### Fourier Analysis

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 .