Fourier Analysis for Beginners

Abstract

Fourier analysis is ubiquitous. In countless areas of science, engineering, and mathematics one finds Fourier analysis routinely used to solve real, important problems. Vision science is no exception: today's graduate student must understand Fourier analysis in order to pursue almost any research topic. This situation has not always been a source of concern. The roots of vision science are in "physiological optics", a term coined by Helmholtz which suggests a field populated more by physicists than by biologists. Indeed, vision science has traditionally attracted students from physics (especially optics) and engineering who were steeped in Fourier analysis as undergraduates. However, these days a vision scientist is just as likely to arrive from a more biological background with no more familiarity with Fourier analysis than with, say, French. Indeed, many of these advanced students are no more conversant with the language of mathematics than they are with other foreign languages, which isn't surprising given the recent demise of foreign language and mathematics requirements at all but the most conservative universities. Consequently, a Fourier analysis course taught in a mathematics, physics, or engineering undergraduate department would be much too difficult for many vision science graduate students simply because of their lack of fluency in the languages of linear algebra, calculus, analytic geometry, and the algebra of complex numbers.