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 $\sqrt{\eta}$, and the $\mathcal{O}(1)$ scale of the jump discontinuities. We use concentration factors which are adjusted to the standard deviation of the noise $\sqrt{\eta} \gg 1/N$ in order to detect the underlying $\mathcal{O}(1)$-edges, which are separated from the noise scale $\sqrt{\eta} \ll 1$.