Wavelets

Wavelets are functions used to analyze data at multiple scales, providing localization in both time and frequency. In signal processing and applied mathematics, a wavelet is a square-integrable function with zero mean, often called a mother wavelet. By translating and dilating this prototype, a family of wavelets is generated to probe signals at different resolutions.

The continuous wavelet transform (CWT) computes coefficients by taking inner products of a signal with scaled and shifted versions of the mother wavelet, revealing how signal content varies with time and scale. The discrete wavelet transform (DWT) uses a dyadic grid of scales and positions for efficient analysis, frequently implemented with filter banks. This multiresolution framework, known as multiresolution analysis (MRA), represents a signal as a coarse approximation plus a sequence of detail components across increasing levels of detail.

Key properties include localization (finite or compact support in time), vanishing moments (which improve ability to represent smooth or polynomial trends), and, depending on the family, orthogonality or tight frame behavior. The choice of wavelet affects time versus frequency localization and the handling of boundaries and noise.

Common families and examples of wavelets include Haar (step-like), Daubechies (compactly supported with varying vanishing moments), Coiflets and Symlets (balanced regularity), and continuous wavelets such as Morlet and Mexican hat. Applications span denoising, compression (notably JPEG 2000), feature extraction,.texture analysis, and medical imaging, among others. The versatility of wavelets arises from their ability to isolate signal components at different scales while preserving temporal information.