Reconstruction of objects containing circular crosssections Lajos Rodek

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

The encountered problem • • • Nondestructive substance examination Neutron mapping Tomography Draft structure of the 3 D object A cross-section to be reconstructed Result of a classic method • Using few projections (acquisition is time consuming and expensive)

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

Reduction to 2 D • Subproblem: reconstruction of 2 D crosssections • Assumptions: – known number of circles – at most four different substances

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

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)

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

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

Choosing a new Adjustment of radius, centre or attenuation coef. of one of the circles, in agreement with the restrictions configuration 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)

Simulation studies

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 Additive noise of uniform distribution 0% 5% 10% 20%

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%

Results on real measurements

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 Result of our method from 60 projections 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 3 D – 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 7 th 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