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SIAM J. SCI. COMPUT. c 2008 Society for Industrial and Applied Mathematics
Vol. 30, No. 2, pp. 785803
A FRAMEWORK FOR DISCRETE INTEGRAL
TRANSFORMATIONS IITHE 2D DISCRETE
RADON TRANSFORM
A. AVERBUCH, R. R. COIFMAN, D. L. DONOHO§, M. ISRAELI¶, Y. SHKOLNISKY,
AND I. SEDELNIKOV
In memory of Moshe Israeli 19402007
Abstract. Although naturally at the heart of many fundamental physical computations, and
potentially useful in many important image processing tasks, the Radon transform lacks a coherent
discrete definition for twodimensional (2D) discrete images which is algebraically exact, invertible,
and rapidly computable. We define a notion of 2D discrete Radon transforms for 2D discrete images,
which is based on summation along lines of absolute slope less than 1. Values at nongrid locations
are defined using trigonometric interpolation on a zeropadded grid. Our definition is shown to
be geometrically faithful: the summation avoids wraparound effects. Our proposal uses a special
collection of lines in R2 for which the transform is rapidly computable and invertible. We describe
a fast algorithm using O(N log N) operations, where N = n2 is the number of pixels in the image.
The fast algorithm exploits a discrete projectionslice theorem, which associates the discrete Radon
transform with the pseudopolar Fourier transform [A. Averbuch et al., SIAM J. Sci. Comput., 30
