A butterfly algorithm for synthetic aperture radar

Abstract

It is not currently known if it is possible to accurately form a synthetic aperture radar image from N data points in provable near-linear complexity, where accuracy is defined as the ℓ₂ error between the full O(N²) backprojection image and the approximate image. To bridge this gap, we present a backprojection algorithm with complexity O(log(1/ϵ)N log N), with ϵ the tunable pixelwise accuracy. It is based on the butterfly scheme, which works for vastly more general oscillatory integrals than the discrete Fourier transform. Unlike previous methods this algorithm allows the user to directly choose the amount of acceptable image error based on a well-defined metric. Additionally, the algorithm does not invoke the far-field approximation or place restrictions on the antenna flight path, nor does it impose the frequency-independent beampattern approximation required by time-domain backprojection techniques.