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Image Reconstruction from Undersampled Fourier Data Using the Polynomial Annihilation Transform

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Abstract

Fourier samples are collected in a variety of applications including magnetic resonance imaging and synthetic aperture radar. The data are typically under-sampled and noisy. In recent years, \(l^1\) regularization has received considerable attention in designing image reconstruction algorithms from under-sampled and noisy Fourier data. The underlying image is assumed to have some sparsity features, that is, some measurable features of the image have sparse representation. The reconstruction algorithm is typically designed to solve a convex optimization problem, which consists of a fidelity term penalized by one or more \(l^1\) regularization terms. The Split Bregman Algorithm provides a fast explicit solution for the case when TV is used for the \(l^1\) regularization terms. Due to its numerical efficiency, it has been widely adopted for a variety of applications. A well known drawback in using TV as an \(l^1\) regularization term is that the reconstructed image will tend to default to a piecewise constant image. This issue has been addressed in several ways. Recently, the polynomial annihilation edge detection method was used to generate a higher order sparsifying transform, and was coined the “polynomial annihilation (PA) transform.” This paper adapts the Split Bregman Algorithm for the case when the PA transform is used as the \(l^1\) regularization term. In so doing, we achieve a more accurate image reconstruction method from under-sampled and noisy Fourier data. Our new method compares favorably to the TV Split Bregman Algorithm, as well as to the popular TGV combined with shearlet approach.

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Notes

  1. Technically, the \(l^0\) norm of an expression is a better measure of sparsity. However, the \(l^0\) norm does not meet the convexity requirements and is very slow to compute. Additional detail on using the \(l^1\) norm in place of the \(l^0\) in order to measure sparsity can be found in [7].

  2. This was a departure from the work done in [25] which assumed the given data were sampled in the physical domain, which would imply simulating the Fourier data of the underlying function rather than collecting them.

  3. In [28] our method compared favorably to multi-wavelet constructed regularization terms [24]. We do not repeat those experiments here.

  4. For simplicity we choose \(2N+1\) equally spaced grid points to match the number of Fourier coefficients. The techniques described in this paper are easily extended to different gridding schemes in the image domain.

  5. In that case, the given data were blurred and/or noisy grid point data in the one-dimensional image domain.

  6. Although designed for MRI data, it is applicable whenever data are sampled in the Fourier domain, especially when dimension reduction is desirable.

  7. We tried a variety of parameters in our experiments to ensure that our comparisons were fair. As it turned out, the default parameters yielded the best results.

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Acknowledgments

This work is supported in part by grants NSF-DMS 1216559, AFOSR FA9550-12-1-0393, and AFOSR FA9550-15-1-0152. The submitted manuscript is based upon work, authored in part by contractors [UT-Battelle LLC, manager of Oak Ridge National Laboratory (ORNL)], and supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program. Accordingly, the U.S. Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes.

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Authors and Affiliations

  1. Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA

    Rick Archibald

  2. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA

    Anne Gelb & Rodrigo B. Platte

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  1. Rick Archibald
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  2. Anne Gelb
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  3. Rodrigo B. Platte
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Correspondence to Anne Gelb.

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Archibald, R., Gelb, A. & Platte, R.B. Image Reconstruction from Undersampled Fourier Data Using the Polynomial Annihilation Transform. J Sci Comput 67, 432–452 (2016). https://doi.org/10.1007/s10915-015-0088-2

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  • DOI: https://doi.org/10.1007/s10915-015-0088-2

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