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Edge Detection from Non-Uniform Fourier Data Using the Convolutional Gridding Algorithm

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Abstract

Detecting edges in images from a finite sampling of Fourier data is important in a variety of applications. For example, internal edge information can be used to identify tissue boundaries of the brain in a magnetic resonance imaging (MRI) scan, which is an essential part of clinical diagnosis. Likewise, it can also be used to identify targets from synthetic aperture radar data. Edge information is also critical in determining regions of smoothness so that high resolution reconstruction algorithms, i.e. those that do not “smear over” the internal boundaries of an image, can be applied. In some applications, such as MRI, the sampling patterns may be designed to oversample the low frequency while more sparsely sampling the high frequency modes. This type of non-uniform sampling creates additional difficulties in processing the image. In particular, there is no fast reconstruction algorithm, since the FFT is not applicable. However, interpolating such highly non-uniform Fourier data to the uniform coefficients (so that the FFT can be employed) may introduce large errors in the high frequency modes, which is especially problematic for edge detection. Convolutional gridding, also referred to as the non-uniform FFT, is a forward method that uses a convolution process to obtain uniform Fourier data so that the FFT can be directly applied to recover the underlying image. Carefully chosen parameters ensure that the algorithm retains accuracy in the high frequency coefficients. Similarly, the convolutional gridding edge detection algorithm developed in this paper provides an efficient and robust way to calculate edges. We demonstrate our technique in one and two dimensional examples.

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Notes

  1. This is assuming that the other parameters of the method, namely the density compensation factors, are suitably chosen, see e.g.  [6, 16, 17, 19].

  2. web.eecs.umich.edu/\(\sim \)fessler/code/.

  3. In this case, although the operations are not commutative, it is clear that there are admissible concentration factors in the method described in [11] that account for these interim interpolating steps.

  4. http://web.eecs.umich.edu/~fessler/code/

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Arizona, Tucson, AZ, 85721, USA

    Adam Martinez

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

    Anne Gelb

  3. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Alexander Gutierrez

Authors
  1. Adam Martinez
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  2. Anne Gelb
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  3. Alexander Gutierrez
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Corresponding author

Correspondence to Anne Gelb.

Additional information

This work is supported in part by grants NSF-DMS 1216559 and AFOSR 12004863.

Appendix

Appendix

The following script implements the convolutional gridding edge detection algorithm in one dimension for given Fourier coefficients of a sawtooth function:

figure c

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Martinez, A., Gelb, A. & Gutierrez, A. Edge Detection from Non-Uniform Fourier Data Using the Convolutional Gridding Algorithm. J Sci Comput 61, 490–512 (2014). https://doi.org/10.1007/s10915-014-9836-y

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  • DOI: https://doi.org/10.1007/s10915-014-9836-y

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