In more and more a data-driven world, mathematical tools known as waves have become an indispensable way to analyze and understand information. Many researchers receive their data in the form of continuous signals, which means a continuous flow of information evolving over time, such as a geophysicist who listens to sound waves bouncing off rock layers underground, or a data scientist who studies electric currents from data obtained by scanning images. These data can take many different forms and patterns, making it difficult to analyze them as a whole or to separate them and study their parts – but waves can help.
Wavelets are representations of short wave oscillations with different frequency ranges and shapes. Because they can take many forms – almost any frequency, wavelength and specific shape is possible – researchers can use them to identify and compare specific wave patterns in almost any continuous signal. Due to their broad flexibility, wavelets have revolutionized the study of complex wave phenomena in image processing, communications, and scientific data streams.
“In fact, few mathematical discoveries have affected our technological society as much as the waves,” he said Amir-Homayon Najmi, a theoretical physicist at Johns Hopkins University. “Wave theory has opened the door to many applications in a unified framework with an emphasis on speed, rarity and accuracy that were simply not available before.”
Wavelets emerged as a kind of update to an extremely useful mathematical technique known as the Fourier transform. In 1807, Joseph Fourier discovered that any periodic function — an equation whose values are repeated cyclically — can be expressed as the sum of trigonometric functions such as sine and cosine. This proved useful as it allowed researchers to divide the signal flow into its constituent parts, allowing the seismologist to identify the nature of the underground structures based on the intensity of the different frequencies in the reflected sound waves.
As a result, the Fourier transform led directly to a number of applications in research and technology. But the waves allow much greater precision. “Wavelets opened the door to many improvements in noise removal, image recovery and image analysis,” they said. Veronica Deluy, an applied mathematician and astrophysicist at the Royal Observatory of Belgium, who uses waves to analyze images of the sun.
This is because Fourier transforms have a great limitation: they provide information only about frequencies present in a signal without saying anything about their time or quantity. It’s as if you have a process for determining what types of accounts are in a pile of money, but not how many of each one actually existed. “Wavelets definitely solved that problem and that’s why they’re so interesting,” he said Martin Wetterley, President of the Swiss Federal Institute of Technology in Lausanne.
The first attempt to solve this problem came from Denis Gabor, a Hungarian physicist who in 1946 proposed cutting the signal into short, time-segmented segments before applying a Fourier transform. However, they were difficult to analyze in more complex signals with highly variable frequency components. This prompted geophysical engineer Jean Morlet to develop the use of time windows to study waves, the length of the windows depending on the frequency: wide windows for low-frequency signal segments and narrow windows for high-frequency segments.
But these windows still contained cluttered real-life frequencies that were difficult to analyze. So Morlet had the idea to match each segment with a similar wave that was well understood in mathematics. This allowed him to grasp the overall structure and timing of these segments and to examine them with much greater accuracy. In the early 1980s, Morlet called these idealized wave models “ondelettes”, French for “wavelets” – literally “small waves” – because of their appearance. In this way, the signal can be cut into smaller areas, each of which is centered around a specific wavelength and analyzed, pairing with the matching wave. Now facing a pile of money to go back to the previous example, we will know how many of each type of banknote it contains.