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

Adam Martinez, Anne Gelb, Alexander Gutierrez

Research output: Contribution to journalArticle

9 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)490-512
Number of pages23
JournalJournal of Scientific Computing
Volume61
Issue number3
DOIs
StatePublished - Oct 14 2014

Fingerprint

Edge Detection
Edge detection
Fast Fourier transforms
Sampling
Magnetic Resonance Imaging
Reconstruction Algorithm
Magnetic resonance
Nonuniform Sampling
Internal
Imaging techniques
Synthetic Aperture
Coefficient
Synthetic aperture radar
Convolution
Radar
Fast Algorithm
Low Frequency
Smoothness
Brain
High Resolution

Keywords

  • Convolutional gridding
  • Edge detection
  • Fourier data
  • Non-uniform fast Fourier transform

ASJC Scopus subject areas

  • Software
  • Computational Theory and Mathematics
  • Theoretical Computer Science
  • Engineering(all)

Cite this

Edge Detection from Non-Uniform Fourier Data Using the Convolutional Gridding Algorithm. / Martinez, Adam; Gelb, Anne; Gutierrez, Alexander.

In: Journal of Scientific Computing, Vol. 61, No. 3, 14.10.2014, p. 490-512.

Research output: Contribution to journalArticle

Martinez, Adam ; Gelb, Anne ; Gutierrez, Alexander. / Edge Detection from Non-Uniform Fourier Data Using the Convolutional Gridding Algorithm. In: Journal of Scientific Computing. 2014 ; Vol. 61, No. 3. pp. 490-512.
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