Edge Detection Using Fourier Coefficients

1. INTRODUCTION. Edge detection and the detection of discontinuities are important in many fields. In image processing, for example, one often needs to determine the boundaries of the items of which a picture is composed. (For more information about edge detection in image processing, see [10].) We consider the problem of detecting the edges present in a function when given the Fourier coefficients of the function. There are numerical methods that estimate the Fourier coefficients of a function of interest rather than directly estimating the solution. The spectral viscosity method, a numerical method used to solve nonlinear partial differential equations (PDEs), is an example of such a method [12]. The method approximates the Fourier coefficients of the solution of a PDE. The Fourier coefficients are then used to calculate an approximation to the solution. The accurate reconstruction of the solution requires that the positions of the discontinuities of the solution be known [5]. In this paper we discuss techniques for using a function's Fourier coefficients to determine the locations and sizes of the jump discontinuities of the function. At first glance the spectral representation of the signal—the Fourier series or transform associated with the signal—does not seem to be the ideal place to look for information about discontinuities in the signal. When a signal is discontinuous the convergence of the Fourier series or transform associated with the signal is not uniform; in such cases the Gibbs phenomenon [11] appears and truncating the series after any finite number of terms always leads to O(1) oscillations in the reconstructed signal. (For a nice, detailed treatment of the Gibbs phenomenon, see [6].) Considering the question again, however, one realizes that if a discontinuity is characterized by a " phenomenon, " then the existence of the discontinuity is indeed encoded in the coefficients. The question becomes how to effectively " decode " the discontinuity. One does not do this by directly summing the series—one uses the spectral representation in a somewhat different way to " concentrate " the function about the discontinuity. In what follows, we explain how this is done. We restrict ourselves to periodic (or compactly supported) functions and only consider Fourier series. (Those interested in seeing a more general theory of concentration factors are referred to [3, 4].) Much of the information in this article is well known [3, 4]. The use of the Euler-Mascheroni constant to improve the performance of …