Spectral edge detection in two dimensions using wavefronts

Leslie Greengard, C. Stucchio

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)69-95
Number of pages27
JournalApplied and Computational Harmonic Analysis
Volume30
Issue number1
DOIs
StatePublished - Jan 2011

Fingerprint

Edge Detection
Edge detection
Wavefronts
Wave Front
Two Dimensions
Wavefront Reconstruction
Approximation Theory
Magnetic Resonance Imaging
Approximation theory
Unit Sphere
Singular Point
Set of points
One Dimension
Higher Dimensions
Image Processing
Magnetic resonance
Fourier transform
Discontinuity
High-dimensional
Segmentation

Keywords

  • Directional filter
  • Edge detection
  • MRI
  • Segmentation
  • Wavefront

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

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

In: Applied and Computational Harmonic Analysis, Vol. 30, No. 1, 01.2011, p. 69-95.

Research output: Contribution to journalArticle

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