## Abstract

The Hilbert transform is of interest for image-processing
applications because it forms an image that is edge enhanced relative
to an input object. Recently a fractional Hilbert transform was
introduced that can select which edges are enhanced and to what degree
the edge enhancement occurs. Although experimental results of this
selective edge enhancement were presented, there was no explanation of
this phenomenon. We analyze a one-dimensional fractional Hilbert
transform acting on a one-dimensional rectangle function and show how
it produces an output image that is selectively edge
enhanced.

© 1998 Optical Society of America

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### Equations (5)

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(1)
$$\tilde{g}\left(x\right)=\left[g\left(x\right)\right]*\left[h\left(x\right)\right],$$
(2)
$${H}_{P}\left(u\right)=exp\left(\mathit{iP}\mathrm{\pi}/2\right)S\left(u\right)+exp\left(-\mathit{iP}\mathrm{\pi}/2\right)S\left(-u\right),$$
(3)
$${H}_{P}\left(u\right)=cos\left(P\mathrm{\pi}/2\right)+isin\left(P\mathrm{\pi}/2\right)\mathrm{sgn}\left(u\right),$$
(4)
$$\tilde{g}\left(x\right)=g\left(x\right)cos\left(P\mathrm{\pi}/2\right)+i\left\{\left[g\left(x\right)\right]*\left[\frac{1}{i\mathrm{\pi}x}\right]\right\}sin\left(P\mathrm{\pi}/2\right),$$
(5)
$$g\left(x\right)=\mathrm{rect}\left(\frac{x}{L}\right).$$