Natural philosophy is written in this grand book the universe, which
stands continually open to our gaze. But the book cannot be understood
unless one first learns to comprehend the language and to read the
alphabet in which it is composed. It is written in the language of
mathematics, and its characters are triangles, circles, and other geometric
figures, without which it is humanly impossible to understand a single
word of it; without these, one wanders about in a dark labyrinth.
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. It is for these students that the present course was developed.
To communicate with the biologically-oriented vision scientist requires a different approach from that typically used to teach Fourier analysis to physics or engineering students. The traditional sequence is to start with an integral equation involving complex exponentials that defines the Fourier transform of a continuous, complex-valued function defined over all time or space. Given this elegant, comprehensive treatment, the real-world problem of describing the frequency content of a sampled waveform obtained in a laboratory experiment is then treated as a trivial, special case of the more general theory. Here we do just the opposite. Catering to the concrete needs of the pragmatic laboratory scientist, we start with the analysis of real-valued, discrete data sampled for a finite period of time. This allows us to use the much friendlier linear algebra, rather than the intimidating calculus, as a vehicle for learning. It also allows us to use simple spreadsheet computer programs (e.g. Excel), or preferably a more scientific platform like Matlab, to solve real-world problems at a very early stage of the course.
With this early success under our belts, we can muster the resolve necessary to tackle the more abstract cases of an infinitely long observation time, complex-valued data, and the analysis of continuous functions. Along the way we review vectors, matrices, and the algebra of complex numbers in preparation for transitioning to the standard Fast Fourier Transform (FFT) algorithm built into Matlab. We also introduce such fundamental concepts as orthogonality, basis functions, convolution, sampling, aliasing, and the statistical reliability of Fourier coefficients computed from real-world data. Ultimately, we aim for students to master not just the tools necessary to solve practical problems and to understand the meaning of the answers, but also to be aware of the limitations of these tools and potential pitfalls if the tools are misapplied.
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