Tomography, approximate reconstruction, and continuous wavelet transforms
type: Item
instance of: scholarly article
title: Tomography, approximate reconstruction, and continuous wavelet transforms
zbMATH Open document ID: 0945.42021
author: W. R. Madych
published in: Applied and Computational Harmonic Analysis
publication date: 2000-06-04T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 92C55
Mathematics Subject Classification ID: 94A12
Mathematics Subject Classification ID: 44A12
zbMATH DE Number: 1330701
zbMATH Keywords: continuous wavelet transforms
zbMATH Keywords: Radon transform
zbMATH Keywords: computerized X-ray tomography
zbMATH Keywords: JFM 46.0436.02
zbMATH Keywords: high-resolution local reconstructions
zbMATH Keywords: mid-pass filtering bivariate scalar density functions
reviewed by: Walter Schempp
describes a project that uses: Matlab
MaRDI profile type: Publication
full work available at URL: acha.1998.0258
OpenAlex ID: W2028076783
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DOI: 10.1006/ACHA.1998.0258
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description: scientific article; zbMATH DE number 1330701