A Computational Approach to Edge Detection
IEEE Transactions on Pattern Analysis and Machine Intelligence
A practical assessment of spectral accuracy for hyperbolic problems with discontinuity
Journal of Scientific Computing
Fundamentals of digital image processing
Fundamentals of digital image processing
Authenticating Edges Produced by Zero-Crossing Algorithms
IEEE Transactions on Pattern Analysis and Machine Intelligence
Refining edges detected by a LoG operator
Computer Vision, Graphics, and Image Processing
Family of spectral filters for discontinuous problems
Journal of Scientific Computing
Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions
Mathematics of Computation
On the Gibbs Phenomenon and Its Resolution
SIAM Review
Determination of the jumps of a bounded function by its Fourier series
Journal of Approximation Theory
On a high order numerical method for functions with singularities
Mathematics of Computation
Journal of Scientific Computing
Enhanced spectral viscosity approximations for conservation laws
Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
Detection of Edges in Spectral Data II. Nonlinear Enhancement
SIAM Journal on Numerical Analysis
Reducing the Effects of Noise in Image Reconstruction
Journal of Scientific Computing
Towards the resolution of the Gibbs phenomena
Journal of Computational and Applied Mathematics
Adaptive Edge Detectors for Piecewise Smooth Data Based on the minmod Limiter
Journal of Scientific Computing
Edge Detection Free Postprocessing for Pseudospectral Approximations
Journal of Scientific Computing
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
Hypothesis Testing for Fourier Based Edge Detection Methods
Journal of Scientific Computing
Iterative Design of Concentration Factors for Jump Detection
Journal of Scientific Computing
Edge detection from truncated Fourier data using spectral mollifiers
Advances in Computational Mathematics
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Edge detection from Fourier spectral data is important in many applications including image processing and the post-processing of solutions to numerical partial differential equations. The concentration method, introduced by Gelb and Tadmor in 1999, locates jump discontinuities in piecewise smooth functions from their Fourier spectral data. However, as is true for all global techniques, the method yields strong oscillations near the jump discontinuities, which makes it difficult to distinguish true discontinuities from artificial oscillations. This paper introduces refinements to the concentration method to reduce the oscillations. These refinements also improve the results in noisy environments. One technique adds filtering to the concentration method. Another uses convolution to determine the strongest correlations between the waveform produced by the concentration method and the one produced by the jump function approximation of an indicator function. A zero crossing based concentration factor, which creates a more localized formulation of the jump function approximation, is also introduced. Finally, the effects of zero-mean white Gaussian noise on the refined concentration method are analyzed. The investigation confirms that by applying the refined techniques, the variance of the concentration method is significantly reduced in the presence of noise.