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Local tomography. (English) Zbl 0758.65081
Tomography produces the reconstruction of a function \(f\) from a large number of line integrals of \(f\). Conventional tomography is a global procedure in that the standard convolution formulas for reconstruction at a single point require the intervals over all lines within some plane containing the point. Local tomography, as introduced initially, produced the reconstruction of a related function \(\Lambda f\), where \(\Lambda\) is the square root of \(-\Delta\), the positive Laplace operator.
The reconstruction of \(\Lambda f\) is local in that the reconstruction at a point requires integrals only over lines passing infinitesimally close to the point, and \(\Lambda f\) has the same smooth regions and boundaries as \(f\). However, \(\Lambda f\) is cupped in regions where \(f\) is constant. \(\Lambda^{-1}f\), also amenable to local reconstruction, is smooth everywhere and contains a countercup.
This article provides a detailed study of the actions of \(\Lambda\) and \(\Lambda^{-1}\), and shows several examples of what can be achieved with a linear combination. It includes results of \(X\)-ray experiments on two- dimensional image intensifiers and fluorescent screens, instead of the usual linear detector arrays.

65R10 Numerical methods for integral transforms
65R30 Numerical methods for ill-posed problems for integral equations
44A12 Radon transform
92C55 Biomedical imaging and signal processing
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