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Gabor wavelet

The motivation for Gabor wavelets comes from finding some function ${\displaystyle f($x)} which minimizes its standard deviation in the time and frequency domains. More formally, the variance in the position domain is:

${\displaystyle (\Delta x)^{2}={\frac {\int _{-\infty }^{\infty }(x-\mu )^{2}f($x)f^{*}(x)\,dx}{\int _{-\infty }^{\infty }f(x)f^{*}(x)\,dx}}}

where ${\displaystyle f^{*}($x)} is the complex conjugate of ${\displaystyle f($x)} and ${\displaystyle \mu }$ is the arithmetic mean, defined as:

${\displaystyle \mu ={\frac {\int _{-\infty }^{\infty }xf($x)f^{*}(x)\,dx}{\int _{-\infty }^{\infty }f(x)f^{*}(x)\,dx}}}

The variance in the wave number domain is:

${\displaystyle (\Delta k)^{2}={\frac {\int _{-\infty }^{\infty }(k-k_{0})^{2}F($k)F^{*}(k)\,dk}{\int _{-\infty }^{\infty }F(k)F^{*}(k)\,dk}}}

Where ${\displaystyle k_{0}}$ is the arithmetic mean of the Fourier Transform of ${\displaystyle f($x)} , ${\displaystyle F($x)} :

${\displaystyle k_{0}={\frac {\int _{-\infty }^{\infty }kF($k)F^{*}(k)\,dk}{\int _{-\infty }^{\infty }F(k)F^{*}(k)\,dk}}}

With these defined, the uncertainty is written as:

${\displaystyle (\Delta x)(\Delta k)}$

This quantity has been shown to have a lower bound of ${\displaystyle {\frac {1}{2}}}$. The quantum mechanics view is to interpret ${\displaystyle (\Delta x)}$ as the uncertainty in position and ${\displaystyle \hbar (\Delta k)}$ as uncertainty in momentum. A function ${\displaystyle f($x)} that has the lowest theoretically possible uncertainty bound is the Gabor Wavelet.^[3]

Equation[edit]

The equation of a 1-D Gabor wavelet is a Gaussian modulated by a complex exponential, described as follows:^[3]

${\displaystyle f($x)=e^{-(x-x_{0})^{2}/a^{2}}e^{-ik_{0}(x-x_{0})}}

As opposed to other functions commonly used as bases in Fourier Transforms such as ${\displaystyle \sin }$ and ${\displaystyle \cos }$, Gabor wavelets have the property that they are localized, meaning that as the distance from the center ${\displaystyle x_{0}}$ increases, the value of the function becomes exponentially suppressed. ${\displaystyle a}$ controls the rate of this exponential drop-off and ${\displaystyle k_{0}}$ controls the rate of modulation.

It is also worth noting the Fourier transform (unitary, angular-frequency convention) of a Gabor wavelet, which is also a Gabor wavelet:

${\displaystyle F($k)=ae^{-(k-k_{0})^{2}a^{2}}e^{-ix_{0}(k-k_{0})}}

An example wavelet is given here:

Time-causal analogue of the Gabor wavelet[edit]

When processing temporal signals, data from the future cannot be accessed, which leads to problems if attempting to use Gabor functions for processing real-time signals that depend upon the temporal dimension. A time-causal analogue of the Gabor filter has been developed in ^[4] based on replacing the Gaussian kernel in the Gabor function with a time-causal and time-recursive smoothing kernel referred to as the time-causal limit kernel. In this way, time-frequency analysis based on the resulting complex-valued extension of the time-causal limit kernel makes it possible to capture essentially similar transformations of a temporal signal as the Gabor wavelets can handle, and corresponding to the Heisenberg group, while carried out with strictly time-causal and time-recursive operations, see ^[4] for further details.

See also[edit]

• Gabor transform
• Gabor filter
• Morlet wavelet

References[edit]

External links[edit]

• MATLAB code for 2D Gabor wavelets and Gabor feature extraction