Detection of Edges in Spectral Data II. Nonlinear Enhancement
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Abstract
We discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where f := f(x+)โf(xโ) โ 0. Our approach is based on two main aspectsโlocalization using appropriate concentration kernels and separation of scales by nonlinear enhancement. To detect such edges, one employs concentration kernels, K_๐(ยท), depending on the small scale ๐. Itis shown that odd kernels, properly scaled, and admissible (in the sense of having small Wโ1,โ- moments of order O(๐)) satisfy K_๐ โ f(x) = f + O(๐), thus recovering both the location and amplitudes of all edges. As an example we consider general concentration kernels of the form KฯN (t) = ๐จฯ(k/N) sin kt to detect edges from the first 1/๐ = N spectral modes of piecewise smooth fโs. Here we improve in generality and simplicity over our previous study in [A. Gelb and E. Tadmor, Appl. Comput. Harmon. Anal., 7 (1999), pp. 101โ135]. Both periodic and nonperiodic spectral projections are considered. We identify, in particular, a new family of exponential factors, ฯexp(ยท), 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_๐ โ f(x) โผ f โ 0, and the smooth regions where K_๐ โ f = O(๐) โผ 0. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors.