Detection of edges in spectral data II. Nonlinear enhancement

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.