Fast calculation of multiple line integrals

A line integral is defined as the integral of two-dimensional data along a (one-dimensional, straight) line of given length and orientation. Line integrals are used in various forms of edge and line detectors in images and in the computation of the Radon transform. We present a recursive algorithm which enables approximation of discretized line integrals at all lengths, orientations, and locations to within a prescribed error bound in at most O(n log n log log n) operations, where n is the number of data points. Furthermore, for most applications (in particular, where even small amounts of noise are present in the data) all of these integrals can be computed to the desired accuracy in about 24n log n operations.