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:

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

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

CT: Reconstructing Image from Projection Data

t

Sinogram

X-rays

Proj

ectio

n

data

:

Measurement

Reconstruction

Image ( )f x

1x

2xt

( , )Rf t

s

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 .

Backprojection

x

( , )L x

0( , ) ,c Rf x d

( )f x Can’t be reconstructed only from

projection data associated with

.x

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)

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)

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

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.

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

:

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.

How to Handle Non-uniqueness of IP

Truncated FBPLambda CT Interior CT (iCT)

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

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

Lambda CT

( ) : Shepp-Logan

Phantom

f x ( )( )f x 1( )( )f x1

( ) 0.15 ( )

0.85( )( )

Lf x f x

f x

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

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

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.

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.

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,

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

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

,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

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

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.

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

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

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.

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

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

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

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)

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)

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

(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.

Thanks for your attention!