Geodesic
Three-dimensional reconstruction from projections
Applications of Mathematics, Tome 30 (1985) no. 2, pp. 92-109
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Computerized tomograhphy is a technique for computation and visualization of density (i.e. X- or $\gamma$-ray absorption coefficients) distribution over a cross-sectional anatomic plane from a set of projections. Three-dimensional reconstruction may be obtained by using a system of parallel planes. For the reconstruction of the transverse section it is necessary to choose an appropriate method taking into account the geometry of the data collection, the noise in projection data, the amount of data, the computer power available, the accuracy required etc. In the paper the theory related to the convolution reconstruction methods is reviewed. The principal contribution consists in the exact mathematical treatment of Radon's inverse transform based on the concepts of the regularization of a function and the generalized function. This approach naturally leads to the employment of the generalized Fourier transform. Reconstructions using simulated projection data are presented for both the parallel and divergent-ray collection geometries.
Computerized tomograhphy is a technique for computation and visualization of density (i.e. X- or $\gamma$-ray absorption coefficients) distribution over a cross-sectional anatomic plane from a set of projections. Three-dimensional reconstruction may be obtained by using a system of parallel planes. For the reconstruction of the transverse section it is necessary to choose an appropriate method taking into account the geometry of the data collection, the noise in projection data, the amount of data, the computer power available, the accuracy required etc. In the paper the theory related to the convolution reconstruction methods is reviewed. The principal contribution consists in the exact mathematical treatment of Radon's inverse transform based on the concepts of the regularization of a function and the generalized function. This approach naturally leads to the employment of the generalized Fourier transform. Reconstructions using simulated projection data are presented for both the parallel and divergent-ray collection geometries.
DOI : 10.21136/AM.1985.104131
Classification : 43A85, 44A15, 65R10, 92A07, 92F05
Keywords: research survey; parallel beam; divergent beam; ill-posed problem; convolution reconstruction methods; computer tomography; Radon’s inverse transform; regularization; generalized Fourier transform; spatial filter; window functions; errors 
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Jelínek, Jiří; Segeth, Karel; Overton, T. R. Three-dimensional reconstruction from projections. Applications of Mathematics, Tome 30 (1985) no. 2, pp. 92-109. doi: 10.21136/AM.1985.104131

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