Recovery of Edges from Spectral Data with Noise - A New Perspective

We consider the problem of detecting edges—jump discontinuities in piecewise smooth functions from their N -degree spectral content, which is assumed to be corrupted by noise. There are three scales involved: the “smoothness” scale of order 1/N , the noise scale of order √ η, and the O(1) scale of the jump discontinuities. We use concentration factors which are adjusted to the standard deviation of the noise √ η 1/N in order to detect the underlying O(1)-edges, which are separated from the noise scale √ η 1.