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| Main Authors: | , |
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| Format: | Preprint |
| Udgivet: |
2001
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| Fag: | |
| Online adgang: | https://arxiv.org/abs/math/0112016 |
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Indholdsfortegnelse:
- 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, $K_ε(\cdot)$, depending on the small scale $ε$. It is shown that odd kernels, properly scaled, and admissible (in the sense of having small $W^{-1,\infty}$-moments of order ${\cal O}(ε)$) satisfy $K_ε*f(x) = [f](x) +{\cal O}(ε)$, thus recovering both the location and amplitudes of all edges.As an example we consider general concentration kernels of the form $K^σ_N(t)=\sumσ(k/N)\sin kt$ 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, $σ^{exp}(\cdot)$, 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 $K_ε*f(x)\sim [f](x) \neq 0$, and the smooth regions where $K_ε*f = {\cal O}(ε) \sim 0$. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors.