## 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.

### MSC:

 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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