Waveletbased

Wavelet-based methods refer to analytical and computational techniques that use wavelets as localized basis functions to decompose, analyze, and reconstruct data across multiple scales. Unlike Fourier-based approaches, which assume stationarity and provide global frequency information, wavelet-based analysis captures transient and localized features by varying resolution in time and frequency.

Central to wavelet-based methods is the idea of multiresolution analysis, which expresses a signal as a sum of components at coarse and progressively finer scales. This is implemented through transforms such as the discrete wavelet transform (DWT) and its variants, or the continuous wavelet transform (CWT). The DWT often uses filter banks to separate approximation and detail coefficients, enabling efficient signal compression and denoising. The CWT provides a highly redundant, scale-dependent representation useful for feature detection and time-scale localization.

Common wavelet families include Haar, Daubechies, Symlets, and Coiflets, which differ in support, smoothness, and vanishing moments, as well as Morlet and Mexican hat wavelets used in continuous transforms. The choice of wavelet affects analysis results and is guided by signal characteristics and the desired balance between locality and smoothness.

Applications of wavelet-based methods span signal processing, image and audio compression (notably wavelet-based JPEG 2000), denoising, texture analysis, and feature extraction. In medicine and geophysics, wavelet-based techniques are employed for denoising EEG data, MRI edge detection, and earthquake signal analysis.

Limitations include boundary effects, parameter choices (wavelet type, decomposition level, padding), and computational demands for large datasets. Ongoing research explores adaptive and biorthogonal wavelets, lifting schemes, and wavelets for irregularly sampled data, expanding the versatility of wavelet-based analysis.