The Polynomial Resampling Method for Non-Uniform Fourier Data

Abstract The reconstruction of piecewise smooth functions from non-uniform Fourier data is an important problem in applications such as sensing (e.g., Magnetic Resonance Imaging). In this talk I present a the polynomial resampling method of approximating the Fourier transform $\hat{f}(\omega)$ of an underlying piecewise smooth function as an asymptotic expansion of mapped Chebyshev polynomials. The method is shown to converge exponentially in the(finite) Fourier transform domain given exact edge information.

Recent edge detection methods from non-uniform Fourier data can provide sufficient initial edge location estimates when information is not known in advance. Our new method then applies an optimization procedure that improves the accuracy of the edges along with that of our approximation of the Fourier transform.