Wavelet introduction (1)

(1) Summary

This article describes the development of the mathematical modeling technique known as wavelets, which is used in computer imaging and animation as well as by the FBI to encode its large database of fingerprints. In the nineteenth century, mathematicians perfected a useful tool known as Fourier analysis. This mathematical technique allows complex periodic and non-periodic functions (or waves) to be summed as a series of simpler functions. It has trouble reproducing transient signals or signals with abrupt changes, such as the spoken word (see Transforming Reality). Over the course of the twentieth century, scientists worked to get around these limitations, in order to allow representations of the data to adapt to the nature of the information. Different groups of researchers in disparate fields developed techniques to decompose signals into pieces that could be localized in time and analyzed at different scales of resolution. These techniques were the precursors of wavelet theory (see An Idea with No Name). In 1981, Jean Morlet, a geologist analyzing seismic signals, developed what are now known as “Morlet wavelets”. Further research showed that his technique worked better than Fourier transforms. Many researchers followed the original idea with refinements of their own which made wavelet analysis much easier and turned the theory into a practical tool (see The Great Synthesis). One prominent application of wavelets has been in digital image compression. Wavelets are central to the new JPEG-2000 digital image standard and the WSQ method that the FBI uses to compress its fingerprint database. They allow us to zoom in on an image without losing resolution, which is common with other techniques (see How Do Wavelets Work?). With the foundations of wavelet theory securely in place, the field has grown rapidly over the last decade. Engineers are trying new applications. Mathematicians continue trying to answer important theoretical questions. Many researchers are interested in expanding the application of wavelets beyond image compression to other areas such as pattern recognition. We will doubtless be reaping the benefits of applications of wavelets for a long time to come (see Wavelets in the Future).