waveletmuunnoksessa

Wavelet transform (Finnish: waveletmuunnos) is a mathematical technique that decomposes a signal into components associated with different scales and localizations in time. Unlike the Fourier transform, which uses infinite-duration sinusoids, the wavelet transform employs wavelets that are localized in both time and frequency, providing a time–frequency representation well suited to transients and non-stationary signals.

There are two main forms: continuous wavelet transform (CWT) and discrete wavelet transform (DWT). The CWT computes coefficients by correlating the signal with scaled and shifted versions of a mother wavelet, producing a redundant representation across scales. The DWT discretizes the scale and translation, yielding an efficient, nonredundant decomposition into approximation and detail coefficients via filter banks or pyramid algorithms. The inverse transform allows exact reconstruction under suitable conditions.

Key concepts include the mother wavelet, a finite-energy function with certain regularity and vanishing moments, and the scaling function used in multiresolution analysis. Multiresolution analysis builds a hierarchy of nested function spaces, enabling analysis at progressively coarser resolutions while preserving detail information.

Applications span image and audio compression, denoising, feature extraction, and biomedical signal processing. The JPEG 2000 image compression standard uses wavelets for efficient representation, while wavelet-based denoising relies on thresholding detail coefficients.

Historically, early continuous-wavelet ideas emerged in the 1980s with Morlet and Grossmann, and the formal multiresolution framework was developed by Mallat and others in the 1980s and 1990s. Today, wavelet methods remain a versatile tool in signal processing, offering flexible time–frequency localization and scalable analysis.