Manipulation of two-dimensionally periodic stimulus patterns

An increasing number of visual studies require precise knowledge of the two-dimensional Fourier components of periodic spatial patterns. For patterns that are more complex than a simple sine-wave grating, however, the frequencies, magnitudes, phases, and even the orientations of these components are not always obvious from inspection of the pattern. Although the fast Fourier transform can provide a finite, sampled approximation to any desired spectrum, blind use of a packaged computer program sacrifices the intuitive insight of the analytic transform, which is particularly important for conceiving new experiments. This paper draws attention to some analytic techniques involving arrays of Dirac delta functions, which the author has found helpful in designing and understanding both real andGedanken experiments in pattern perception.