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

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

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

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

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

Parametres of F Number of circlesNumber of circles Attenuation coefficientsAttenuation coefficients RadiiRadii CentresCentres Restrictions:Restrictions: –disjointness –minimal & maximal radii –circles are within the ring

Mathematical description Given: few projections with known number of circles & beamsGiven: 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)Sought solution: configuration of parametres, which determines a function having projections of the best approximation of input data (p)

Difficulties Switching componentsSwitching components Superposition of projectionsSuperposition of projections Noisy input dataNoisy input data

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

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

Optimization Random choice of a new configurationRandom choice of a new configuration If, will be acceptedIf, will be accepted Else choosing anotherElse choosing another Termination, if or no better solution is found in a certain number of iteration stepsTermination, if or no better solution is found in a certain number of iteration steps Objective function:Objective function:

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

Simulation studies

Initial conf. Reconstructed conf. Difference Real conf. 2 projs 3 projs 4 projs Effects of changing the number of projections using 2, 3 & 4 noiseless projections

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

Results from noisy projections Real conf. Initial conf. Difference Reconstructed conf. using 4 projections, in case of 5, 20 & 40% of noise 5% 20% 40% Noise

Results on real measurements

Encountered problems on real data Precessing axis of revolutionPrecessing axis of revolution Distorted, noisy projectionsDistorted, noisy projections Low resolutionLow resolution Too few quantization levelsToo few quantization levels Attenuation coefs are unknown  they should be estimated automaticallyAttenuation coefs are unknown  they should be estimated automatically

Data from Berliner Hahn-Meitner Institut Result of convolution backprojection from 60 projections Result of our method from 4 projections seen above 0000 45  90  135 

Summary A new reconstruction method has been implemented based on real physical measurements: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 noiseGood results may be achieved from 4 projections even in case of greater amount of noise Future plans:Future plans: –extension to 3D –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, 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 Homepage of DIRECT :