How Wavelets Let Researchers Transform and Understand Data

Wavelets, mathematical tools that can be used to analyze and comprehend information in a data-driven world have been essential. Many researchers receive data as continuous signals. This means that a stream of information is constantly evolving. For example, a geophysicist listening for sound waves to bounce off of the rock layers below, or a data scientist looking at the electrical data streams generated by scanning images. These data can come in many forms and patterns making it difficult to analyze or take apart the pieces. However, wavelets are a way to help.
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Wavelets are short wavelike oscillations that can be represented in wavelets. They come in different frequencies and shapes. Wavelets can be used to match and identify specific wave patterns in continuous signals. They can also take on multiple forms, including wavelengths and frequencies. Wavelets are a revolutionary tool for studying complex wave phenomena in image processing and communications.

Amir-Homayoon Najmi is a Johns Hopkins University theoretical physicist. He said that wavelets are one of the most influential mathematical discoveries in modern technology. Wavelet theory has allowed for many new applications, with a focus on speed, sparsity and accuracy that was previously unimaginable.

Wavelets were created as an update to the Fourier transform, a highly useful mathematical technique. Joseph Fourier, an 1807 mathematician, discovered that any periodic functionan equation with values that repeat cyclically could be expressed as the sum trigonometric functions sine and cosine. This was useful because it allowed researchers to break down a signal stream into its constituent components, which allows, for example, a seismologist, to determine the nature of underground structures using the intensity of reflected sound waves.

The Fourier transform has been used in a variety of scientific research and technology applications. Wavelets, however, allow for greater precision. Wavelets allow for many improvements in image restoration and image analysis. Vronique Delouille is an applied mathematician at the Royal Observatory of Belgium and a physicist who uses wavelets to analyze images of the sun.

Fourier transforms are limited in that they only provide information about frequencies present in a signal. They do not give any details about their timing or quantities. As if there was a way to determine the types of bills in a stack of cash but not how many. This problem was solved by Wavelets, which is why they are so fascinating, according to Martin Vetterli (president of the Swiss Federal Institute of Technology Lausanne).

Dennis Gabor, a Hungarian scientist, was the first to solve this problem. He suggested that in 1946, the signal be cut into shorter, more time-localized segments and then applying Fourier transforms. These were not easy to analyze for complex signals that had frequency components that changed rapidly. Jean Morlet, a geophysical engineer, developed time windows to study waves. The windows are lengthened depending on frequency. Wide windows are for low-frequency signals and narrow windows for high frequencies.

These windows contained real-life frequencies that were difficult to analyze. Morlet came up with the idea of matching each segment to a similar wave mathematically understood. Morlet was able to understand the structure and timing of each segment and explored them more accurately. These idealized wave patterns were called ondelettes in the 1980s. This is French for "wavelets" meaning little waves. This allowed a signal to be broken up into smaller areas that were centered on a particular wavelength. The signals could then be analyzed by being paired together with the matching wavelet. We now have a stack of cash. To return to the previous example, we would know how many bills it contained.