Detection of Edges in Spectral Data II. Nonlinear Enhancement

dc.contributor.authorGELB, ANNE
dc.contributor.authorTADMOR, EITAN
dc.date.accessioned2008-10-20T17:59:29Z
dc.date.available2008-10-20T17:59:29Z
dc.date.issued2000
dc.description.abstractWe 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. Itis shown that odd kernels, properly scaled, and admissible (in the sense of having small W−1,∞- moments of order O(𝛆)) satisfy K_𝛆 ∗ f(x) = [f](x) + O(𝛆), thus recovering both the location and amplitudes of all edges. As an example we consider general concentration kernels of the form KσN (t) = 𝝨σ(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 K_𝛆 ∗ f(x) ∼ [f](x) ≠ 0, and the smooth regions where K_𝛆 ∗ f = O(𝛆) ∼ 0. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors.en
dc.format.extent245473 bytes
dc.format.mimetypeapplication/pdf
dc.identifier.citationA. Gelb & E. Tadmor (2000). Detection of Edges in Spectral Data II. Nonlinear Enhancement. SIAM Journal on Numerical Analysis 38 (2000) 1389-1408.en
dc.identifier.urihttp://hdl.handle.net/1903/8646
dc.language.isoen_USen
dc.publisherCopyright: Society for Industrial and Applied Mathematicsen
dc.relation.isAvailableAtCollege of Computer, Mathematical & Physical Sciencesen_us
dc.relation.isAvailableAtMathematicsen_us
dc.relation.isAvailableAtDigital Repository at the University of Marylanden_us
dc.relation.isAvailableAtUniversity of Maryland (College Park, MD)en_us
dc.subjectpiecewise smoothnessen
dc.subjectconcentration kernelsen
dc.subjectspectral expansionsen
dc.titleDetection of Edges in Spectral Data II. Nonlinear Enhancementen
dc.typeArticleen
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