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Fast Slant Stack: A notion of Radon Transform for Data in a Cartesian Grid

Summary: Fast Slant Stack:
A notion of Radon Transform for Data in a Cartesian Grid
which is Rapidly Computible, Algebraically Exact, Geometrically Faithful and Invertible
A. Averbuch, R.R. Coifman, D.L. Donoho, M. Israeli, J. WaldŽen
Abstract. We define a notion of Radon Transform for data in an n by n grid. It is based on summation
along lines of absolute slope less than 1 (as a function either of x or of y), with values at non-Cartesian
locations defined using trigonometric interpolation on a zero-padded grid. The definition is geometrically
faithful: the lines exhibit no `wraparound effects'.
For a special set of lines equispaced in slope (rather than angle), we describe an exact algorithm which uses
O(N log N) flops, where N = n2
is the number of pixels. This relies on a discrete projection-slice theorem
relating this Radon transform and what we call the Pseudopolar Fourier transform. The Pseudopolar FT
evaluates the 2-D Fourier transform on a non-Cartesian pointset, which we call the pseudopolar grid. Fast
Pseudopolar FT ­ the process of rapid exact evaluation of the 2-D Fourier transform at these non-Cartesian
grid points ­ is possible using chirp-Z transforms.
This Radon transform is one-to-one and hence invertible on its range; it is rapidly invertible to any degree
of desired accuracy using a preconditioned conjugate gradient solver. Empirically, the numerical conditioning
is superb; the singular value spread of the preconditioned Radon transform turns out numerically to be less
than 10%, and three iterations of the conjugate gradient solver typically suffice for 6 digit accuracy.
We also describe a 3-D version of the transform.


Source: Averbuch, Amir - School of Computer Science, Tel Aviv University
Donoho, David - Department of Statistics, Stanford University


Collections: Computer Technologies and Information Sciences; Mathematics