{
  "id": "math/0112016",
  "version": "v1",
  "published": "2001-12-03T07:30:32.000Z",
  "updated": "2001-12-03T07:30:32.000Z",
  "title": "Detection of Edges in Spectral Data II. Nonlinear Enhancement",
  "authors": [
    "Anne Gelb",
    "Eitan Tadmor"
  ],
  "journal": "SIAM Journal of Numerical Analysis 38(4), (2000), 1389-1408",
  "categories": [
    "math.NA"
  ],
  "abstract": "We discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where $[f](x):=f(x+)-f(x-) \\neq 0$. 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, $K_\\epsilon(\\cdot)$, depending on the small scale $\\epsilon$. It is shown that odd kernels, properly scaled, and admissible (in the sense of having small $W^{-1,\\infty}$-moments of order ${\\cal O}(\\epsilon)$) satisfy $K_\\epsilon*f(x) = [f](x) +{\\cal O}(\\epsilon)$, thus recovering both the location and amplitudes of all edges.As an example we consider general concentration kernels of the form $K^\\sigma_N(t)=\\sum\\sigma(k/N)\\sin kt$ to detect edges from the first $1/\\epsilon=N$ 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, $\\sigma^{exp}(\\cdot)$, 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 $K_\\epsilon*f(x)\\sim [f](x) \\neq 0$, and the smooth regions where $K_\\epsilon*f = {\\cal O}(\\epsilon) \\sim 0$. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-12-03T07:30:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A10",
      "42A50",
      "65T10"
    ],
    "keywords": [
      "spectral data",
      "nonlinear enhancement procedure",
      "appropriate concentration kernels",
      "employs concentration kernels",
      "nonperiodic spectral projections"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math.....12016G"
    }
  }
}