How Wavelets Allow Researchers to Change and Understand


In an increasing number data-driven world, mathematical tools known as wavelets have become an essential way to analyze and understand information. Many researchers receive their data in the form of continuous signals, meaning an unbreakable stream of information that changes over time, like a geophysicist listening to the sound waves that overlap. underground rock layers, or a data scientist studying streams of electrical data obtained by scanning images. This data can take many different forms and patterns, making it difficult to analyze them as a whole or take them separately and study their pieces-but wavelets can help.

Wavelets are a representation of short wavelike oscillations with different frequency ranges and shapes. Because they can take many shapes – almost any frequency, wavelength, and specific shape possible – they can be used by researchers to identify and match specific wave patterns to almost any continuous signal. Because of their vast knowledge, wavelets have revolutionized the study of complex wave phenomena in image processing, communication, and scientific data flow.

“Actually, some mathematical discoveries have influenced our technological society as much as wavelets did,” he said. Amir-Homayoon Najmi, a theoretical physicist at Johns Hopkins University. “Wavelet theory opens doors to many applications in a unified framework with an emphasis on speed, variability, and accuracy that has not been used before.”

Wavelets originated as a kind of update to a multi-beneficial mathematical method known as the Fourier transform. In 1807, Joseph Fourier discovered that any periodic function – an equation whose repetitive values ​​are secretive – could be expressed as the value of trigonometric functions such as sine and cosine. This has proven useful because it allows researchers to divide a signal stream into its constituent components, where, for example, a seismologist can identify the nature of underground structures based on intensity. at different frequencies of the displayed sound waves.

As a result, the Fourier transform leads directly to many applications in scientific and technological research. But the wavelets allow for greater accuracy. “Wavelets open the door to many advances in sound loss, image restoration, and image analysis,” he said. Veronique Delouille, an applied mathematician and astrophysicist at the Royal Observatory in Belgium who uses wavelets for analyzing images of the sun.

That’s because Fourier transformations have one fundamental limitation: They only provide information about frequencies are on the same signal, nothing is said about their time or number. It’s as if you have a process to figure out what types of fees are in a pile of cash, but not how many are in there. “Wavelets have definitely solved this problem, and that’s why they’re so much more interesting,” he said. Martin Vetterli, the president of the Swiss Federal Institute of Technology Lausanne.

The first attempt to fix this problem came from Dennis Gabor, a Hungarian physicist who in 1946 proposed to cut the signal to short, placed in local parts before applying the Fourier transform. However, it is difficult to analyze more complex signals with strong changes in frequency components. It led geophysical engineer Jean Morlet to develop the use of time windows to investigate waves, whose length of windows depends largely: wide windows for low-frequency windows. of signal and narrow windows for high-frequency segments.

However these windows also contain negative real -life frequencies, which are difficult to analyze. So Morlet had the idea to match each part to a similar wave that could be understood mathematically. He was allowed to understand the overall structure and timing of these parts and examine them with greater accuracy. In the early 1980s Morlet named these ideal wave patterns “ondelettes,” French for “wavelets” – literally, “small waves” – because of their appearance. A signal can be cut into small areas, each centered around a specific wavelength and analyzed by pairing the same wavelet. Now facing a pile of money, to go back to the previous example, we will find out how much of each type of research is involved in it.



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