 
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 nonCartesian
locations defined using trigonometric interpolation on a zeropadded 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 projectionslice theorem
relating this Radon transform and what we call the Pseudopolar Fourier transform. The Pseudopolar FT
evaluates the 2D Fourier transform on a nonCartesian pointset, which we call the pseudopolar grid. Fast
Pseudopolar FT the process of rapid exact evaluation of the 2D Fourier transform at these nonCartesian
grid points is possible using chirpZ transforms.
This Radon transform is onetoone 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 3D version of the transform.
