A Hybrid Approach to Spectral Reconstruction of Piecewise Smooth Functions
Journal of Scientific Computing
Reducing the Effects of Noise in Image Reconstruction
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The spectral signal processing suite
ACM Transactions on Mathematical Software (TOMS)
Chebyshev super spectral viscosity method for a fluidized bed model
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Towards the resolution of the Gibbs phenomena
Journal of Computational and Applied Mathematics
Parameter Optimization and Reduction of Round Off Error for the Gegenbauer Reconstruction Method
Journal of Scientific Computing
On inverse methods for the resolution of the Gibbs phenomenon
Journal of Computational and Applied Mathematics
Reduction of the Gibbs phenomenon for smooth functions with jumps by the ε-algorithm
Journal of Computational and Applied Mathematics
Detection of Edges in Spectral Data III--Refinement of the Concentration Method
Journal of Scientific Computing
A wavelet-based method for overcoming the Gibbs phenomenon
MATH'08 Proceedings of the American Conference on Applied Mathematics
Applied Numerical Mathematics
Edge Detection Free Postprocessing for Pseudospectral Approximations
Journal of Scientific Computing
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ACM Transactions on Mathematical Software (TOMS)
Iterative methods based on spline approximations to detect discontinuities from Fourier data
Journal of Computational and Applied Mathematics
On Reconstruction from Non-uniform Spectral Data
Journal of Scientific Computing
Iterative adaptive RBF methods for detection of edges in two-dimensional functions
Applied Numerical Mathematics
Edge Detection by Adaptive Splitting
Journal of Scientific Computing
Detecting discontinuity points from spectral data with the quotient-difference (qd) algorithm
Journal of Computational and Applied Mathematics
Sparsity Enforcing Edge Detection Method for Blurred and Noisy Fourier data
Journal of Scientific Computing
A note on singularity approximation
Mathematical and Computer Modelling: An International Journal
Hypothesis Testing for Fourier Based Edge Detection Methods
Journal of Scientific Computing
Iterative Design of Concentration Factors for Jump Detection
Journal of Scientific Computing
Adaptive Spectral Viscosity for Hyperbolic Conservation Laws
SIAM Journal on Scientific Computing
Edge detection from truncated Fourier data using spectral mollifiers
Advances in Computational Mathematics
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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.