ll
h h h
x x x x
211
10 1 2
0nn
l n ll
h
x x
211
10 1 2
0;nn
l n ll
u
x x
1 2
1 2
1 2
, 1 20
( )n
k kk k
k k
u x c x x
( ) 0u x
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HOT Minimization Method:An unified
ApproachAssume that the object image
0 ( )f x is piecewisepolynomial in .
Theorem 8n-th
Then
0
0 n+1,
arg min HOT ( ).f f u
u U
f f
Let be a Linear function
space on U
. If satisfies U
(1) Every is analytic;
u U(2) Any can’t be polynomial unless . u U 0u
(Null space)
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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)
0 50 100 150 200 250 3000
200
400
600
800
FullRecIterNum=5IterNum=10IterNum=15IterNum=20
0 50 100 150 200 250 3000
200
400
600
800
FullRecIterNum=5IterNum=10IterNum=15IterNum=20
Raw data from GE Medical Systems, 2011
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(a) (b) (c) (d)
(e) (f)8 -6 -4 -2 0 2 4 6 8
0
0.2
0.4
0.6
0.8
1
Phantom ImageInterior SPECT
cm-8 -6 -2-4 0 2 4 6 80
0.2
0.4
0.6
0.8
1
Phantom ImageInterior SPECT
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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Thanks for your attention!