Reconstruction of objects containing circular cross sections
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Reconstruction of objects containing circular cross-sections. Lajos Rodek [email protected] Supervisor: Attila Kuba Ph.D. University of Szeged, Hungary, Department of Applied Informatics. Zoltán Kiss [email protected] SSIP 2003. The encountered problem. Tomography.

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Reconstruction of objects containing circular cross sections l.jpg

Reconstruction of objects containing circular cross-sections

Lajos Rodek

[email protected]

Supervisor: Attila Kuba Ph.D.

University of Szeged, Hungary, Department of Applied Informatics

Zoltán Kiss

[email protected]

SSIP 2003


The encountered problem l.jpg
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 l.jpg
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 l.jpg
Reduction to 2D

  • Subproblem: reconstruction of 2D cross-sections

  • Assumptions:

    • known number of circles

    • at most four different substances


Acquisition of projections l.jpg
Acquisition of projections

  • Projection:

  • Given: projections (p), directions, number of beams

  • Unknown: F, implicit parametric function to be reconstructed (4-valued)


Parametres of f l.jpg
Parametres of F

  • Number of circles

  • Attenuation coefficients

  • Radii

  • Centres

  • Restrictions:

    • disjointness

    • minimal & maximal radii

    • circles are within the ring


Mathematical description l.jpg
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)

?


Difficulties l.jpg
Difficulties

  • Switching components

  • Superposition of projections

  • Noisy input data


Implemented algorithm l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
Simulated annealing

  • Fundaments: thermodynamic cooling process

  • Boltzmann-distribution:

(1)

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



Effects of changing the number of projections l.jpg
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


Effects of noise l.jpg
Effects of noise

Additive noise of uniform distribution

0% 5% 10% 20%


Results from noisy projections l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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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