Namespaces
Variants
Actions

Difference between revisions of "Gabor transform"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (AUTOMATIC EDIT (latexlist): Replaced 17 formulas out of 17 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.)
 
Line 1: Line 1:
An [[Integral transform|integral transform]] introduced by D. Gabor, the Hungarian-born Nobel laureate in physics, who, in his paper [[#References|[a3]]], modified the well-known [[Fourier transform|Fourier transform]] of a function (or a signal) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200101.png" /> by introducing a time-localization window function (also called a time-frequency window). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200102.png" /> denote the Fourier transform
+
<!--This article has been texified automatically. Since there was no Nroff source code for this article,  
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200103.png" /></td> </tr></table>
+
Out of 17 formulas, 17 were replaced by TEX code.-->
  
and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200104.png" /> denote the Gaussian function
+
{{TEX|semi-auto}}{{TEX|done}}
 +
An [[Integral transform|integral transform]] introduced by D. Gabor, the Hungarian-born Nobel laureate in physics, who, in his paper [[#References|[a3]]], modified the well-known [[Fourier transform|Fourier transform]] of a function (or a signal) $f \in L ^ { 2 } ( \mathbf{R} )$ by introducing a time-localization window function (also called a time-frequency window). Let $\hat { f } ( \omega )$ denote the Fourier transform
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200105.png" /></td> </tr></table>
+
\begin{equation*} \hat { f } ( \omega ) = \int _ { - \infty } ^ { \infty } e ^ { - i \omega t } f ( t ) d t, \end{equation*}
  
Then the Gabor transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200106.png" /> is defined by
+
and let $g _ { \alpha } ( t )$ denote the Gaussian function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200107.png" /></td> </tr></table>
+
\begin{equation*} g _ { \alpha } ( t ) = \frac { 1 } { 2 \sqrt { \pi \alpha } } e ^ { - t ^ { 2 } / ( 4 \alpha ) } , \alpha &gt; 0. \end{equation*}
  
where the real parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200108.png" /> is used to translate the  "window"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g1200109.png" />. The Gabor transform localizes the Fourier transform at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g12001010.png" />. A similar transform can be introduced for Fourier series.
+
Then the Gabor transform of $f \in L ^ { 2 } ( \mathbf{R} )$ is defined by
  
By choosing more general windows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g12001011.png" />, the transforms are called short-time Fourier transform and the Gabor transform is a special case, based on the Gaussian window. One property of the special choice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g12001012.png" /> is
+
\begin{equation*} ( G _ { b } ^ { \alpha } f ) ( \omega ) = \int _ { - \infty } ^ { \infty } \left[ e ^ { - i \omega t } f ( t ) \right] g _ { \alpha } ( t - b ) d t, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g12001013.png" /></td> </tr></table>
+
where the real parameter $b$ is used to translate the  "window" $g _ { \alpha } ( t )$. The Gabor transform localizes the Fourier transform at $t = b$. A similar transform can be introduced for Fourier series.
  
which says that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g12001014.png" /> of Gabor transforms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g12001015.png" /> decomposes the Fourier transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g12001016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120010/g12001017.png" /> exactly.
+
By choosing more general windows $g$, the transforms are called short-time Fourier transform and the Gabor transform is a special case, based on the Gaussian window. One property of the special choice $g _ { \alpha } ( t )$ is
 +
 
 +
\begin{equation*} \int _ { - \infty } ^ { \infty } ( G _ { b } ^ { \alpha } f ) ( \omega ) d  b  = \hat { f } ( \omega ), \end{equation*}
 +
 
 +
which says that the set $\{ G _ { b } ^ { \alpha } f : b \in \mathbf{R} \}$ of Gabor transforms of $f$ decomposes the Fourier transform $\hat { f }$ of $f$ exactly.
  
 
Gabor transforms (and related topics based on the Gabor transform) are applied in numerous engineering applications, many of them without obvious connection to the traditional field of time-frequency analysis for deterministic signals. Detailed information (including many references) about the use of Gabor transforms in such diverse fields as image analysis, object recognition, optics, filter banks, or signal detection can be found in [[#References|[a4]]], the first book devoted to Gabor transforms and related analysis.
 
Gabor transforms (and related topics based on the Gabor transform) are applied in numerous engineering applications, many of them without obvious connection to the traditional field of time-frequency analysis for deterministic signals. Detailed information (including many references) about the use of Gabor transforms in such diverse fields as image analysis, object recognition, optics, filter banks, or signal detection can be found in [[#References|[a4]]], the first book devoted to Gabor transforms and related analysis.
Line 24: Line 32:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Ch.K. Chui,  "An introduction to wavelets" , Acad. Press  (1992)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I. Daubechies,  "Ten lectures on wavelets" , SIAM (Soc. Industrial Applied Math.)  (1992)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Gabor,  "Theory of communication"  ''J. IEE'' , '''93'''  (1946)  pp. 429–457</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H.G. Feichtinger,  Th. Strohmer,  "Gabor analysis and algorithms" , Birkhäuser  (1998)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  Ch.K. Chui,  "An introduction to wavelets" , Acad. Press  (1992)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  I. Daubechies,  "Ten lectures on wavelets" , SIAM (Soc. Industrial Applied Math.)  (1992)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  D. Gabor,  "Theory of communication"  ''J. IEE'' , '''93'''  (1946)  pp. 429–457</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  H.G. Feichtinger,  Th. Strohmer,  "Gabor analysis and algorithms" , Birkhäuser  (1998)</td></tr></table>

Latest revision as of 17:00, 1 July 2020

An integral transform introduced by D. Gabor, the Hungarian-born Nobel laureate in physics, who, in his paper [a3], modified the well-known Fourier transform of a function (or a signal) $f \in L ^ { 2 } ( \mathbf{R} )$ by introducing a time-localization window function (also called a time-frequency window). Let $\hat { f } ( \omega )$ denote the Fourier transform

\begin{equation*} \hat { f } ( \omega ) = \int _ { - \infty } ^ { \infty } e ^ { - i \omega t } f ( t ) d t, \end{equation*}

and let $g _ { \alpha } ( t )$ denote the Gaussian function

\begin{equation*} g _ { \alpha } ( t ) = \frac { 1 } { 2 \sqrt { \pi \alpha } } e ^ { - t ^ { 2 } / ( 4 \alpha ) } , \alpha > 0. \end{equation*}

Then the Gabor transform of $f \in L ^ { 2 } ( \mathbf{R} )$ is defined by

\begin{equation*} ( G _ { b } ^ { \alpha } f ) ( \omega ) = \int _ { - \infty } ^ { \infty } \left[ e ^ { - i \omega t } f ( t ) \right] g _ { \alpha } ( t - b ) d t, \end{equation*}

where the real parameter $b$ is used to translate the "window" $g _ { \alpha } ( t )$. The Gabor transform localizes the Fourier transform at $t = b$. A similar transform can be introduced for Fourier series.

By choosing more general windows $g$, the transforms are called short-time Fourier transform and the Gabor transform is a special case, based on the Gaussian window. One property of the special choice $g _ { \alpha } ( t )$ is

\begin{equation*} \int _ { - \infty } ^ { \infty } ( G _ { b } ^ { \alpha } f ) ( \omega ) d b = \hat { f } ( \omega ), \end{equation*}

which says that the set $\{ G _ { b } ^ { \alpha } f : b \in \mathbf{R} \}$ of Gabor transforms of $f$ decomposes the Fourier transform $\hat { f }$ of $f$ exactly.

Gabor transforms (and related topics based on the Gabor transform) are applied in numerous engineering applications, many of them without obvious connection to the traditional field of time-frequency analysis for deterministic signals. Detailed information (including many references) about the use of Gabor transforms in such diverse fields as image analysis, object recognition, optics, filter banks, or signal detection can be found in [a4], the first book devoted to Gabor transforms and related analysis.

A recent development (starting at 1992) that is more effective for analyzing signals with sharp variations is based on wavelets (see [a2] or Wavelet analysis); for the relation between wavelets and the Gabor transform, see [a1]. The Gabor transform can also be viewed in connection with "coherent states" associated with the Weyl–Heisenberg group; see [a4].

References

[a1] Ch.K. Chui, "An introduction to wavelets" , Acad. Press (1992)
[a2] I. Daubechies, "Ten lectures on wavelets" , SIAM (Soc. Industrial Applied Math.) (1992)
[a3] D. Gabor, "Theory of communication" J. IEE , 93 (1946) pp. 429–457
[a4] H.G. Feichtinger, Th. Strohmer, "Gabor analysis and algorithms" , Birkhäuser (1998)
How to Cite This Entry:
Gabor transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gabor_transform&oldid=50367
This article was adapted from an original article by N.M. Temme (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article