Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions
Mathematics of Computation
On the Gibbs Phenomenon and Its Resolution
SIAM Review
On a high order numerical method for functions with singularities
Mathematics of Computation
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
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
Inverse Polynomial Reconstruction of Two Dimensional Fourier Images
Journal of Scientific Computing
Reconstruction of Piecewise Smooth Functions from Non-uniform Grid Point Data
Journal of Scientific Computing
Detection of Edges in Spectral Data III--Refinement of the Concentration Method
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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Fourier spectral methods have proven to be powerful tools that are frequently employed in image reconstruction. However, since images can be typically viewed as piecewise smooth functions, the Gibbs phenomenon often hinders accurate reconstruction. Recently, numerical edge detection and reconstruction methods have been developed that effectively reduce the Gibbs oscillations while maintaining high resolution accuracy at the edges. While the Gibbs phenomenon is a standard obstacle for the recovery of all piecewise smooth functions, in many image reconstruction problems there is the additional impediment of random noise existing within the spectral data. This paper addresses the issue of noise in image reconstruction and its effects on the ability to locate the edges and recover the image. The resulting numerical method not only recovers piecewise smooth functions with very high accuracy, but it is also robust in the presence of noise.