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Concept# Discrete wavelet transform

Summary

In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information (location in time).
Haar wavelet
The first DWT was invented by Hungarian mathematician Alfréd Haar. For an input represented by a list of numbers, the Haar wavelet transform may be considered to pair up input values, storing the difference and passing the sum. This process is repeated recursively, pairing up the sums to prove the next scale, which leads to differences and a final sum.
Daubechies wavelet
The most commonly used set of discrete wavelet transforms was formulated by the Belgian mathematician Ingrid Daubechies in 1988. This formulation is based on the use of recurrence relations to generate progressively finer discrete samplings of an implicit mother wavelet function; each resolution is twice that of the previous scale. In her seminal paper, Daubechies derives a family of wavelets, the first of which is the Haar wavelet. Interest in this field has exploded since then, and many variations of Daubechies' original wavelets were developed.
Complex wavelet transform
The dual-tree complex wavelet transform (WT) is a relatively recent enhancement to the discrete wavelet transform (DWT), with important additional properties: It is nearly shift invariant and directionally selective in two and higher dimensions. It achieves this with a redundancy factor of only , substantially lower than the undecimated DWT. The multidimensional (M-D) dual-tree WT is nonseparable but is based on a computationally efficient, separable filter bank (FB).
Other forms of discrete wavelet transform include the Le Gall–Tabatabai (LGT) 5/3 wavelet developed by Didier Le Gall and Ali J. Tabatabai in 1988 (used in JPEG 2000 or JPEG XS ), the Binomial QMF developed by Ali Naci Akansu in 1990, the set partitioning in hierarchical trees (SPIHT) algorithm developed by Amir Said with William A.

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