  We discuss a general framework for recovering edges in piecewise smooth
functions with finitely many jump discontinuities, where $[f](x):=f(x+)-f(x-)
\neq 0$. 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, $K_\epsilon(\cdot)$,
depending on the small scale $\epsilon$. It is shown that odd kernels, properly
scaled, and admissible (in the sense of having small $W^{-1,\infty}$-moments of
order ${\cal O}(\epsilon)$) satisfy $K_\epsilon*f(x) = [f](x) +{\cal
O}(\epsilon)$, thus recovering both the location and amplitudes of all edges.As
an example we consider general concentration kernels of the form
$K^\sigma_N(t)=\sum\sigma(k/N)\sin kt$ 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. We identify, in particular, a
new family of exponential factors, $\sigma^{exp}(\cdot)$, 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
$K_\epsilon*f(x)\sim [f](x) \neq 0$, and the smooth regions where $K_\epsilon*f
= {\cal O}(\epsilon) \sim 0$. Numerical examples demonstrate that by coupling
concentration kernels with nonlinear enhancement one arrives at effective edge
detectors.
