Detection of Edges in Spectral Data II. Nonlinear Enhancement

Details Detection of Edges in Spectral Data II. Nonlinear Enhancement Detection of Edges in Spectral Data II. Nonlinear Enhancement

2001-12-03

arxiv.org/abs/math/0112016v1

SIAM Journal of Numerical Analysis 38(4), (2000), 1389-1408

Mathematics

Numerical Analysis

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Detection of Edges in Spectral Data II. Nonlinear Enhancement does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Detection of Edges in Spectral Data II. Nonlinear Enhancement, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Detection of Edges in Spectral Data II. Nonlinear Enhancement will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-235265