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Reconstruction of objects containing circular cross-sections

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### Reconstruction of objects containing circular cross-sections

Lajos Rodek

Supervisor: Attila Kuba Ph.D.

University of Szeged, Hungary, Department of Applied Informatics

Zoltán Kiss

SSIP 2003

The encountered problem

- Tomography

- Nondestructive substance examination
- Neutron mapping

Draft structure of the 3D object

A cross-section to be reconstructed

Result of a classic method

- Using few projections (acquisition is time consuming and expensive)

Reconstruction of 3D objects

- Discrete tomography
- Input: few projections (2-4)
- a priori information:
- geometrical structure (spheres)
- range (attenuation coefficients)
- Output: 3D model

Reduction to 2D

- Subproblem: reconstruction of 2D cross-sections
- Assumptions:
- known number of circles
- at most four different substances

Acquisition of projections

- Projection:

- Given: projections (p), directions, number of beams
- Unknown: F, implicit parametric function to be reconstructed (4-valued)

Parametres of F

- Number of circles
- Attenuation coefficients
- Radii
- Centres
- Restrictions:
- disjointness
- minimal & maximal radii
- circles are within the ring

Mathematical description

- Given:few projections with known number of circles & beams
- Sought solution: configuration of parametres, which determines a function having projections of the best approximation of input data (p)

?

Implemented algorithm

- Considered as optimization problem
- Iteratively looking for a global optimum by random modification of parametres from an initial configuration

Choosing a new configuration

Adjustment of radius, centre or attenuation coef. of one of the circles, in agreement with the restrictions

radius

centre

attenuation coefficient

Optimization

- Objective function:

- Random choice of a new configuration
- If , will be accepted
- Else choosing another
- Termination, if or no better solution is found in a certain number of iteration steps

Simulated annealing

- Fundaments: thermodynamic cooling process
- Boltzmann-distribution:

(1)

- If , will be accepted in accordance with (1)

Effects of changing the number of projections

using 2, 3 & 4 noiseless projections

Real conf.

Initial conf.

Reconstructed conf.

Difference

2 projs

3 projs

4 projs

Results from noisy projections

using 4 projections, in case of 5, 20 & 40% of noise

Real conf.

Initial conf.

Reconstructed conf.

Difference

Noise

5%

20%

40%

Encountered problems on real data

- Precessing axis of revolution
- Distorted, noisy projections
- Low resolution
- Too few quantization levels
- Attenuation coefs are unknown they should be estimated automatically

Data from Berliner Hahn-Meitner Institut

0

45

90

135

Result of convolution backprojection

from 60 projections

Result of our method

from 4 projections seen above

Summary

- A new reconstruction method has been implemented based on real physical measurements:
- the effects of increasing the number of circles, projections & the amount of noise have been examined
- Good results may be achieved from 4 projections even in case of greater amount of noise
- Future plans:
- extension to 3D
- deformable models

References

- A. Kuba, L. Ruskó, Z. Kiss, L. Rodek, E. Balogh, S. Zopf, A. Tanács: Preliminary Results in Discrete Tomography Applied for Neutron Tomography, COST Meeting on Neutron Radiography, Loughborogh, England, 2002.
- A. Kuba, L. Ruskó, L. Rodek, Z. Kiss: Preliminary Studies of Discrete Tomography in Neutron Imaging, IEEE Trans. on Nuclear Sciences, submitted
- A. Kuba, L. Ruskó, L. Rodek, Z. Kiss: Application of Discrete Tomography in Neutron Imaging, Proc. of 7th World Conference on Neutron Radiography, Rome, Italy, 2002., accepted
- Kiss Z., Kuba A., Rodek L.: Körmetszeteket tartalmazó tárgyak rekonstrukciója néhány vetületből, KÉPAF Konferencia kiadvány, Domaszék, Hungary 2002.

Homepage of DIRECT:

http://www.inf.u-szeged.hu/~direct

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