﻿ Wavelet_transform : definition of Wavelet_transform and synonyms of Wavelet_transform (English)
 »
Arabic Bulgarian Chinese Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hindi Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Malagasy Norwegian Persian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Thai Turkish Vietnamese
Arabic Bulgarian Chinese Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hindi Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Malagasy Norwegian Persian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Thai Turkish Vietnamese

# definition - Wavelet_transform

definition of Wikipedia

# analogical dictionary

Editeurs (fr)[Domaine]

Domaines (fr)[Domaine]

wavelet transform (n.)

# Wavelet transform

An example of the 2D discrete wavelet transform that is used in JPEG2000.

In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform.

## Formal definition

A function $\psi\in L^2(\mathbb{R})$ is called an orthonormal wavelet if it can be used to define a Hilbert basis, that is a complete orthonormal system, for the Hilbert space $L^2(\mathbb{R})$ of square integrable functions. The Hilbert basis is constructed as the family of functions $\{\psi_{jk}:j,k\in\Z\}$ by means of dyadic translations and dilations of $\psi\,$,

$\psi_{jk}(x) = 2^{j/2} \psi(2^jx-k)\,$

for integers $j,k\in \mathbb{Z}$. This family is an orthonormal system if it is orthonormal under the inner product

$\langle\psi_{jk},\psi_{lm}\rangle = \delta_{jl}\delta_{km}$

where $\delta_{jl}\,$ is the Kronecker delta and $\langle f,g\rangle$ is the standard inner product $\langle f,g\rangle = \int_{-\infty}^\infty f(x)\overline{g(x)}dx$ on $L^2(\mathbb{R}).$ The requirement of completeness is that every function $h\in L^2(\mathbb{R})$ may be expanded in the basis as

$h(x)=\sum_{j,k=-\infty}^\infty c_{jk} \psi_{jk}(x)$

with convergence of the series understood to be convergence in norm. Such a representation of a function f is known as a wavelet series. This implies that an orthonormal wavelet is self-dual.

## Wavelet transform

The integral wavelet transform is the integral transform defined as

$\left[W_\psi f\right](a,b) = \frac{1}{\sqrt{|a|}} \int_{-\infty}^\infty \overline{\psi\left(\frac{x-b}{a}\right)}f(x)dx\,$

The wavelet coefficients $c_{jk}$ are then given by

$c_{jk}= \left[W_\psi f\right](2^{-j}, k2^{-j})$

Here, $a=2^{-j}$ is called the binary dilation or dyadic dilation, and $b=k2^{-j}$ is the binary or dyadic position.

## Wavelet compression

Wavelet compression is a form of data compression well suited for image compression (sometimes also video compression and audio compression). Notable implementations are JPEG 2000, DjVu and ECW for still images, REDCODE, CineForm, the BBC's Dirac, and Ogg Tarkin for video. The goal is to store image data in as little space as possible in a file. Wavelet compression can be either lossless or lossy.[1]

Using a wavelet transform, the wavelet compression methods are adequate for representing transients, such as percussion sounds in audio, or high-frequency components in two-dimensional images, for example an image of stars on a night sky. This means that the transient elements of a data signal can be represented by a smaller amount of information than would be the case if some other transform, such as the more widespread discrete cosine transform, had been used.

Wavelet compression is not good for all kinds of data: transient signal characteristics mean good wavelet compression, while smooth, periodic signals are better compressed by other methods, particularly traditional harmonic compression (frequency domain, as by Fourier transforms and related). Data statistically indistinguishable from random noise is not compressible by any means.

See Diary Of An x264 Developer: The problems with wavelets (2010) for discussion of practical issues of current methods using wavelets for video compression.

### Method

First a wavelet transform is applied. This produces as many coefficients as there are pixels in the image (i.e.: there is no compression yet since it is only a transform). These coefficients can then be compressed more easily because the information is statistically concentrated in just a few coefficients. This principle is called transform coding. After that, the coefficients are quantized and the quantized values are entropy encoded and/or run length encoded.

A few 1D and 2D applications of wavelet compression use a technique called "wavelet footprints".[2][3]

## Other practical applications

The wavelet transform can provide us with the frequency of the signals and the time associated to those frequencies, making it very convenient for its application in numerous fields. For instance, signal processing of accelerations for gait analysis[4].

## References

• Chui, Charles K. (1992). An Introduction to Wavelets. San Diego: Academic Press. ISBN 0-12-174584-8.
1. ^ JPEG 2000, for example, may use a 5/3 wavelet for lossless (reversible) transform and a 9/7 wavelet for lossy (irreversible) transform.
2. ^ N. Malmurugan, A. Shanmugam, S. Jayaraman and V. V. Dinesh Chander. "A New and Novel Image Compression Algorithm Using Wavelet Footprints"
3. ^ Ho Tatt Wei and Jeoti, V. "A wavelet footprints-based compression scheme for ECG signals". Ho Tatt Wei; Jeoti, V. (2004). A wavelet footprints-based compression scheme for ECG signals. A. pp. 283. DOI:10.1109/TENCON.2004.1414412.  edit
4. ^ "Novel method for stride length estimation with body area network accelerometers", IEEE BioWireless 2011, pp. 79-82

sensagent's content

• definitions
• synonyms
• antonyms
• encyclopedia

Dictionary and translator for handheld

New : sensagent is now available on your handheld

sensagent's office

Shortkey or widget. Free.

Windows Shortkey: . Free.

Vista Widget : . Free.

Webmaster Solution

Alexandria

A windows (pop-into) of information (full-content of Sensagent) triggered by double-clicking any word on your webpage. Give contextual explanation and translation from your sites !

Try here  or   get the code

SensagentBox

With a SensagentBox, visitors to your site can access reliable information on over 5 million pages provided by Sensagent.com. Choose the design that fits your site.

WordGame

The English word games are:
○   Anagrams
○   Wildcard, crossword
○   Lettris
○   Boggle.

Lettris

Lettris is a curious tetris-clone game where all the bricks have the same square shape but different content. Each square carries a letter. To make squares disappear and save space for other squares you have to assemble English words (left, right, up, down) from the falling squares.

boggle

Boggle gives you 3 minutes to find as many words (3 letters or more) as you can in a grid of 16 letters. You can also try the grid of 16 letters. Letters must be adjacent and longer words score better. See if you can get into the grid Hall of Fame !

English dictionary
Main references

Most English definitions are provided by WordNet .
English thesaurus is mainly derived from The Integral Dictionary (TID).
English Encyclopedia is licensed by Wikipedia (GNU).

The wordgames anagrams, crossword, Lettris and Boggle are provided by Memodata.
The web service Alexandria is granted from Memodata for the Ebay search.
The SensagentBox are offered by sensAgent.

Translation

Change the target language to find translations.
Tips: browse the semantic fields (see From ideas to words) in two languages to learn more.

last searches on the dictionary :

4569 online visitors

computed in 0.047s

I would like to report:
section :
a spelling or a grammatical mistake
an offensive content(racist, pornographic, injurious, etc.)
an error
a missing statement
other

My account

registration