Detection of Edges in Spectral Data II. Nonlinear Enhancement
Detection of Edges in Spectral Data II. Nonlinear Enhancement
Loading...
Files
Publication or External Link
Date
2000
Authors
GELB, ANNE
TADMOR, EITAN
Advisor
Citation
A. Gelb & E. Tadmor (2000). Detection of Edges in Spectral Data II. Nonlinear Enhancement. SIAM Journal on Numerical Analysis 38 (2000) 1389-1408.
DRUM DOI
Abstract
We discuss a general framework for recovering edges in piecewise smooth functions
with finitely many jump discontinuities, where [f](x) := f(x+)โf(xโ) โ 0. Our approach is based
on two main aspectsโlocalization using appropriate concentration kernels and separation of scales
by nonlinear enhancement.
To detect such edges, one employs concentration kernels, K_๐(ยท), depending on the small scale ๐.
Itis shown that odd kernels, properly scaled, and admissible (in the sense of having small Wโ1,โ-
moments of order O(๐)) satisfy K_๐ โ f(x) = [f](x) + O(๐), thus recovering both the location and
amplitudes of all edges. As an example we consider general concentration kernels of the form KฯN
(t) =
๐จฯ(k/N) sin kt to detect edges from the first 1/๐ = N spectral modes of piecewise smooth fโs. Here
we improve in generality and simplicity over our previous study in [A. Gelb and E. Tadmor, Appl.
Comput. Harmon. Anal., 7 (1999), pp. 101โ135]. Both periodic and nonperiodic spectral projections
are considered. We identify, in particular, a new family of exponential factors, ฯexp(ยท), 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_๐ โ f(x) โผ [f](x) โ 0, and the smooth
regions where K_๐ โ f = O(๐) โผ 0. Numerical examples demonstrate that by coupling concentration
kernels with nonlinear enhancement one arrives at effective edge detectors.