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Generalized Discrete Radon Transforms and Their Use in the Ridgelet Transform

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Abstract

We introduce and study a new class of Radon transforms in a discrete setting for the purpose of applying them to the ridgelet and curvelet transforms. We give a detailed analysis of the p-adic case and provide a closed-form formula for an inverse of the p-adic Radon transform. We give conditions for a scaled version of the generalized discrete Radon transform to yield a tight frame, and discuss a direct Radon matrix method for the implementation of a local ridgelet transform. We then study the effectiveness of some types of the generalized Radon transforms in reducing a type of noise known as speckle that is present in synthetic aperture radar (SAR) imagery.

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Correspondence to Flavia Colonna.

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Flavia Colonna received the M.A. degree and the Ph.D degree in mathematics from the University of Maryland (College Park) in 1980 and 1985, respectively. She was an assistant professor at the University of Bari (Italy) until she joined the faculty of George Mason University in 1986, where she is currently professor of mathematics. Her research interests include discrete harmonic analysis, integral geometry, potential theory, and classical complex function theory.

Glenn R. Easley received the B.S. degree (with honors) and the M.A. degree in mathematics from the University of Maryland, College Park, in 1993 and 1996, respectively, and the Ph.D. degree in computational science and informatics from George Mason University in 2000. Since 2000, he has been working for System Planning Corporation in signal and image processing. His research interests include computational harmonic analysis, wavelet analysis, synthetic aperture radar, deconvolution and computer vision.

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Colonna, F., Easley, G. Generalized Discrete Radon Transforms and Their Use in the Ridgelet Transform. J Math Imaging Vis 23, 145–165 (2005). https://doi.org/10.1007/s10851-005-6463-0

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