waveletperheen

A wavelet family, or waveletperheen, is a collection of wavelets generated by dilating and translating a single function known as the mother wavelet. In the continuous wavelet transform, the family consists of functions ψ_{a,b}(t) = (1/√a) ψ((t−b)/a) for all a > 0 and b ∈ R. This family provides a time–frequency representation of signals at different scales, and, for an admissible mother wavelet, the transform is invertible.

In practice, discrete wavelet transforms use a dyadic subset of the continuous family, with a = 2^j and b = k·2^j. Depending on the chosen mother wavelet, the resulting wavelet system can form an orthonormal or biorthogonal basis for L^2(R). The properties of the wavelet family—such as orthogonality, compact support, regularity, and the number of vanishing moments—determine how well signals can be represented and reconstructed.

Common wavelet families arise from specific mother wavelets. The Haar family uses the simplest step function and yields very fast, but coarse representations. Daubechies wavelets (with N vanishing moments) are compactly supported and offer a balance between locality and smoothness. Symlets are near-symmetric variants of Daubechies with improved symmetry. Coiflets feature a higher number of vanishing moments for both wavelet and scaling functions. Biorthogonal families, such as those used in image compression, allow symmetric filters with separate analysis and reconstruction functions. Other well-known examples include Meyer, Morlet, and the Mexican hat wavelets, which are often discussed in the context of continuous transforms.

Applications of wavelet families include signal and image denoising, compression, feature extraction, and multiresolution analysis. The choice of family depends on the desired trade-offs between support, regularity, and symmetry for a given task.