Title:    Detection of Edges in Spectral Data

Authors:  Anne Gelb, Eitan Tadmor

Date:     April 1998

Keywords: 

Fourier expansion, conjugate partial sums, piecewise smoothness,
concentration factors.

Abstract:

We are interested in the detection of jump discontinuities in piecewise
smooth functions which are realized by their spectral data. Specifically,
given the Fourier coefficients, $\{\hat{f}_k=a_k+ib_k\}_{k=1}^N$, we form
the generalized conjugate partial sum $\Sconj^\sigma[f](x)=\sum_{k=1}^N
\sigma(\frac{k}{N})(a_k\sin kx -b_k\cos kx)$.  The classical conjugate
partial sum, $\Sconj[f](x)$, corresponds to $\sigma\equiv 1$ and it is
known that $\frac{-\pi}{\log N}\Sconj[f](x)$ converges to the jump function
$[f](x):= f(x+)-f(x-)$; thus, $\frac{-\pi}{\log N}\Sconj[f](x)$
tends to 'concentrate' near the edges of $f$. The convergence, however,
is at the unacceptably slow rate of order ${\cal O}(1/\log N)$.

To accelerate the convergence, thereby creating an effective edge detector,
we introduce the so called 'concentration factors',
$\sigma_{k,N}=\sigma(\frac{k}{N})$. Our main result shows that an arbitrary
$C^2[0,1]$ non-decreasing $\sigma(x)$ satisfying
$\int_{\frac{1}{N}}^1 \frac{\sigma(x)}{x}dx \arrowinfty -\pi$,
leads to the summability kernel which admits the desired concentration
property.  To improve over the slowly convergent conjugate Dirichlet kernel
(-- corresponding to the admissible $\sigma_N(x)\equiv \frac{-\pi}{\log N}$),
we demonstrate the examples of two families of concentration functions
(depending on free parameters $p$ and $\alpha$):  the so-called Fourier
factors, $\sigma^F_\alpha(x)=\frac{-\pi}{Si(\alpha)}\sin \alpha x$, and
polynomial factors, $\sigma^p(x)=-p\pi x^p$. These yield effective
detectors of (one or more) edges, where both the location and the amplitude
of the discontinuities are recovered.

Publication History:

  Submitted to:  Applied Harmonic Analysis, January 1998

  Published in: