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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

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
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
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
Reduction to 2D
  • Subproblem: reconstruction of 2D cross-sections
  • Assumptions:
    • known number of circles
    • at most four different substances
acquisition of projections
Acquisition of projections
  • Projection:
  • Given: projections (p), directions, number of beams
  • Unknown: F, implicit parametric function to be reconstructed (4-valued)
parametres of f
Parametres of F
  • Number of circles
  • Attenuation coefficients
  • Radii
  • Centres
  • Restrictions:
    • disjointness
    • minimal & maximal radii
    • circles are within the ring
mathematical description
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
Difficulties
  • Switching components
  • Superposition of projections
  • Noisy input data
implemented algorithm
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
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
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
Simulated annealing
  • Fundaments: thermodynamic cooling process
  • Boltzmann-distribution:

(1)

  • If , will be accepted in accordance with (1)
effects of changing the number of projections
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
Effects of noise

Additive noise of uniform distribution

0% 5% 10% 20%

results from noisy projections
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
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
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
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
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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