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Tomography Reconstruction : Introduction and new results on Region of

Interest reconstruction

-Catherine Mennessier- Rolf Clackdoyle-Moctar Ould Mohamed

Laboratoire Hubert Curien, St Etienne

Bucharest, May 2008

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Table of contents

1. Introduction

2. Reconstruction in 2D tomography : standard algorithms

3. Reconstruction of a Region Of Interest from truncated data : new results.

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1. Introduction

Computer Tomography : a non-destructive imaging technique for interior inspection.

Waste inspection CT scanner

Some applications…

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1. Introduction

Domains of application:

• Medical image processing :– Anatomic imaging (CT, Image Guided Surgery, Diagnostic..) density– Functional imaging (SPECT, PET…search for tumour, heart muscle

viable…) radioactive tracer

• Industrial :– Non destructive techniques for characterization (drum nuclear waste..),

defect detection (on production lines)…

• Archaeology :– Interior reconstruction (of amphora…)

• Astronomy : – Doppler imaging

• Geology :– Seismic studies (wave tomography)

• …

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1. Introduction

In transmission tomography, the X ray (or gamma ray…) are attenuated. The degree of attenuation depends on the density of the object. The absorption of the X-ray is measured, from different positions of the source/detector system.

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1. Introduction

x

N0 f NxNfNNN 0

Beer-Lambert law:

X-ray and matter interaction:

• Photoelectric absorption

• Compton scattering

• Rayleigh scattering

X-ray attenuation

Macroscopic scale

Microscopic scale

The absorption coefficient f depends on the material. For instance, at 60KeV, water(0,203/cm), white matter(0,210/cm), gray matter(0,212/cm) …

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1. Introduction

Lout

in

X

X

N

N

xfdxN

Nsodxxf

N

dN out

in

out

in

)(log,)(

xout

xin LX-ray source

X-ray sensor

Patient

X-ray and matter interaction :

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1. Introduction

?

s

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2. Reconstruction in 2D tomography : standard algorithms

Notations

)sin,(cos),sin,(coswith ,)(),(

dtstfsp

s t

p(,s)

f(x)

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2. Reconstruction in 2D tomography : standard algorithms

fp R

The Radon transform :

s t

p(,s)

f(x)

dtstfsp

SS

)(),(

)],([)(:R

02

We note :

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2. Reconstruction in 2D tomography : the Fourier slice theorem

p(,s)

Fourier domainDirect domain

f(x)

F()

1D Fourier transform

2D Fourier transform

=

)(),( FP

F()

P(, )

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2. Reconstruction in 2D tomography : the BackProjection

],[

).,()(

)()],([:R

0

20

dxpxb

SS

xx

p(1,s)

p(2,s)

p(3,s)

pb RWe note :

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2. Reconstruction in 2D tomography : the BackProjection

Backprojection of the Radon transform of a centred disk of constant intensity :

N=

1

N=

2

N=

4

N=

180

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2. Reconstruction in 2D tomography : the FBP algorithm

1. Projection filtering

For k=1:N

pf(,s)=(pr ) (,s) where R()=| |

End

2. Backprojection

f=R* pf

)( with wherepf Rsrsgspf f )(),(),(R*

Ramp filter

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2. Reconstruction in 2D tomography : the FBP algorithm

)( with wherepf Rrppf f*R

Comments :

• To compute the single value f(x) at x, all the projections are needed as the filtering step is not local if one data is missing, all the reconstruction (for all x) is affected by the FBP algorithm.

• FBP is very efficient (standard from 30 years).

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3. Reconstruction of a ROI from truncated data : new results

Truncated data : only the lines that intersect the circle are measured

Not measuredmeasured

Is it possible to reconstruct exactly a part of the object from the incomplete set of data?

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3. Reconstruction of a ROI from truncated data : new results

Solution : the answer is

• no if FBP is used

• yes for some ROI using

- virtual fan-beam algorithm (2004)

-Differentiated Backprojection with truncated Hilbert Inverse (2004) (two-step, DBP, chord…)

Is it possible to reconstruct exactly a part of the object from an incomplete set of data?

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3. Reconstruction of a ROI from truncated data : new results

1. Virtual fan-beam

1. The ramp filter and the Hilbert transform

2. Fan-beam projection

3. Rebining (the Hilbert transform)

2. DBP

1. Differentiated Backprojection

2. Truncated Hilbert Inverse

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3. Reconstruction of a ROI from truncated data : virtual fan-beam

Inverse Radon transform and the Hilbert transform : the filtering step

Remind : )(with where Rrpppfs

ff **R

ansformHilbert tr theis s

1*pHp Where

where2

1

s

),(),(

),(),(

ss

sHps

sp f

Then

sssri

iR

1

2

12

2

1 )(

22

)())()sgn(

()sgn(

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3. Reconstruction of a ROI from truncated data : virtual fan-beam

Rebinning formula:

Let us introduce :

dttafag )(),(

a

a.sat ),(),( agsp

a

s

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3. Reconstruction of a ROI from truncated data : virtual fan-beam

Rebinning formula:

Let us define : )sin(

),(),( *

1agaHg

a

Hilbert rebinning formula :

a.sfor ),(),( aHgsHp

a

s

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3. Reconstruction of a ROI from truncated data : new results

a.sat ),(),( aHgsHp

Not measuredmeasured

Is it possible to reconstruct exactly a part of the object from the incomplete set of data?

Yes, by selecting a switable virtual fan-beam projection

s

a

).,()( computeThen ],0[

dxHps

xf

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3. Reconstruction of a ROI from truncated data : new results

The ROI that can be exactly reconstructed using the virtual fan-beam algorithm

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3. Reconstruction of a ROI from truncated data : new results

),()()()(

),(R)( *

21220

0

11 where

then(x)s

Define

xxfxfxHfxb

spxb

xx

The DBP algorithm : Differentiated backprojection

Remind ),(),( sps

Hsp f

2

1

x1

xs

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3. Reconstruction of a ROI from truncated data : new results

thenneeded, is for only t reconstruc To

Supp( Assume

2221

2

1

1

],[)(),(

],[))(

LLxxHfxf

LLxfxx

x

The DBP algorithm

fx1(x2) can be reconstructed where a vertical line, crossing the support of f, can be found, assuming backprojection of the line points is possible.

NB: Generalization for all the direction (not only the vertical line)

-L

+L

fx1(x2)

x2

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Merci de votre attention…

Any questions?