Tomography, approximate reconstruction, and continuous wavelet transforms
Adopting the viewpoint of continuous wavelet transforms as a tool for approximating planar density reconstructions, the paper under review studies the Radon transform of computerized X-ray tomography which is defined on a circular cylinder. Notice that the classical Radon transform performs projectively averages by integration over line bundles [\textit{J. Radon}, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Naturw. Kl. 69, 262-277 (1917; JFM 46.0436.02); \textit{S. Helgason}; ``The Radon transform'' (1980; Zbl 0453.43011; 2nd ed. 1999; Zbl 0932.43010)]. The objective of the article is to introduce continuous wavelet transforms appropriate to the generation of high-resolution local reconstructions from local Radon transform data. This is accomplished by showing that the Radon inversion formula actually represents a summability method. The corresponding convolution kernels then allow the generation of analyzing wavelets with numerical efficient properties of mid-pass filtering bivariate scalar density functions. In this way, modifications of the backprojection algorithms are defined to analyze the local frequency content. MATLAB software is used to implement the algorithms and display the results with the Shepp-Logan phantom.
mid-pass filtering bivariate scalar density functions
scientific article; zbMATH DE number 1330701
scientific article