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# Reconstruction of objects containing circular cross-sections

Reconstruction of objects containing circular cross-sections. Lajos Rodek rodek@inf.u-szeged.hu. Supervisor: Attila Kuba Ph.D. University of Szeged, Hungary, Department of Applied Informatics. Zoltán Kiss kissz@inf.u-szeged.hu. SSIP 2003. The encountered problem. Tomography.

## Reconstruction of objects containing circular cross-sections

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1. Reconstruction of objects containing circular cross-sections Lajos Rodek rodek@inf.u-szeged.hu Supervisor: Attila Kuba Ph.D. University of Szeged, Hungary, Department of Applied Informatics Zoltán Kiss kissz@inf.u-szeged.hu SSIP 2003

2. 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)

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

4. Reduction to 2D • Subproblem: reconstruction of 2D cross-sections • Assumptions: • known number of circles • at most four different substances

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

6. Parametres of F • Number of circles • Attenuation coefficients • Radii • Centres • Restrictions: • disjointness • minimal & maximal radii • circles are within the ring

7. 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) ?

8. Difficulties • Switching components • Superposition of projections • Noisy input data

9. Implemented algorithm • Considered as optimization problem • Iteratively looking for a global optimum by random modification of parametres from an initial configuration

10. 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

11. 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

12. Simulated annealing • Fundaments: thermodynamic cooling process • Boltzmann-distribution: (1) • If , will be accepted in accordance with (1)

13. Simulation studies

14. 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

15. Effects of noise Additive noise of uniform distribution 0% 5% 10% 20%

16. 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%

17. Results on real measurements

18. 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

19. 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

20. 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

21. 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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