Detection of edges in spectral data II. Nonlinear enhancement

discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where [f](x) : = f(x+) - f(x-) not equal 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 (.), depending on the small scale epsilon. It is shown that odd kernels, properly scaled, and admissible ( in the sense of having small W--1,W-infinity - moments of order O(epsilon)) satisfy K-epsilon * f(x) = [f](x) + O(epsilon), thus recovering both the location and amplitudes of all edges. As an example we consider general concentration kernels of the form K-N(sigma) (t) = Sigma 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. e identify, in particular, a new family of exponential factors, sigma (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-epsilon * f(x) similar to [f](x) not equal 0, and the smooth regions where K-epsilon * f = O(epsilon) similar to 0. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors.