Plotting and Scheming with Wavelets

Abstract

Wavelets are acquiring a visibility and popularity that may soon be on the scale first enjoyed by fractals a few years back. Like fractals, wavelets have attractive and novel features, both as mathematical entities and in numerous applications. They are often touted as worthwhile alternatives to classical Fourier analysis, which works best when applied to periodic data: wavelet methods make no such assumptions. However, the mathematics of wavelets can seem intractable to the novice. Indeed, most introductions to wavelets assume that the reader is already well versed in Fourier techniques. Our main goal is simple: to convince the reader that at their most basic level, wavelets are fun, easy, and ideal for livening up dull conversations. We demonstrate how elementary linear algebra makes accessible this exciting and relatively new area at the border of pure and applied mathematics. In Plotting, we explore several ways of visually representing data, with the help of Matlab software. In Scheming, we discuss a simple wavelet-based compression technique, whose generalizations are being used today in signal and image processing, as well as in computer graphics and animation. The basic technique uses only addition, subtraction, and division by two! Only later, in Wavelets, do we come clean and reveal what wavelets are, while unveiling the multiresolution setting implicit in any such analysis. In Averaging and Differencing with Matrices, which may be read independently of Wavelets, we provide a matrix formulation of the compression scheme. In Wavelets on the World Wide Web we mention a natural form of progressive image transmission that lends itself to use by the emerging generation of web browsers (such wavelet-enhanced software is already on the market). In Wavelet Details, we attempt to put everything in context, while hinting at the more sophisticated mathematics that must be mastered if one wishes to delve deeper into the subject. Finally, in Closing Remarks, we mention some other common applications of wavelets. Along the way we find ourselves trying out an adaptive plotting technique for ordinary functions of one variable that differs from those currently employed by many of today's popular computer algebra packages. While this technique, as described here, is limited in its usefulness, it can be modified to produce acceptable results. We were much inspired by Stollnitz, DeRose, and Salesin's fine wavelets primers ([15, 16]), which, along with [17], [18], [19], we recommend heartily to beginners wlho