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]

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 of a Gabor wavelet, which is also a Gabor wavelet:

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

An example wavelet is given here:

A Gabor wavelet with a = 2, x₀ = 0, and k₀ = 1

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.

• Gabor transform
• Gabor filter
• Morlet wavelet

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