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# Reconstruction of objects containing circular cross-sections - PowerPoint PPT Presentation

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

Lajos Rodek

Supervisor: Attila Kuba Ph.D.

University of Szeged, Hungary, Department of Applied Informatics

Zoltán Kiss

SSIP 2003

• 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)

• Discrete tomography

• Input: few projections (2-4)

• a priori information:

• geometrical structure (spheres)

• range (attenuation coefficients)

• Output: 3D model

• Subproblem: reconstruction of 2D cross-sections

• Assumptions:

• known number of circles

• at most four different substances

• Projection:

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

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

• Number of circles

• Attenuation coefficients

• Radii

• Centres

• Restrictions:

• disjointness

• minimal & maximal radii

• circles are within the ring

• 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)

?

• Switching components

• Superposition of projections

• Noisy input data

• Considered as optimization problem

• Iteratively looking for a global optimum by random modification of parametres from an initial configuration

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

radius

centre

attenuation coefficient

• 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

• Fundaments: thermodynamic cooling process

• Boltzmann-distribution:

(1)

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

using 2, 3 & 4 noiseless projections

Real conf.

Initial conf.

Reconstructed conf.

Difference

2 projs

3 projs

4 projs

Additive noise of uniform distribution

0% 5% 10% 20%

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

Real conf.

Initial conf.

Reconstructed conf.

Difference

Noise

5%

20%

40%

• Precessing axis of revolution

• Distorted, noisy projections

• Low resolution

• Too few quantization levels

• Attenuation coefs are unknown  they should be estimated automatically

0

45

90

135

Result of convolution backprojection

from 60 projections

Result of our method

from 4 projections seen above

• 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

• 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