A formulation in concordance with sampling theorem for band-limited images reconstruction from projections
Image reconstruction from sampled projections has been extensively used in the field of medical imaging. Imaging means visualization, not to make visible and reconstruct what is already visible. Influenced by the classical backprojection technique of computerized tomography (CT) and coherent optical image processing, magnetic resonance tomography (MRI), for instance, has applied this method before the symplectic Fourier transform method in the super-encoding projective space became available for routine clinical examinations [\textit{W. Schempp}, Magnetic resonance imaging: Mathematical foundations and applications. J. Wiley \& Sons, New York (to appear)]. The paper under review contributes to the method of semantic filtering by presenting a filter formulation of the imaging reconstruction problem. The optimal reconstruction from a finite number of samples is performed in two steps: (i) one-dimensional convolution/backprojection, and (ii) solution of a convolution equation involving a large positive definite Toeplitz-block-Toeplitz matrix \(A\) via a conjugate gradient method. Due to the circulant block structure of \(A\), the implementation is efficiently performed by parallel processors for the multidimensional fast Fourier transform (FFT).
scientific article; zbMATH DE number 871413
scientific article