Fourier analysis of spatial and temporal visual stimuli has become common in the last 35 years. For many people interested in vision but not trained in mathematics this causes some confusion. It is hoped that this brief tutorial, although incomplete and simplified, will assist the reader in understanding the rudiments of this analytic method.

*At the outset, I would like to acknowledge the valuable
e-mail exchanges I had with **Dr. D.H. Kelly.**
Although I obviously must take responsibility for any errors or
misconceptions that still remain, I am grateful to Dr. Kelly for helping
me to present these difficult concepts intuitively and
accurately.*

The purpose of this section of the book is to familiarize readers with these concepts so that they will not be entirely new and strange when encountered in hardcopy textbooks. A second reason, aimed at students in the early stages of their educational career, is to encourage them to take the appropriate mathematics courses so they can become proficient in the use of Fourier and allied methods.

Before proceeding, let's understand one important point. The use of these Fourier methods does not mean that the visual system performs a Fourier analysis. At present it should be understood that this approach is a convenient way to analyze visual stimuli.

Readers who would like a brief, albeit intense, summary of the
details involved with the Fourier approach are invited to see
section 1.8.1 in *Human Brain Electrophysiology*, by David
Regan. A somewhat more elementary tutorial can be found in
Chapter XII of Cornsweet's *Visual Perception*. Donald Kelly
presented an advanced tutorial related to flicker in the *Handbook
of Sensory Physiology VII/7, Visual Psychophysics* edited by
Jameson and Hurvich. More recently Beau Watson attempted an
intuitive explanation which was published in the *Handbook of
Perception and Human Performance edited by Boff, Kaufman &
Thomas*.

Jean Baptiste Fourier, a mathematician, showed that any repetitive waveform can be broken down into a series of sine waves at appropriate amplitudes and phases.

For the time being we will assume that the reader knows little
about Fourier analysis and the properties of sine waves and that
the following is useful. A sine wave is a wave of a single
frequency. It has a given frequency, amplitude and phase. Click
on **sine wave** for a
graphical representation.

To illustrate the power of Fourier's discovery I will show how
it is possible to take an appropriately chosen set of sine waves
and add them together to produce a **square
wave****. **

Clearly, if it is possible to construct a
wave of a particular pattern by adding together appropriately
chosen sine waves then the reverse is true as well. The building
of complex waves by combining appropriately chosen sine waves is
called Fourier synthesis. The breaking apart of a complex wave
into its component sine waves is call Fourier analysis. In the **Fourier synthesis**
illustration the curve in figure A is added to the curve in
figure B to produce the curve in figure C. Figure C is reproduced
at the top right of the illustration and added to figure D to
produce figure E. You can see as the number of sine waves at
appropriate amplitudes and frequencies are added together the
result appears increasingly like a **square
wave****.**

The Fourier approach to analyzing visual stimuli actually
comes under the heading of "Linear Systems Analysis."
Another concept that falls under linear systems is **point spread functions** .

**Table of Contents
Subject Index
Table of Contents [When not using frames]
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