### Abstract

We discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where [f](x) := f(x+) - f(x-) ≠ 0. Our approach is based on two main aspects - localization using appropriate concentration kernels and separation of scales by nonlinear enhancement. To detect such edges, one employs concentration kernels, K_{ε}(·), depending on the small scale ε. It is shown that odd kernels, properly scaled, and admissible (in the sense of having small W^{-1,∞}-moments of order script O sign(ε)) satisfy Kε * f(x) = [f](x) + script O sign(ε), thus recovering both the location and amplitudes of all edges. As an example we consider general concentration kernels of the form K^{σ}
_{N}(t) = Σ σ(k/N) sin kt to detect edges from the first 1/ε = N spectral modes of piecewise smooth f's. Here we improve in generality and simplicity over our previous study in [A. Gelb and E. Tadmor, Appl. Comput. Harmon. Anal., 7 (1999), pp. 101-135]. Both periodic and nonperiodic spectral projections are considered. We identify, in particular, a new family of exponential factors, σ^{exp}(·), with superior localization properties. The other aspect of our edge detection involves a nonlinear enhancement procedure which is based on separation of scales between the edges, where K_{ε} * f(x) ∼ [f](x) ≠ 0, and the smooth regions where K_{ε} * f = script O sign(ε) ∼ 0. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors.

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

Pages (from-to) | 1389-1408 |

Number of pages | 20 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 38 |

Issue number | 4 |

DOIs | |

State | Published - 2001 |

### Fingerprint

### Keywords

- Concentration kernels
- Piecewise smoothness
- Spectral expansions

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics
- Computational Mathematics

### Cite this

*SIAM Journal on Numerical Analysis*,

*38*(4), 1389-1408. https://doi.org/10.1137/S0036142999359153

**Detection of edges in spectral data II. Nonlinear enhancement.** / Gelb, Anne; Tadmor, Eitan.

Research output: Contribution to journal › Article

*SIAM Journal on Numerical Analysis*, vol. 38, no. 4, pp. 1389-1408. https://doi.org/10.1137/S0036142999359153

}

TY - JOUR

T1 - Detection of edges in spectral data II. Nonlinear enhancement

AU - Gelb, Anne

AU - Tadmor, Eitan

PY - 2001

Y1 - 2001

N2 - We discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where [f](x) := f(x+) - f(x-) ≠ 0. Our approach is based on two main aspects - localization using appropriate concentration kernels and separation of scales by nonlinear enhancement. To detect such edges, one employs concentration kernels, Kε(·), depending on the small scale ε. It is shown that odd kernels, properly scaled, and admissible (in the sense of having small W-1,∞-moments of order script O sign(ε)) satisfy Kε * f(x) = [f](x) + script O sign(ε), thus recovering both the location and amplitudes of all edges. As an example we consider general concentration kernels of the form Kσ N(t) = Σ σ(k/N) sin kt to detect edges from the first 1/ε = N spectral modes of piecewise smooth f's. Here we improve in generality and simplicity over our previous study in [A. Gelb and E. Tadmor, Appl. Comput. Harmon. Anal., 7 (1999), pp. 101-135]. Both periodic and nonperiodic spectral projections are considered. We identify, in particular, a new family of exponential factors, σexp(·), with superior localization properties. The other aspect of our edge detection involves a nonlinear enhancement procedure which is based on separation of scales between the edges, where Kε * f(x) ∼ [f](x) ≠ 0, and the smooth regions where Kε * f = script O sign(ε) ∼ 0. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors.

AB - We discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where [f](x) := f(x+) - f(x-) ≠ 0. Our approach is based on two main aspects - localization using appropriate concentration kernels and separation of scales by nonlinear enhancement. To detect such edges, one employs concentration kernels, Kε(·), depending on the small scale ε. It is shown that odd kernels, properly scaled, and admissible (in the sense of having small W-1,∞-moments of order script O sign(ε)) satisfy Kε * f(x) = [f](x) + script O sign(ε), thus recovering both the location and amplitudes of all edges. As an example we consider general concentration kernels of the form Kσ N(t) = Σ σ(k/N) sin kt to detect edges from the first 1/ε = N spectral modes of piecewise smooth f's. Here we improve in generality and simplicity over our previous study in [A. Gelb and E. Tadmor, Appl. Comput. Harmon. Anal., 7 (1999), pp. 101-135]. Both periodic and nonperiodic spectral projections are considered. We identify, in particular, a new family of exponential factors, σexp(·), with superior localization properties. The other aspect of our edge detection involves a nonlinear enhancement procedure which is based on separation of scales between the edges, where Kε * f(x) ∼ [f](x) ≠ 0, and the smooth regions where Kε * f = script O sign(ε) ∼ 0. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors.

KW - Concentration kernels

KW - Piecewise smoothness

KW - Spectral expansions

UR - http://www.scopus.com/inward/record.url?scp=0034431580&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034431580&partnerID=8YFLogxK

U2 - 10.1137/S0036142999359153

DO - 10.1137/S0036142999359153

M3 - Article

AN - SCOPUS:0034431580

VL - 38

SP - 1389

EP - 1408

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 4

ER -