Adaptive edge detectors for piecewise smooth data based on the minmod limiter

Anne Gelb, E. Tadmor

Research output: Contribution to journalArticle

41 Citations (Scopus)

Abstract

We are concerned with the detection of edges-the location and amplitudes of jump discontinuities of piecewise smooth data realized in terms of its discrete grid values. We discuss the interplay between two approaches. One approach, realized in the physical space, is based on local differences and is typically limited to low-order of accuracy. An alternative approach developed in our previous work [Gelb and Tadmor, Appl. Comp. Harmonic Anal., 7, 101-135 (1999)] and realized in the dual Fourier space, is based on concentration factors; with a proper choice of concentration factors one can achieve higher-orders-in fact in [Gelb and Tadmor, SIAM J. Numer. Anal., 38, 1389-1408 (2001)] we constructed exponentially accurate edge detectors. Since the stencil of these highly-accurate detectors is global, an outside threshold parameter is required to avoid oscillations in the immediate neighborhood of discontinuities. In this paper we introduce an adaptive edge detection procedure based on a cross-breading between the local and global detectors. This is achieved by using the minmod limiter to suppress spurious oscillations near discontinuities while retaining high-order accuracy away from the jumps. The resulting method provides a family of robust, parameter-free edge-detectors for piecewise smooth data. We conclude with a series of one- and two-dimensional simulations.

Original languageEnglish (US)
Pages (from-to)279-306
Number of pages28
JournalJournal of Scientific Computing
Volume28
Issue number2-3
DOIs
StatePublished - Sep 2006

Fingerprint

Limiter
Limiters
Detector
Detectors
Discontinuity
Jump
Oscillation
Threshold Parameter
High Order Accuracy
Edge Detection
Edge detection
Harmonic
Higher Order
Grid
Series
Alternatives
Simulation

Keywords

  • Concentration method
  • Edge detection
  • Local difference formulas
  • Minmod algorithm
  • Piecewise smoothness

ASJC Scopus subject areas

  • Computer Networks and Communications
  • Hardware and Architecture
  • Software

Cite this

Adaptive edge detectors for piecewise smooth data based on the minmod limiter. / Gelb, Anne; Tadmor, E.

In: Journal of Scientific Computing, Vol. 28, No. 2-3, 09.2006, p. 279-306.

Research output: Contribution to journalArticle

@article{37240bdcda0e463595a9c891678acbac,
title = "Adaptive edge detectors for piecewise smooth data based on the minmod limiter",
abstract = "We are concerned with the detection of edges-the location and amplitudes of jump discontinuities of piecewise smooth data realized in terms of its discrete grid values. We discuss the interplay between two approaches. One approach, realized in the physical space, is based on local differences and is typically limited to low-order of accuracy. An alternative approach developed in our previous work [Gelb and Tadmor, Appl. Comp. Harmonic Anal., 7, 101-135 (1999)] and realized in the dual Fourier space, is based on concentration factors; with a proper choice of concentration factors one can achieve higher-orders-in fact in [Gelb and Tadmor, SIAM J. Numer. Anal., 38, 1389-1408 (2001)] we constructed exponentially accurate edge detectors. Since the stencil of these highly-accurate detectors is global, an outside threshold parameter is required to avoid oscillations in the immediate neighborhood of discontinuities. In this paper we introduce an adaptive edge detection procedure based on a cross-breading between the local and global detectors. This is achieved by using the minmod limiter to suppress spurious oscillations near discontinuities while retaining high-order accuracy away from the jumps. The resulting method provides a family of robust, parameter-free edge-detectors for piecewise smooth data. We conclude with a series of one- and two-dimensional simulations.",
keywords = "Concentration method, Edge detection, Local difference formulas, Minmod algorithm, Piecewise smoothness",
author = "Anne Gelb and E. Tadmor",
year = "2006",
month = "9",
doi = "10.1007/s10915-006-9088-6",
language = "English (US)",
volume = "28",
pages = "279--306",
journal = "Journal of Scientific Computing",
issn = "0885-7474",
publisher = "Springer New York",
number = "2-3",

}

TY - JOUR

T1 - Adaptive edge detectors for piecewise smooth data based on the minmod limiter

AU - Gelb, Anne

AU - Tadmor, E.

PY - 2006/9

Y1 - 2006/9

N2 - We are concerned with the detection of edges-the location and amplitudes of jump discontinuities of piecewise smooth data realized in terms of its discrete grid values. We discuss the interplay between two approaches. One approach, realized in the physical space, is based on local differences and is typically limited to low-order of accuracy. An alternative approach developed in our previous work [Gelb and Tadmor, Appl. Comp. Harmonic Anal., 7, 101-135 (1999)] and realized in the dual Fourier space, is based on concentration factors; with a proper choice of concentration factors one can achieve higher-orders-in fact in [Gelb and Tadmor, SIAM J. Numer. Anal., 38, 1389-1408 (2001)] we constructed exponentially accurate edge detectors. Since the stencil of these highly-accurate detectors is global, an outside threshold parameter is required to avoid oscillations in the immediate neighborhood of discontinuities. In this paper we introduce an adaptive edge detection procedure based on a cross-breading between the local and global detectors. This is achieved by using the minmod limiter to suppress spurious oscillations near discontinuities while retaining high-order accuracy away from the jumps. The resulting method provides a family of robust, parameter-free edge-detectors for piecewise smooth data. We conclude with a series of one- and two-dimensional simulations.

AB - We are concerned with the detection of edges-the location and amplitudes of jump discontinuities of piecewise smooth data realized in terms of its discrete grid values. We discuss the interplay between two approaches. One approach, realized in the physical space, is based on local differences and is typically limited to low-order of accuracy. An alternative approach developed in our previous work [Gelb and Tadmor, Appl. Comp. Harmonic Anal., 7, 101-135 (1999)] and realized in the dual Fourier space, is based on concentration factors; with a proper choice of concentration factors one can achieve higher-orders-in fact in [Gelb and Tadmor, SIAM J. Numer. Anal., 38, 1389-1408 (2001)] we constructed exponentially accurate edge detectors. Since the stencil of these highly-accurate detectors is global, an outside threshold parameter is required to avoid oscillations in the immediate neighborhood of discontinuities. In this paper we introduce an adaptive edge detection procedure based on a cross-breading between the local and global detectors. This is achieved by using the minmod limiter to suppress spurious oscillations near discontinuities while retaining high-order accuracy away from the jumps. The resulting method provides a family of robust, parameter-free edge-detectors for piecewise smooth data. We conclude with a series of one- and two-dimensional simulations.

KW - Concentration method

KW - Edge detection

KW - Local difference formulas

KW - Minmod algorithm

KW - Piecewise smoothness

UR - http://www.scopus.com/inward/record.url?scp=33747884037&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33747884037&partnerID=8YFLogxK

U2 - 10.1007/s10915-006-9088-6

DO - 10.1007/s10915-006-9088-6

M3 - Article

AN - SCOPUS:33747884037

VL - 28

SP - 279

EP - 306

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

IS - 2-3

ER -