TABLE OF CONTENTS In addition to the Fourier and Hankel transforms, there are other integral transforms that are useful in optics and optical signal processing, which will be introduced here from a purely mathematical perspective. These transforms will be useful when we discuss the physical theory of diffraction, as well as image processing and pattern recognition applications of optics. These transforms consist of the
Fresnel transform
Hilbert transform
Radon transform
Mellin transform
Wavelet transform
We will first introduce and briefly describe each of these transforms in order to familiarize ourselves with their appearance, notation, and area of application.
The Fresnel transform is given by
and arises from consideration of the diffraction of light through an aperture in the so-called near-field (or Fresnel zone), which is close to the aperture compared to its size. x and y are spatial coordinates, as are x ? and y ?. In the limit as the plane of observation, defined by ( x, y), physically approaches the input aperture plane ( x ?, y ?), the observed function F ( x, y) approaches f ( x, y), which can be considered an image projection operation. In the other extreme, when the ( x, y) plane recedes far away, F( x, y) approaches the Fourier transform of f( x, y).
The Hilbert transform, given by
(shown in one dimension), arises when considering single-sided (causal and/or positive frequency) signals. In optics...
