On the Gibbs Phenomenon and Its Resolution
SIAM Review
Detection of Edges in Spectral Data II. Nonlinear Enhancement
SIAM Journal on Numerical Analysis
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
Detection of Edges in Spectral Data III--Refinement of the Concentration Method
Journal of Scientific Computing
Recovery of Edges from Spectral Data with Noise—A New Perspective
SIAM Journal on Numerical Analysis
A New Alternating Minimization Algorithm for Total Variation Image Reconstruction
SIAM Journal on Imaging Sciences
Improved Total Variation-Type Regularization Using Higher Order Edge Detectors
SIAM Journal on Imaging Sciences
Nonuniform fast Fourier transforms using min-max interpolation
IEEE Transactions on Signal Processing
IEEE Transactions on Information Theory
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
IEEE Transactions on Information Theory
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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We present a new method for estimating the edges in a piecewise smooth function from blurred and noisy Fourier data. The proposed method is constructed by combining the so called concentration factor edge detection method, which uses a finite number of Fourier coefficients to approximate the jump function of a piecewise smooth function, with compressed sensing ideas. Due to the global nature of the concentration factor method, Gibbs oscillations feature prominently near the jump discontinuities. This can cause the misidentification of edges when simple thresholding techniques are used. In fact, the true jump function is sparse, i.e. zero almost everywhere with non-zero values only at the edge locations. Hence we adopt an idea from compressed sensing and propose a method that uses a regularized deconvolution to remove the artifacts. Our new method is fast, in the sense that it only needs the solution of a single l 1 minimization. Numerical examples demonstrate the accuracy and robustness of the method in the presence of noise and blur.