morfunksjon

Morfunksjon, or mother function, is the base function used to generate a family of wavelets in wavelet theory. The most common form is the mother wavelet ψ(t), from which a continuum of wavelets is formed by scaling and translation. The continuous wavelet family is written as ψ_{a,b}(t) = (1/√|a|) ψ((t − b)/a), where a > 0 is the scale and b ∈ R is the translation. This family is used in the continuous wavelet transform to analyze signals at different time–frequency resolutions.

A key requirement in many wavelet constructions is admissibility. A practical condition is that ψ has finite energy, ∫ |ψ(t)|^2 dt < ∞, and often zero mean, ∫ ψ(t) dt = 0. The zero-mean condition helps ensure that the wavelet detects changes rather than constant components, while finite energy guarantees a well-defined transform. In discrete implementations, scales are often dyadic, with a = 2^j and b = k 2^j, giving ψ_{j,k}(t) = 2^{j/2} ψ(2^{j} t − k).

Common examples of mother wavelets include the Morlet, Mexican hat (Ricker), Daubechies, and others. The choice of morfunksjon affects time and frequency localization and is guided by the signal characteristics and application, such as feature detection, denoising, or compression.

In multiresolution analysis, the morfunksjon is distinct from the scaling function φ, which is used to construct approximation spaces. Together, they enable representation of signals at multiple resolutions. Applications span signal processing, image analysis, and data compression.