Introduction to wavelet and the wavelet transformation

[객원 에디터 4기 / 한동민 기자] Wavelets, a modern tool for signal processing, are wave-like oscillations used to divide a signal into different scale components. Wavelet has two basic properties: scale and location. Scale defines how stretched each wavelet is, while location defines where the wavelet is located in terms of space and time. There are four types of wavelets: minimum, mixed, zero, and maximum phase, each with unique displays and spectrums. For instance, in zero phases, the wave of the phase is ideally symmetrical and is a noncasual system, meaning that it produces an output before any inputs are made. Furthermore, because the zero phase has a relatively shorter duration, it can better distinguish two objects. As shown in the diagram below, different phases’ wavelengths and frequencies differentiate them.

Using these different wavelets, scientists can transform them to yield many advantages. There are two types of wavelet transformations: continuous wavelet transform (CWT) and discrete wavelet transform (DWT). CWT uses wavelets over a range of scales and locations continuously, whereas DWT only uses a set of wavelets over a specific area of data. Transforming wavelets offers simultaneous localization in the time and frequency domain in a fast and detailed manner. Small wavelets allow researchers to isolate unmeasurable details from a signal, whereas large wavelets can only depict general details. In addition, wavelet theory enables scientists to reveal different aspects of each data that other signal analysis techniques cannot distinguish, including trends, breakdown points, and discontinuities in a single data. 

Wavelet technology is primarily used in electric power systems analysis. The U.S. Federal Bureau of Investigation used wavelet transformation to compress images of astronomical pictures and minimize noises and blurs to make more reliable approximations of the data collected. In the three figures below, researchers used wavelet transformation to separate the first photo into two separate data to illustrate the stars and the vapor surrounding them. 

Sources: ScienceDirect, Intechopen, NYU Engineering, TowardsDataScience