Tomography Reconstruction : Introduction and new results on Region of Interest reconstruction

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Tomography Reconstruction Introduction and new
results on Region of Interest reconstruction
Laboratoire Hubert Curien, St Etienne

• Catherine Mennessier
• Rolf Clackdoyle
• Moctar Ould Mohamed

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, PETsearch 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-ray and matter interaction
• Photoelectric absorption
• Compton scattering
• Rayleigh scattering

Microscopic scale
X-ray attenuation
Beer-Lambert law
Macroscopic 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
X-ray and matter interaction
Patient
X-ray sensor
X-ray source
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1. Introduction
?
?
s
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2. Reconstruction in 2D tomography standard
algorithms
Notations
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2. Reconstruction in 2D tomography standard
algorithms
The Radon transform
p(?,s)
We note
s
t
?
f(x)
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2. Reconstruction in 2D tomography the Fourier
slice theorem
Fourier domain
Direct domain
P(?, ?)
1D Fourier transform

p(?,s)
F(??)
??
??
F(?)
2D Fourier transform
f(x)
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2. Reconstruction in 2D tomography the
BackProjection
p(?2,s)
p(?1,s)
p(?3,s)
x
x
We 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?180
N?4
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2. Reconstruction in 2D tomography the FBP
algorithm
1. Projection filtering For k1N? pf(?,s)(p?r
) (?,s) where R(?) ? End 2.
Backprojection fR pf
Ramp filter
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2. Reconstruction in 2D tomography the FBP
algorithm

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

• 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)

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3. Reconstruction of a ROI from truncated data
new results

1. Virtual fan-beam
2. The ramp filter and the Hilbert transform
3. Fan-beam projection
4. Rebining (the Hilbert transform)
5. DBP
6. Differentiated Backprojection
7. 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
Then
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3. Reconstruction of a ROI from truncated data
virtual fan-beam
Rebinning formula
Let us introduce
?
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3. Reconstruction of a ROI from truncated data
virtual fan-beam
Rebinning formula
Let us define
?
Hilbert rebinning formula
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3. Reconstruction of a ROI from truncated data
new results
Is it possible to reconstruct exactly a part of
the object from the incomplete set of data?
a
s
Yes, by selecting a switable virtual fan-beam
projection
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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
The DBP algorithm Differentiated backprojection
xs
Remind
x1
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3. Reconstruction of a ROI from truncated data
new results
x2
The DBP algorithm
L
fx1(x2)
-L
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)
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Merci de votre attentionAny questions?