A representation of a function or signal by coefficients obtained from inner products with a family of localized, dilated and translated basis functions (wavelets), encoding both scale and location information.

Principle

Decompose a signal into components at multiple scales by applying dilation and translation operators to a compactly supported or rapidly decaying prototype; coefficients reveal localized features across resolutions.

Demonstration

Compute the continuous wavelet transform of a transient audio pulse using a Morlet wavelet: the transform yields high-magnitude coefficients localized near the pulse time for small scales and a sustained low-frequency coefficient for large scales.

Misapplication

Applying thresholding to wavelet coefficients as if they were independent white-noise coefficients when the noise is correlated, producing biased denoising and spurious artifacts.

Consequence

Provides sparse, multiresolution representations that enable localized denoising, compression, edge detection and scale-dependent feature extraction.

Reversal

A purely global frequency basis (stationary sine/cosine basis) prioritizes frequency resolution at the expense of temporal or spatial localization.

Boundary

Defined for functions or discrete signals in L2 or suitable sequence spaces and for transforms implemented continuously or discretely; not applicable to objects without a notion of scale or translation.

Semantic Tension

Competes with the short-time Fourier transform: both localize in time and frequency, but wavelets adapt resolution with scale while the windowed Fourier uses fixed resolution.

Synthesis

A wavelet transform is a scale-localized linear decomposition that expresses a signal as a sum of localized building blocks, trading between time (or space) and scale to reveal multiresolution structure.