Detection of Edges in Spectral Data II. Nonlinear Enhancement

Abstract

We discuss a general framework for recovering edges in piecewise smooth functionswith finitely many jump discontinuities, where [f](x) := f(x+)−f(x−) = 0. Our approach is basedon two main aspects—localization using appropriate concentration kernels and separation of scalesby nonlinear enhancement.To detect such edges, one employs concentration kernels, K(·), depending on the small scale .Itis 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 andamplitudes of all edges. As an example we consider general concentration kernels of the form KσN(t) =σ(k/N) sin kt to detect edges from the first 1/ = N spectral modes of piecewise smooth f’s. Herewe 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 projectionsare considered. We identify, in particular, a new family of exponential factors, σexp(·), with superiorlocalization properties.The other aspect of our edge detection involves a nonlinear enhancement procedure which isbased on separation of scales between the edges, where K ∗ f(x) ∼ [f](x) = 0, and the smoothregions where K ∗ f = O() ∼ 0. Numerical examples demonstrate that by coupling concentrationkernels with nonlinear enhancement one arrives at effective edge detectors.