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*From*: brian whatcott <inet@INTELLISYS.NET>*Date*: Fri, 22 Jun 2001 14:31:45 -0500

It is helpful to students who lose traction on mathematical

treatments to consider optical processes that offer much the

same effect.

1D Fourier Transform.

The idea of analysing continuous waveforms like music,

into amplitude contributions at different frequencies,

is familiar to people who use graphic equalizers in their

car stereo systems.

The intensity versus time graph is transformed to an

intensity versus frequency graph.

It is a short step to considering the information in an image -

it might be a 35mm transparency - for its spatial frequencies,

by converting the light intensity versus X and Y co ordinates

to a light intensity versus spatial frequency transform

in two axes.

2D Fourier Transform.

The optical method of generating a 2D Fourier transform of the

spatial information in a slide is ridiculously easy.

It goes like this:

On an optical bench set out a laser of a few milliwatts and a

beam expander in order to illuminate a 35 mm slide.

Set out a converging lens of convenient focal length, say

15cm, at one focal length behind the slide.

Then one focal length behind the lens, set out a white card.

The card displays a fourier transform of the spatial frequencies

present in the slide.

Consider a small hole as the image to analyze.

Its FT is a large circle of light, with two visible haloes of

reduced intensity surrounding it.

If we consider a slice through this original image, it is a square

shouldered pulse in shape, so it is no surprise that examination

of the FT at the transform plane shows a zero frequency component

at the center, surrounded by third and fifth spatial harmonics

in all radial directions. These are the haloes that I mentioned.

If we place a second similar converging lens at a focal length behind

this transform plane, we see the original image once more at a focal length

behind this second lens - the FT is symmetrical.

Optical Signal Processing

Placing a camera less lens at the FT plane enables us to take

a record of the FT. This in turn can serve as an optical filter.

If, for example, a small round aperture is placed at the transform

plane, the second lens transforms to an image with only low

spatial frequencies present.

A starting image of a silhouette of mickey mouse with a thin

belt, if passed through such a 'low pass' filter, transforms

back to a rounder image, whose thin belt has disappeared,

for example.

By contrast, a high pass optical filter (a central opaque dot)

placed at the (second) fourier transform plane results in a

mickey mouse outline only at the final focal plane.

You can see, I hope, how accessible this experimental topic is.

An atlas of optical transforms, like

Harburn, Taylor & Welberry, Bell Pub 75 ISBN 07135 1760 3

shows many more.

This was an impetus of a UNESCO pilot project of the

Commission on Crystallographic Teaching of IUC, where

X ray images were a pressing topic for visual transformation.

brian whatcott <inet@intellisys.net> Altus OK

Eureka!

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