An example of the 2D discrete wavelet transformthat is used in JPEG2000. The original image is

high-pass filtered, yielding the three large images,each describing local changes in brightness

(details) in the original image. It is then low-passfiltered and downscaled, yielding an

approximation image; this image is high-passfiltered to produce the three smaller detail

images, and low-pass filtered to produce the finalapproximation image in the upper-left.

Discrete wavelet transformFrom Wikipedia, the free encyclopedia

In numerical analysis and functionalanalysis, a discrete wavelet transform(DWT) is any wavelet transform forwhich the wavelets are discretelysampled. As with other wavelettransforms, a key advantage it has overFourier transforms is temporal resolution:it captures both frequency and locationinformation (location in time).

Contents

1 Examples1.1 Haar wavelets1.2 Daubechies wavelets1.3 Others

2 Properties3 Applications4 Comparison with Fouriertransform5 Definition

5.1 One level of thetransform5.2 Cascading and Filterbanks

6 Other transforms7 Code examples8 See also9 Notes10 References

Examples

Haar wavelets

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Main article: Haar wavelet

The first DWT was invented by the Hungarian mathematician Alfréd Haar. For an input

represented by a list of 2n numbers, the Haar wavelet transform may be considered to simplypair up input values, storing the difference and passing the sum. This process is repeated

recursively, pairing up the sums to provide the next scale: finally resulting in 2n − 1differences and one final sum.

Daubechies wavelets

Main article: Daubechies wavelet

The most commonly used set of discrete wavelet transforms was formulated by the Belgianmathematician Ingrid Daubechies in 1988. This formulation is based on the use of recurrencerelations to generate progressively finer discrete samplings of an implicit mother waveletfunction; each resolution is twice that of the previous scale. In her seminal paper, Daubechiesderives a family of wavelets, the first of which is the Haar wavelet. Interest in this field hasexploded since then, and many variations of Daubechies' original wavelets were developed.

Others

Other forms of discrete wavelet transform include the non- or undecimated wavelet transform(where downsampling is omitted), the Newland transform (where an orthonormal basis ofwavelets is formed from appropriately constructed top-hat filters in frequency space). Waveletpacket transforms are also related to the discrete wavelet transform. Complex wavelettransform is another form.

Properties

The Haar DWT illustrates the desirable properties of wavelets in general. First, it can beperformed in O(n) operations; second, it captures not only a notion of the frequency contentof the input, by examining it at different scales, but also temporal content, i.e. the times atwhich these frequencies occur. Combined, these two properties make the Fast wavelettransform (FWT) an alternative to the conventional Fast Fourier Transform (FFT).

Applications

The discrete wavelet transform has a huge number of applications in science, engineering,mathematics and computer science. Most notably, it is used for signal coding, to represent adiscrete signal in a more redundant form, often as a preconditioning for data compression.

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