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
•
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:
c
 cl0
I1  I 0 e
(Beer's Law )
I 0 , I 1 : radiation pow er
c : absorption density
I0
l
I1
I
l 0 : path length
dI I   cdl (differential form )
l0
f ( x ) : the distribution of absorption density of a cross-section of the object
dI I   f ( x ) dl

x1
x0
f ( x ) dl   ln( I 1 I 0 )
I0 x
0
x
f (x)
I (x)
x1 I 1
Projection Data:Line Integral
of Image
x2
x  t  s


(  sin  , cos  )  

  (cos  , sin  )
t

x1
L ( , t )

L ( , t )  {t  s :   s   }
R f ( , t ) 

L ( ,t )
f ( x ) dl 




f ( t  s ) ds
CT: Reconstructing Image from
Projection Data
Measurement
x2
s


Sinogram
t
p

x1

Rf ( , t )
Image f ( x )
t
X-rays
Reconstruction
Projection data corresponding
to all line which pass through
any given point x
x2
x  t  s


(  sin  , cos  )  


  (cos  , sin  )
t

x
x1
L ( , t ) t  x  
L ( , x   )
Projection data associated with x :
R f ( , x   ),
0  p.
Backprojection
p
c  R f ( , x   ) d  ,

0
f ( x ) Can’t be reconstructed only from
projection data associated with
x.
x
L ( , x   )
Complete Projection Data and
Radon Inversion Formula
Radon transform (complete projection data)
Rf ( , t ) 




f ( t  s ) ds ,
  (cos  , sin  ),    (  sin  , cos  ),
0    p ,   t   .
Radon inversion formula
f ( x) 
2p
 t Rf ( , t )
p
1
2
x  ( x1 , x 2 )
 
0
R
x  t
dtd  .
Filtered-Backprojection (FBP)
Incomplete Projection Data and
Imaging Region of Interest(ROI)
ROI
ROI
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)
H g (s)= H  0 f ( x )  
(a  s  b)

H  f ( x) 
1
p
1

2p

PV
 0 p

b
 t R f ( , x   )d 
0

f ( x  s )
a
ds
s


 (  sin  , co s  )

g ( s )  f ( s 0  t 0 )
Filtering
( b  s )( s  a ) g ( s ) 
s  (a, b)

0

  (cos  , sin  ),
0
1
p
b
1
 g ( s ) ds  p
a
b
PV 
a
su p p g  [ a , b ]
( b  s  )( s   a )H g ( s )d s 
s  s
(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
f0 ( x)
is compactly supported in a disc
 A : | x | A ,
Seek to reconstruct
f0 ( x)
in a region of interest (ROI)
: | x | a
only from projection data
corresponding to the lines
which go through the ROI:
a
R f 0 ( , t ), 0    p ,  a  t  a
ROI
suppf 0
Non-uniqueness of IP
Theorem 1 (Non-uniqueness of IP)
an image

u  C0 (
2
)
There exists
satisfying
(1) S upp u   A ;
(2) Ru ( , t )  0, 0    p ,  a  t  a ;
(3) u ( x )  0, x   a .
Both f 0 and f 0  u 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
Ta f ( x ) 
p
1
2p
2
 
0
|t |  a
f ( x ) : S hepp-Logan
P hantom
 t R f ( , t )
x  t
dtd  , x   a
Ta f ( x )
Lambda CT
Lambda operator:
(  f )( x )  c1 
2p
0
Sharpened image
 s R f (  , x  )d 
2
Inverse Lambda operator:
(
1
f )( x )  c 2 
2p
Rf ( , x  )d 
0
Combination of both:
L f  (1   )  f   
1
f
(  f )( )  |  | fˆ ( )
Blurred image
(
1
f )( )  |  |
1
fˆ ( )
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
f ( x ) : S hepp-Logan
P hantom
(  f )( x )
(
1
f )( x )
L f ( x )  0.15  f ( x )
 0.85( 
1
f )( x )
Interior CT (iCT)


Landmark-based iCT
The object image f 0 ( x ) is known in a
small sub-region of the ROI
Sparsity-based iCT
The object image f 0 ( x ) in the ROI is
piecewise constant or polynomial
Candidate Images
Any solution of IP f ( x ) satisfies
(1) S upp f   A ;
(2) R f ( , t )  R f 0 ( , t ), 0    p ,  a  t  a .
and is called a candidate image. f ( x ) can be written as
f ( x)  f0 ( x)  u ( x)
where u ( x ) is called an ambiguity image and satisfies
(1) S uppu   A ;
u  N ull Space
(2) Ru ( , t )  0, 0    p ,  a  t  a .
Property of Ambiguity Image
Theorem 2
then u |
a
If u ( x )
is an arbitrary ambiguity image,
is analytic, that is,  | can be written as
a
u ( x) 

bn1 , n 2 x1 1 x 2 2 , x   a .
n
n
n1 , n 2
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
If a candidate image f  f 0  u
f | sm all  f 0 | sm all ,
satisfies
suppf 0
u | small  0.
we have
Therefore,
u |R O I  0
and
f |R O I  f 0 |R O I .
Method: Analytic Continuation
Sub-region
sm all
ROI
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
If
then
Let u ( x ) be an arbitrary ambiguity image.
u ( x) 

n1  n 2  n
bn1 , n 2 x1 1 x 2 2 , x   a ,
n
n
u ( x )  0, x   a .
That is, u |
a
cannot be polynomial unless u |  0.
a
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 f 0 ( x ) is
piecewise constant in ROI
 , that is

can be
1
partitioned into finite subregions
 
such that
2
m
5
4
i
3
i 1
f 0 | i  c i , 1  i  m .
ROI 
suppf 0
Total Variation (TV)
For a smooth function f
on 
2
TV( f ) 

2
 f 
 f 

 
 dx1 dx 2 .
  x1 
 x2 


In general, for any distribution f on 
T V ( f )  sup




f div  dx :   C 0 (  ) ,|  | 1 ,
2
where
  (1 ,  2 ) ,
div  
 1
 x1

2
x2
,
|  |
1   2 .
2
2
W. P. Ziemer, Weakly differential function , Graduate Texts in Mathematics, Springer-Verlag, 1989.
TV of Candidate Images
Theorem 4 Assuming that the object
image f 0 ( x ) is piecewise constant in
the ROI. For any candidate image:
f  f 0  u , we have
TV ( f ) 

| ci  c j | |  i , j |
1 i  j  m


(
u
 x1
) (
2
u
x2
5
4
 1,4
2
2
) dx
1
 2 ,3
 1,5
3
where  i , j is the boundary between
neighboring sub-regions  i and  j . suppf 0
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.
ROI 
TV-based iCT
Theorem 5 Assume that the object image f 0 ( x ) is piecewise
constant in the ROI. For any candidate image: h  f 0  u ,
if
T V ( h )  m in T V ( f ),
then
 |  0 and h |  f 0 | .
That is
f 0  arg min T V ( f ).
f  f0  u 
f  f0  u
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 f 0 ( x ) is piecewise n-th order polynomial
in the ROI  ; that is,  can be
partitioned
into finite subregions
m
 
i
1
i 1
such that
f 0 | i ( x ) 
5
4
n

b k1 , k 2 x1 1 x 2 2  Pi ( x ) ,
k
i
k1  k 2  0
k
2
3
1 i  m .
Where any b k
i
1
,k2
could be 0.
ROI 
suppf 0
How to Define High Order TV?
For any distribution f on  , if n-th (n  2) order TV
of f is trivially defined by


n

n 1
H T Vn ( f )  sup   f div n  dx :   ( r ) r  0  C 0 (  ) , |  | 1 in  
 

where
 r
n
n
div n  
 x x
r
1
r 0
nr
2
n
|  |
,

l0
for a smooth function f on  ,


2
,

2
  f

   x l  x n  l  dx1 dx 2 .
l0 
1
2

n
H T Vn ( f ) 
| l |
n
But for a piecewise smooth function f on  , It is most likely
H T V n ( f )   .
Counter Example
f ( x)
  (0, 2)
2
1, x  (0,1]
f ( x)  
 2, x  (1, 2 )
1
TV ( f )  1

O
1

2



H T V 2 ( f )  sup   f  dx :   C 0 (  ), |  | 1 in  


 sup     (1) :   C 0 (  ), |  | 1 in  

 
x
High Order TV (HOT)
Definition 1
TV of
For any distribution f on  , the n-th order
f is defined by
M
H O Tn ( f ) 
lim sup

 m ax diam { Q }   0 k 1
1 k  M
n
Ik ( f )
k
where {Q k } k 1 is an arbitrary partition
M
of  , diam ( Q k ) is the diameter of Q k ,
and
I k ( f )  m in{T V ( f |Q k ), H T V n ( f |Q k )} .
n
Qk

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
Theorem 6 If the object image f 0 ( x ) is
piecewise n-th polynomial in the ROI.
For any candidate image f  f 0  u ,
we have
H O T ( f )    P  P ds
n+ 1
i
j
1 i  j  m 
i, j


  m in 


  n 1 f
   x l  x n 1 l
l0 
1
2
n 1


 f 
 f  
 , 
 
  dx

  x1 
 x2  

2
2
2
5
4
 1,4
2
1
 2 ,3
 1,5
3
ROI 
where f |  Pi (1  i  m ) is n-th Poly- suppf 0
nomial and  i , j is the boundary between subregions  i and  j .
i
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 f 0 ( x ) is piecewise
n-th polynomial in the ROI. For any candidate image h  f 0  u ,
if
H O Tn+1 ( h )  m in H O T n+1 ( f ),
then
u |  0 and h |  f 0 | .
That is,
f 0  arg m in H O Tn+1 ( f ).
f  f0  u 
f  f0  u
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
m in H O T n+ 1 ( f )  H O T n+ 1 ( f 0 ) 
f  f0   


m in 

n 1
n 1


 h 
 h  
 , 
 
   0;

  x1 
 x2  

2
2
u ( x)  0
2
n 1
  h 
   x l  x n 1 l   0
l0 
1
2

n 1
2
2
  u 
   x l  x n 1 l   0;
l0 
1
2

n 1
Pi  P j ds
1 i  j  m 
i, j
  h
   x l  x n 1 l
l0 
1
2
n 1
 
n
u ( x) 

k1  k 2  0
k
c k1 , k 2 x1 1 x 2
k2
HOT Minimization Method:An
unified Approach
Theorem 8 Assume that the object image f 0 ( x ) is piecewise
n-th polynomial in  . Let U be a Linear function space on
(Null space)
 . If U satisfies
(1) Every u  U is analytic;
(2) Any u  U can’t be polynomial unless u  0 .
Then
f 0  arg m in H O Tn+1 ( f ).
f  f0  u ,
u U
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 Phasecontrast 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
800
800
600
600
400
400
200
0
0
50
100
FullRec
IterNum=5
IterNum=10
IterNum=15
IterNum=20
150
200
200
250
300
0
0
50
100
FullRec
IterNum=5
IterNum=10
IterNum=15
IterNum=20
150
200
250
300
(a)
(b)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
Phantom Image
Interior SPECT
0.2
0
-8
-6
-4
-2
0
(e)
2
(d)
(c)
Phantom Image
Interior SPECT
0.2
4
6
8 cm
0
8
-6
-4
-2
0
(f)
2
4
6
8
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.
Thanks for your attention!