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 (·), depending on the small scale . It is shown that odd kernels, properly scaled, and admissible (in the sense of having small W−1,∞moments of order O( )) satisfy K ∗ f(x) = [f ](x) + O( ), thus recovering both the location and amplitudes of all edges. As an example we consider general concentration kernels of the form Kσ N (t) = ∑ σ(k/N) sin kt to detect edges from the first 1/ = 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 ∗ f(x) ∼ [f ](x) = 0, and the smooth regions where K ∗ f = O( ) ∼ 0. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors.