Wavelet

A wavelet is a mathematical function used to decompose another function or signal into different frequency components. Unlike traditional Fourier analysis, wavelets provide both frequency and spatial (or temporal) information, allowing for a more detailed analysis of localized variations. They are characterized by their oscillation (wave-like behavior) and rapid decay. Wavelets are particularly useful in analyzing non-stationary signals where frequency content changes over time, providing a time-frequency representation and enabling signal compression and noise reduction. The flexibility of wavelet transforms makes them applicable across many scientific fields.

Wavelet meaning with examples

• In image processing, wavelets are used to compress images, reduce noise, and enhance details. JPEG 2000 is a common image compression standard that leverages the efficiency of wavelet transforms. The wavelet transform decomposes an image into different frequency components, allowing for selective removal of information. It is crucial for efficient storage, transmission, and enhancement.
• Audio engineers use wavelet transforms for audio compression, noise reduction, and sound analysis. Lossy compression formats such as MP3 utilize this for compacting audio data. Wavelets enable extraction of features like fundamental frequencies and harmonics from audio signals, which can then be manipulated and utilized for various audio effect purposes.
• Seismologists employ wavelets to analyze seismic signals, identify different types of seismic events, and interpret the earth's structure. Analyzing the time-frequency structure of seismic waves provides insights into subsurface geology. This helps in petroleum exploration or in studying the propagation of earthquakes, offering data on fault lines and ground motion.
• Medical imaging, such as MRI and CT scans, uses wavelets for image reconstruction, noise reduction, and feature extraction. wavelet transforms improve image quality by reducing artifacts and enhancing detail. This aids doctors in diagnosis of diseases, by permitting better visualization and enabling segmentation and quantification of anatomical structures.