A digital phase manipulation for images processing on the algorithm similar to the optical Foucault knife considers by
means of computer modeling method. The images quality preservation at decomposition in Walsh-Hadamard and Haar
spectrum was investigated. It is shown, that Foucault knife for the digital image can be simulated on the basis of
calculations in real numbers space. In this case there is a possibility to select the image contours, too.
Phase-contrast x-ray computed tomography (CT) is an emerging imaging technique that can be implemented
at third generation synchrotron radiation sources or by using a microfocus x-ray tube. Promising experimental
results have recently been obtained in material science and biological applications. At the same time, the lack of
a mathematical theory comparable to that of conventional absorption-based CT limits the progress in this field.
We suggest such a theory and prove a fundamental theorem that plays the same role for phase-contrast CT as
the Fourier slice theorem does for absorption-based CT. The fundamental theorem allows us to derive fast image
reconstruction algorithms in the form of filtered backprojection (FBP).
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