Image Reconstruction

Atam P Dhawan

Image Formation

z

Image Formation System

hObject Domain

Image Domain

x

y

Radiating Object f() Image g(x,y,z)

dddfzyxhzyxg ),,(),,,,,(),,(

dddfzyxhzyxg ),,(),,(),,(

Image Formation: External Source

Reconstructed Cross-Sectional Image

Radiation Source

z

Image Formation

Systemh

Object Domain

Image Domain

Selected Cross-Section

x

yRadiating Object

Image

Image Formation: Internal Source

z

Reconstructed Cross-Sectional Image

Image Formation

Systemh

Object Domain

Image Domain

Selected Cross-Section

x

y

Radiating Object

Image

Fourier Transform

dydxeyxgyxgFTvuG vyuxj ,),()},({),( )(2

Radon Transform

x

y

q

p

p

f(x,y)

P(p,)

Line integral projection P(p,) of the two-dimensional Radon transform.

Radon Transform

Projection p1

Projection p2

Projection p3

Reconstruction Space

A

B

dqqpqpfpJyxfR )cossin,sincos()()},({

Fourier Slice Theorem

• X-y coordinate system rotated to p-q

cossin

sincos

yxq

yxp

cossin

sincos

qpy

qpx

dpdqeqpqpfpJFyxfRF pj

2)cossin,sincos()}({)}},({{

dpepJS pj

2)()(

),(),()( )sincos(2 FdxdyeyxfS yxj

u = cos v= sin

Fourier Slice Theorem…

u

v

F(u,v)

Sk() S2()

S1()

Inverse Radon Transform

2 ( cos sin )

0

( , ) ( , ) j x yf r F e d d

),()},({),(ˆ )(21 dudvevuFvuFFyxf vyxuj

deSpJ

dpJ

ddeSrf

yxj

yxj

)sincos( 2*

0

*

0

)sincos( 2

)( )(

where

)(

)( ),(ˆ

Filtered Backprojection

)(

)(

)(

)( )(

1

1

2

)sincos( 2*

pJF

SF

deS

deSpJ

pj

yxj

to

),( yxf

The integration over the spatial frequency variable should be carried out from

But in practice, the projections are considered to be bandlimited.

This means that any spectral energy beyond a spatial frequency, say must be ignored. can be computed as

)()( 1

),(0

pphpJpddyxf

otherwise 0

if LRH

LRH )( ph LR is the Fourier transform of the filter kernel function in the spatial domain and is bandlimited.

)()( 1

),(0

pphpJpddyxf

otherwise 0

if LRH

LRH )( ph LR is the Fourier transform of the filter kernel function in the spatial domain and is bandlimited.

)( )( BH

otherwise 0

if 1)(

B

deHph pj2)()(

H()

1/2-1/2

1/2

If the projections are sampled with a time interval of t, the projections can be represented as )( kJ

Using the Sampling theorem and the bandlimited constraint, all spatial frequency components beyond are ignored such that

2

1

2

22 2/

)2/ ( sin

4

1 -

/

)/ ( sin

2

1)(

p

p

p

pph

For the bandlimited projections with a sampling interval of

Filter Function

H()

1/2-1/2

h(

hR-L(p)

HHamming(p)

The Final Algorithm: FBP

*

1

( ) ( ) ( )

( , ) * ( )i

L

i

J p J p h p p dp

f x y J pL

Iterative ART

MifwpN

jjjii ,...,1for

1,

MiwfqN

lli

kl

ki ,,,1 allfor

1,

1

jiN

lli

kiik

jkj w

w

qpff ,

1

2,

1

Raywith ray sum pi

f1 f2 f3

fN

Overlapping area for defining wi,j

PET ML Image ReconstructionLet us assume that the object to be reconstructed has an emission density function

tBzyx ],....,[),,( 21

with a Poisson process over a matrix of B pixels. The

emitted photons (in case of SPECT) or photon pairs (in case of PET) are detected by the

detectors with the measurement vector tDJJJJ ],...,,[ 21

with D measurements. The

problem is then to estimate Bbb ,...,1);( from the measurement vector.

Each emission in box b is detected by the detector d (SPECT) or the detector tube d

(PET) with probability p b d P( , ) (detection in d | photon emitted in b). The transition

matrix p b d( , ) is derived from the geometry of the detector array and the reconstruction

space.

Let ),( dbj denotes the number of emissions in box b detected in the detector or detector

tube d are independent Poisson variables with the expected value

),()(),()],([ dbpbdbdbjE .

A

DdBb

dbjdb

dbj

dbejPL

,...,1,...,1

),(),(

)!,(

),()|()(

ML-EM Algorithm

Bbdbpb

dbpdjbb

D

dB

b

old

oldnew ,...,1;),()(ˆ

),()()(ˆ)(ˆ

1

1

Multi-Grid EM Algorithm InitializeGrid level, k = 0Iterations, i = 0

i = i +1

NO

YES

NO

YES

i = 0

WaveletDecomposition

Final ReconstructedImage

NO

YESIs grid

optimizationmeasure

satisfied ?

Is intra-levelperformance

measuresatisfied ?

k = k + 1

Final ReconstructedImage

Is currentgrid resolution

>detector

resolution?

Initial Solution0

0

= EM( ,n)ki+1 i

k

= INTERP( )Wavelet Interpolation

i

kk+10

MGEM Reconstruction

Reconstruction in MRI

dxdydzezyxfMS zyxi zyx )(0 ),,( )(

Fourier Transform Reconstruction Method