##### Abstract

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.
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](x) + 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](x) ≠ 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.