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 break them down and study their parts. However, wavelets are able to help.
Wavelets are short wavelike oscillations that can be represented as wavelets. They come in a variety of frequency ranges and different shapes. Wavelets can be represented in a variety of forms, including specific shapes and frequencies. Researchers can use them to match specific wave patterns within continuous signals. Wavelets are a versatile tool that revolutionizes the study of complex wave phenomena, such as image processing, communication, and scientific data streams.
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 simply not possible before.
Wavelets were created as an update to the Fourier transform, a highly useful mathematical technique. Joseph Fourier, an 1807 mathematician, discovered that any periodic function of an equation whose values repeated cyclically could simply 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, de-noising and image analysis. Vronique Delouille is an applied mathematician at the Royal Observatory of Belgium and an astrophysicist who uses wavelets to analyze images of the sun.
Fourier transforms are limited in that they only provide information about frequencies 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 by Morlet, French for "wavelets", which are little waves due to their appearance. 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.
Imagine that you slide a wavelet of a certain frequency and shape over the raw signal. If you find a good match, the mathematical operation known as "the dot product" between them becomes zero or very close to zero. You can scan the entire signal train with wavelets at different frequencies to get a complete picture. This allows for an extensive analysis.
Wavelets research has evolved rapidly. Yves Meyer was a French mathematician and professor at the cole Normale Suprieure. He was waiting to be scanned at a photocopier, when a colleague presented him with a paper about wavelets by Alex Grossmann and Morlet. Meyer was instantly captivated and took the first train to Marseille. There he worked with Grossman, Morlet, and the mathematician, Ingrid Daubechies (now at Duke University). Meyer won the Abel Prize for his work in wavelet theory.
A few years later, Stphane Mallat, a Pennsylvania State University graduate student studying computer vision/image analysis, ran into an old friend on the beach. Mallat was told by the friend, who was a Paris graduate student studying wavelets. Mallat quickly realized the importance Meyer's research for his own research and teamed up immediately with Meyer. They published a paper in 1986 on wavelets and image analysis. This work eventually led to the creation of JPEG2000, an image compression that is widely used around the globe. Wavelets are used to analyze the signal from a scan and produce a small number of pixels. This allows for reconstruction of the original image at the original resolution. This technique was useful when transmission limitations prevented the transmission of large data sets.
Wavelets are versatile, and can decode nearly any type of data. Wavelets come in many forms, so you can stretch, squish, or adapt them to your image. Wave patterns can vary in many ways in digital images. However, wavelets can be stretched and compressed to match signals with lower frequencies or higher frequencies. Wave patterns can change in shape, and mathematicians have created different types or families of wavelets that are different in wavelength scales and shapes to match this variability.
The Daubechies mother wavelet is one of the most well-known wavelet families. It has a self-similar structure with large asymmetrical peak that mimic smaller replicas of the peaks. Experts have used these wavelets to identify fake Vincent van Gogh paintings by using image analysis. The Mexican hat has one maximum and two minima. The Coiflet wavelet, named after Ronald Coifman, Yale University mathematician, is similar to the Mexican Hat but has sharp peaks and not flat areas. These can be used to capture and eliminate unwanted noise spikes in images and sound signals, as well as data streams that are generated by scientific instruments.
Wavelets can be used in basic research as well as in sound signal analysis and image processing. Wavelets can be used to help scientists find patterns in scientific data. They allow researchers to simultaneously analyze large data sets. Huybrechs said, "It always strikes me how varied the applications are." Wavelets are the best way to view data. This is true regardless of what data it may be.
