Gabor transformation

The Gabor transformation (after Dennis Gábor ) is a special (and in a certain way optimal) windowed Fourier transformation . It is closely related to the wavelet theory and is used in many areas of digital signal and image processing . It is a special case of the short-term Fourier transform .

General

Two-dimensional Gabor wavelet

Every local change in a signal causes a change in the Fourier transform (FT) over the entire frequency axis. For example, the FT graph of the delta distribution (Dirac function) covers the entire frequency range. The FT therefore does not contain any local information about the signal . On the other hand, this means that the information in the frequency spectrum does not directly indicate the local area in which the frequency occurs. One way of locating the FT in the spatial domain is the short-time Fourier transform ( English short-time Fourier transform , short STFT ), the local frequency content in a window around the point describes. A function that drops quickly to 0 is usually selected for this so that it acts as a window. ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle \tau }$ ${\displaystyle g}$

${\displaystyle F^{\mathrm {Fen} }(\omega ,\tau )=\int \limits _{-\infty }^{+\infty }f(t)g(t-\tau )e^{-i\omega t}dt}$

The Fourier transform with window is therefore dependent on two parameters, the frequency and the center of the localization . One therefore speaks of a representation in the spatial / frequency space . ${\displaystyle \omega }$ ${\displaystyle \tau }$

The STFT with a Gaussian function as a window function was used by Dennis Gábor in 1946: ${\displaystyle g_{\sigma }(t)}$

${\displaystyle g_{\sigma }(t)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {t^{2}}{2\sigma ^{2}}}}}$

This special STFT is called the Gabor transformation . If one denotes the result of the Gave transformation of with, then because of the symmetry of ${\displaystyle f}$ ${\displaystyle G_{f}}$ ${\displaystyle g_{\sigma }}$

${\displaystyle {\begin{aligned}G_{f}(\omega ,\tau )&=\int \limits _{-\infty }^{+\infty }f(t)g_{\sigma }(t-\tau )e^{-i\omega t}dt\\&=e^{-i\omega \tau }\int \limits _{-\infty }^{+\infty }f(t)g_{\sigma }(\tau -t)e^{i\omega (\tau -t)}dt\\&=e^{-i\omega \tau }(f(\tau )\ast (g_{\sigma }(\tau )e^{i\omega \tau }))\\&=e^{-i\omega \tau }(f(\tau )\ast h(\tau ))\end{aligned}}}$

In the local space, Gabor filtering therefore represents a convolution up to the factor . However, this factor only causes a phase shift and can therefore be neglected in applications that only take the amplitude of the result into account. ${\displaystyle e^{-i\omega \tau }}$

Since the Fourier transformation of a Gaussian function again produces a Gaussian function, the result of the Gave transformation represents local information in both the spatial and the frequency space. The filter can cover any elliptical region of the frequency or spatial space. Furthermore, the Gave transformation achieves - regardless of the arrangement - maximum simultaneous resolution in the spatial and frequency space, i.e. the Gaussian function reaches the minimum of the uncertainty relation as the (only) window function , whereby the variance of the window function in the spatial space (spatial uncertainty) and correspondingly the in Specifies frequency space (frequency uncertainty). This directly results in the reciprocal connection between the blurring and thus an important trade-off . This means that in order to double the resolution in the spatial area, halving the resolution in the frequency area must be accepted, and vice versa. ${\displaystyle \sigma _{g}^{2}\cdot \sigma _{G}^{2}\geq {\tfrac {\pi }{2}}}$ ${\displaystyle \sigma _{g}^{2}}$ ${\displaystyle \sigma _{G}^{2}}$

Filters with a low bandwidth in the frequency domain are desirable because they allow a fine distinction between different textures . On the other hand, filters that have a narrow bandwidth in the spatial area are required for precise detection of texture boundaries.

Another interesting property of Gabor filters is that they appear to be a good approximation of the sensitivity profiles of neurons in the visual cortex , in that they process frequency and direction-specific signals.

See also

• Laplace transform
• Discrete Fourier Transform
• Discrete cosine transformation
• Wavelet transform

literature

• Hans G. Feichtinger, Thomas Strohmer: "Gabor Analysis and Algorithms", Birkhäuser, 1998; ISBN 0817639594
• Hans G. Feichtinger, Thomas Strohmer: "Advances in Gabor Analysis", Birkhäuser, 2003; ISBN 0817642390
• Karlheinz Gröchenig: "Foundations of Time-Frequency Analysis", Birkhäuser, 2001; ISBN 0817640223

Web links

• NuHAG homepage: many more links
• Gabor server