Detection of Edges in Spectral Data II. Nonlinear Enhancement

Abstract

We discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where f := 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. Itis shown that odd kernels, properly scaled, and admissible (in the sense of having small W−1,∞- moments of order O(𝛆)) satisfy K_𝛆 ∗ f(x) = f + O(𝛆), thus recovering both the location and amplitudes of all edges. As an example we consider general concentration kernels of the form KσN (t) = 𝝨σ(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, 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_𝛆 ∗ f(x) ∼ f ≠ 0, and the smooth regions where K_𝛆 ∗ f = O(𝛆) ∼ 0. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors.