Wavelet introduction (3)

(3) Transforming Reality Wavelet analysis allows researchers to isolate and manipulate specific types of patterns hidden in masses of data, in much the same way our eyes can pick out the trees in a forest, or our ears can pick out the flute in a symphony. One approach to understanding how wavelets do this is to start with the difference between two kinds of sounds—a tuning fork and the human voice. Strike a tuning fork and you get a pure tone that lasts for a very long time. In mathematical theory, such a tone is said to be “localized” in frequency; that is, it consists of a single note with no higher-frequency overtones. A spoken word, by contrast, lasts for only a second, and thus is “localized” in time. It is not localized in frequency because the word is not a single tone but a combination of many different frequencies. Graphs of the sound waves produced by the tuning fork and human voice highlight the difference, as illustrated here. The vibrations of the tuning fork trace out what mathematicians call a sine wave, a smoothly undulating curve that, in theory, could repeat forever. In contrast, the graph of the word “greasy” contains a series of sharp spikes; there are no oscillations. In the nineteenth century, mathematicians perfected what might be called the “tuning fork” version of reality, a theory known as Fourier analysis. Jean Baptiste Joseph Fourier, a French mathematician, claimed in 1807 that any repeating waveform (or periodic function), like the tuning fork sound wave, can be expressed as an infinite sum of sine waves and cosine waves of various frequencies. (A cosine wave is a sine wave shifted forward a quarter cycle.) A familiar demonstration of Fourier’s theory occurs in music. When a musician plays a note, he or she creates an irregularly shaped sound wave. The same shape repeats itself for as long as the musician holds the note. Therefore, according to Fourier, the note can be separated into a sum of sine and cosine waves. The lowest-frequency wave is called the fundamental frequency of the note, and the higher-frequency ones are called overtones. For example, the note A, played on a violin or a flute, has a fundamental frequency of 440 cycles per second and overtones with frequencies of 880, 1320, and so on. Even if a violin and a flute are playing the same note, they will sound different because their overtones have different strengths or “amplitudes.” As music synthesizers demonstrated in the 1960s, a very convincing imitation of a violin or a flute can be obtained by re-combining pure sine waves with the appropriate amplitudes. That, of course, is exactly what Fourier predicted back in 1807.Mathematicians later extended Fourier’s idea to non-periodic functions (or waves) that change over time, rather than repeating in the same shape forever. Most real-world waves are of this type: say, the sound of a motor that speeds up, slows down, and hiccups now and then. In images, too, the distinction between repeating and non-repeating patterns is important. A repeating pattern may be seen as a texture or background, while a non-repeating one is picked out by the eye as an object. Periodic or repeating waves composed of a discrete series of overtones can be used to represent repeating (background) patterns in an image. Non-periodic features can be resolved into a much more complex spectrum of frequencies, called the “Fourier transform,” just as sunlight can be separated into a spectrum of colors. The Fourier transform portrays the structure of a periodic wave in a much more revealing and concentrated form than a traditional graph of a wave would. For example, a rattle in a motor will show up as a peak at an unusual frequency in the Fourier transform.Fourier transforms have been a hit. During the nineteenth century they solved many problems in physics and engineering. This prominence led scientists and engineers to think of them as the preferred way to analyze phenomena of all kinds. This ubiquity forced a close examination of the method. As a result, throughout the twentieth century, mathematicians, physicists, and engineers came to realize a drawback of the Fourier transform: they have trouble reproducing transient signals or signals with abrupt changes, such as the spoken word or the rap of a snare drum. Music synthesizers, as good as they are, still do not match the sound of concert violinists, because the playing of a violinist contains transient features--such as the contact of the bow on the string--that are poorly imitated by representations based on sine waves.The principle underlying this problem can be illustrated by what is known as the Heisenberg Indeterminacy Principle. In 1927, the physicist Werner Heisenberg stated that the position and the velocity of an object cannot both be measured exactly at the same time even in theory. In signal processing terms, this means it is impossible to know simultaneously the exact frequency and the exact time of occurrence of this frequency in a signal. In order to know its frequency, the signal must be spread in time, or vice versa. In musical terms, the trade-off means that any signal with a short duration must have a complicated frequency spectrum made of a rich variety of sine waves, whereas any signal made from a simple combination of a few sine waves must have a complicated appearance in the time domain. Thus, we can’t expect to reproduce the sound of a drum with an orchestra of tuning forks.