Abstract

Wavelets have gained the attention of the signal processing community for their usefulness in analyzing nonstationary signals, for their mathematical elegance, and for their relative ease of computation. This paper is intended to introduce the audience to the basic principles of wavelet analysis and to consider where optical techniques can be applied advantageously. A signal is decomposed on a set of basis functions created by scaling and shifting a single fundamental wavelet. The space and frequency localization of the resulting wavelet transform, spanning the range from pure Nyquist sampling (no frequency localization) to Fourier transforms (no spatial localization), is determined by the choice of this fundamental wavelet. A common choice for the fundamental wavelet has compact support in the signal domain and bandpass-like behavior in the frequency domain. With this choice, rapidly varying information is well localized in the signal domain while slowly varying information is well localized in the frequency domain. We will discuss what constitutes an allowed fundamental wavelet, orthogonal and nonorthogonal wavelet bases, and the choice of sampling intervals in shift and scale. We will also discuss some of our theoretical results on filtering noise from nonstationary signals by using the wavelet transform for nonstationary spectrum estimation. In considering the use of optical techniques for wavelet computations, it is important to be aware of the digital competition. One reason for the popularity of wavelets is that O(N) algorithms exist for their digital computation. This is stiff competition for an optical system if the only advantage that can be claimed is speed. We will discuss the possible advantages of optical systems related to continuous rather than discretely sampled shift coordinates and the ease of implementing arbitrary scaling factors and nonseparable 2D wavelet functions. Finally, we will present an optical system for computing 2D wavelet transforms.

© 1992 Optical Society of America