### Abstract

A recurring task in image processing, approximation theory, and the numerical solution of partial differential equations is to reconstruct a piecewise-smooth real-valued function f(x), where xℝN, from its truncated Fourier transform (its truncated spectrum). An essential step is edge detection for which a variety of one-dimensional schemes have been developed over the last few decades. Most higher-dimensional edge detection algorithms consist of applying one-dimensional detectors in each component direction in order to recover the locations in RN where f(x) is singular (the singular support). In this paper, we present a multidimensional algorithm which identifies the wavefront of a function from spectral data. The wavefront of f is the set of points (x,k→)ℝN×(SN-1/{±1}) which encode both the location of the singular points of a function and the orientation of the singularities. (Here SN-1 denotes the unit sphere in N dimensions.) More precisely, k→ is the direction of the normal line to the curve or surface of discontinuity at x. Note that the singular support is simply the projection of the wavefront onto its x-component. In one dimension, the wavefront is a subset of R1×(S0/{±1})=R, and it coincides with the singular support. In higher dimensions, geometry comes into play and they are distinct. We discuss the advantages of wavefront reconstruction and indicate how it can be used for segmentation in magnetic resonance imaging (MRI).

Original language | English (US) |
---|---|

Pages (from-to) | 69-95 |

Number of pages | 27 |

Journal | Applied and Computational Harmonic Analysis |

Volume | 30 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2011 |

### Fingerprint

### Keywords

- Directional filter
- Edge detection
- MRI
- Segmentation
- Wavefront

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Applied and Computational Harmonic Analysis*,

*30*(1), 69-95. https://doi.org/10.1016/j.acha.2010.02.007

**Spectral edge detection in two dimensions using wavefronts.** / Greengard, Leslie; Stucchio, C.

Research output: Contribution to journal › Article

*Applied and Computational Harmonic Analysis*, vol. 30, no. 1, pp. 69-95. https://doi.org/10.1016/j.acha.2010.02.007

}

TY - JOUR

T1 - Spectral edge detection in two dimensions using wavefronts

AU - Greengard, Leslie

AU - Stucchio, C.

PY - 2011/1

Y1 - 2011/1

N2 - A recurring task in image processing, approximation theory, and the numerical solution of partial differential equations is to reconstruct a piecewise-smooth real-valued function f(x), where xℝN, from its truncated Fourier transform (its truncated spectrum). An essential step is edge detection for which a variety of one-dimensional schemes have been developed over the last few decades. Most higher-dimensional edge detection algorithms consist of applying one-dimensional detectors in each component direction in order to recover the locations in RN where f(x) is singular (the singular support). In this paper, we present a multidimensional algorithm which identifies the wavefront of a function from spectral data. The wavefront of f is the set of points (x,k→)ℝN×(SN-1/{±1}) which encode both the location of the singular points of a function and the orientation of the singularities. (Here SN-1 denotes the unit sphere in N dimensions.) More precisely, k→ is the direction of the normal line to the curve or surface of discontinuity at x. Note that the singular support is simply the projection of the wavefront onto its x-component. In one dimension, the wavefront is a subset of R1×(S0/{±1})=R, and it coincides with the singular support. In higher dimensions, geometry comes into play and they are distinct. We discuss the advantages of wavefront reconstruction and indicate how it can be used for segmentation in magnetic resonance imaging (MRI).

AB - A recurring task in image processing, approximation theory, and the numerical solution of partial differential equations is to reconstruct a piecewise-smooth real-valued function f(x), where xℝN, from its truncated Fourier transform (its truncated spectrum). An essential step is edge detection for which a variety of one-dimensional schemes have been developed over the last few decades. Most higher-dimensional edge detection algorithms consist of applying one-dimensional detectors in each component direction in order to recover the locations in RN where f(x) is singular (the singular support). In this paper, we present a multidimensional algorithm which identifies the wavefront of a function from spectral data. The wavefront of f is the set of points (x,k→)ℝN×(SN-1/{±1}) which encode both the location of the singular points of a function and the orientation of the singularities. (Here SN-1 denotes the unit sphere in N dimensions.) More precisely, k→ is the direction of the normal line to the curve or surface of discontinuity at x. Note that the singular support is simply the projection of the wavefront onto its x-component. In one dimension, the wavefront is a subset of R1×(S0/{±1})=R, and it coincides with the singular support. In higher dimensions, geometry comes into play and they are distinct. We discuss the advantages of wavefront reconstruction and indicate how it can be used for segmentation in magnetic resonance imaging (MRI).

KW - Directional filter

KW - Edge detection

KW - MRI

KW - Segmentation

KW - Wavefront

UR - http://www.scopus.com/inward/record.url?scp=78649335852&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78649335852&partnerID=8YFLogxK

U2 - 10.1016/j.acha.2010.02.007

DO - 10.1016/j.acha.2010.02.007

M3 - Article

VL - 30

SP - 69

EP - 95

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

SN - 1063-5203

IS - 1

ER -