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Finite Fourier Frame Approximation Using the Inverse Polynomial Reconstruction Method

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Abstract

In several applications, data are collected in the frequency (Fourier) domain non-uniformly, either by design or as a consequence of inexact measurements. The two major bottlenecks for image reconstruction from non-uniform Fourier data are (i) there is no obvious way to perform the numerical approximation, as the non-uniform Fourier data is not amenable to fast transform techniques and resampling the data first to uniform spacing is often neither accurate or robust; and (ii) the Gibbs phenomenon is apparent when the underlying function (image) is piecewise smooth, an occurrence in nearly every application. Recent investigations suggest that it may be useful to view the non-uniform Fourier samples as Fourier frame coefficients when designing reconstruction algorithms that attempt to mitigate either of these fundamental problems. The inverse polynomial reconstruction method (IPRM) was developed to resolve the Gibbs phenomenon in the reconstruction of piecewise analytic functions from spectral data, notably Fourier data. This paper demonstrates that the IPRM is also suitable for approximating the finite inverse Fourier frame operator as a projection onto the weighted \(L_2\) space of orthogonal polynomials. Moreover, the IPRM can also be used to remove the Gibbs phenomenon from the Fourier frame approximation when the underlying function is piecewise smooth. The one-dimensional numerical results presented here demonstrate that using the IPRM in this way yields a robust, stable, and accurate approximation from non-uniform Fourier data.

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Notes

  1. We note that iterative procedures using \(l^1\) regularization are also commonly employed, see e.g. [30]. While these methods are often very effective, the purpose of our paper is to develop an analytical framework for a direct solver. Indeed, since the IPRM is a linear construction, it might serve to produce an objective funtion to which a regularization term can be applied.

  2. The original theorem is written for the interval \([-\pi ,\pi ]\).

  3. Since \(e_k = e^{i\lambda _k \pi x}\) and \(\lambda _k\) are sampling the conditioning and invertibility of T depends on the sampling pattern.

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Acknowledgements

Anne Gelb was partially supported by Grants NSF-DMS 1216559, NSF-DMS 1521600, NSF 1502640, and AFOSR FA9550-15-1-0152.

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Authors and Affiliations

  1. Department of Mathematics, Jimei University, Xiamen, Fujian Province, China

    Xinjuan Chen

  2. Department of Mathematics, University at Buffalo, The State University of New York, Buffalo, NY, 14260-2900, USA

    Jae-Hun Jung

  3. Department of Data Science, Ajou University, Yeongtong-gu, Suwon, 16499, Korea

    Jae-Hun Jung

  4. Department of Mathematics, Dartmouth College, Hanover, NH, 03755-3551, USA

    Anne Gelb

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  1. Xinjuan Chen
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  2. Jae-Hun Jung
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Correspondence to Jae-Hun Jung.

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Chen, X., Jung, JH. & Gelb, A. Finite Fourier Frame Approximation Using the Inverse Polynomial Reconstruction Method. J Sci Comput 76, 1127–1147 (2018). https://doi.org/10.1007/s10915-018-0655-4

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