Mathematics
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Item RECOVERY OF EDGES FROM SPECTRAL DATA WITH NOISE—A NEW PERSPECTIVE(Copyright: Society for Industrial and Applied Mathematics, 2008) ENGELBERG, SHLOMO; TADMOR, EITANWe consider the problem of detecting edges—jump discontinuities in piecewise smooth functions from their N-degree spectral content, which is assumed to be corrupted by noise. There are three scales involved: the “smoothness” scale of order 1/N, the noise scale of order √η, and the O(1) scale of the jump discontinuities. We use concentration factors which are adjusted to the standard deviation of the noise √η ≫ 1/N in order to detect the underlying O(1)-edges, which are separated from the noise scale √η ≪ 1.Item Detection of Edges in Spectral Data II. Nonlinear Enhancement(Copyright: Society for Industrial and Applied Mathematics, 2000) GELB, ANNE; TADMOR, EITANWe 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.