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Detection of Edges in Spectral Data II. Nonlinear Enhancement | | Anne Gelb
; Eitan Tadmor
; | | Date: |
3 Dec 2001 | | Journal: | SIAM Journal of Numerical Analysis 38(4), (2000), 1389-1408 | | Subject: | Numerical Analysis MSC-class: 42A10; 42A50; 65T10 | math.NA | | Abstract: | We discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where $[f](x):=f(x+)-f(x-)
eq 0$. 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 $K^sigma_N(t)=sumsigma(k/N)sin kt$ 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 $K_epsilon*f(x)sim [f](x)
eq 0$, 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. | | Source: | arXiv, math.NA/0112016 | | Services: | Forum | Review | PDF | Favorites | |
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