Symmlet

Symmlet, short for symmetric wavelet, refers to a family of orthogonal, compactly supported wavelets that are nearly symmetric. They were developed as a variation of the Daubechies wavelets to combine the favorable localization and vanishing moment properties of Daubechies with greater symmetry in the wavelet and scaling functions. The family is typically denoted symN, where N indicates the number of vanishing moments and roughly corresponds to the filter length; common members include sym2, sym4, sym6, and sym8.

Construction and properties: Symmlets are built from a finite impulse response scaling filter and a corresponding wavelet filter designed to satisfy orthogonality and regularity conditions. The defining feature is near symmetry of the wavelets, achieved by imposing a linear phase constraint on the filter coefficients. This yields improved phase characteristics and more symmetric behavior in the time domain while preserving compact support and the desired number of vanishing moments.

Key characteristics include compact support, orthogonality, and a prescribed number of vanishing moments. The degree of symmetry is approximate rather than exact for a given combination of orthogonality and vanishing moments, but symmlets are designed to maximize symmetry within those constraints. Higher-order members offer smoother reconstruction at the cost of longer support.

Applications: Symmlets are used in signal and image processing tasks such as denoising, compression, texture analysis, and multiresolution analysis. They are available in common wavelet toolboxes and are widely employed as alternatives to Daubechies wavelets when symmetry is advantageous.

See also: Daubechies wavelets, wavelet transform, orthogonal wavelets, vanishing moments.