A Computational Approach to Edge Detection
IEEE Transactions on Pattern Analysis and Machine Intelligence
On the Gibbs Phenomenon and Its Resolution
SIAM Review
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
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
Hypothesis Testing for Fourier Based Edge Detection Methods
Journal of Scientific Computing
Sparsity Enforcing Edge Detection Method for Blurred and Noisy Fourier data
Journal of Scientific Computing
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Edge detection from a finite number of Fourier coefficients is challenging as it requires extracting local information from global data. The problem is exacerbated when the input data is noisy since accurate high frequency information is critical for detecting edges. The noise furthermore increases oscillations in the Fourier reconstruction of piecewise smooth functions, especially near the discontinuities. The edge detection method in Gelb and Tadmor (Appl Comput Harmon Anal 7:101---135, 1999, SIAM J Numer Anal 38(4):1389---1408, 2000) introduced the idea of "concentration kernels" as a way of converging to the singular support of a piecewise smooth function. The kernels used there, however, and subsequent modifications to reduce the impact of noise, were generally oscillatory, and as a result oscillations were always prevalent in the neighborhoods of the jump discontinuities. This paper revisits concentration kernels, but insists on uniform convergence to the "sharp peaks" of the function, that is, the edge detection method converges to zero away from the jumps without introducing new oscillations near them. We show that this is achievable via an admissible class of spectral mollifiers. Our method furthermore suppresses the oscillations caused by added noise.