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Detecting Edges from Non-uniform Fourier Data Using Fourier Frames

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Abstract

Edge detection plays an important role in identifying regions of interest in an underlying signal or image. In some applications, such as magnetic resonance imaging (MRI) or synthetic aperture radar (SAR), data are sampled in the Fourier domain. Many algorithms have been developed to efficiently extract edges of images when uniform Fourier data are acquired. However, in cases where the data are sampled non-uniformly, such as in non-Cartesian MRI or SAR, standard inverse Fourier transformation techniques are no longer suitable. Methods exist for handling these types of sampling patterns, but are often ill-equipped for cases where data are highly non-uniform or when the data are corrupted or otherwise not usable in certain parts of the frequency domain. This investigation further develops an existing approach to discontinuity detection, and involves the use of concentration factors. Previous research shows that the concentration factor technique can successfully determine jump discontinuities in non-uniform data. However, as the distribution diverges further away from uniformity so does the efficacy of the identification. Thus we propose a method that employs the finite Fourier approximation to specifically tailor the design of concentration factors. We also adapt the algorithm to incorporate appropriate smoothness assumptions in the piecewise smooth regions of the function. Numerical results indicate that our new design method produces concentration factors which can more precisely identify jump locations than those previously developed in both one and two dimensions.

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Notes

  1. This idea was first investigated in [23].

  2. Recall that we assume that the discontinuities occur only on grid points \(x_j\). For convenience we choose \(x_j = \frac{j}{J}\), \(-J \le j \le J\) so that the value \(x = 0\) falls on the grid point \(x_0\). The system can be designed for any chosen gridpoints, however.

  3. The numerical results using Algorithm 1 were first reported in [23].

  4. Indeed, a related idea was examined in [30] for suppressing higher order terms in (36) given uniform samples, but in this case we design s(x) to more closely resemble the smooth part of the underlying function.

  5. Algorithm 4 closely follows the one provided in [28] for non-uniform coefficients, although the values obtained in (46) and (47) are substantially refined by Algorithms 1 and 2.

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Acknowledgements

This work is supported in part by Grants NSF-DMS 1216559 (AG), NSF-DMS 1521600 (AG), NSF-DMS 1521661 (GS), NSF 1502640 (AG), and AFOSR FA9550-15-1-0152 (AG).

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Authors and Affiliations

  1. Department of Mathematics, Dartmouth College, 27 N. Main St., Hanover, NH, 03755, USA

    Anne Gelb

  2. Department of Mathematics, Clarkson University, Potsdam, NY, 13699, USA

    Guohui Song

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  1. Anne Gelb
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  2. Guohui Song
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Correspondence to Anne Gelb.

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Gelb, A., Song, G. Detecting Edges from Non-uniform Fourier Data Using Fourier Frames. J Sci Comput 71, 737–758 (2017). https://doi.org/10.1007/s10915-016-0320-8

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