1 The Temporal (1-D) Gabor Filter
Gabor filters can serve as excellent band-pass filters for unidimensionnal signals (e.g., speech)
A complex Gabor filter is defined as the product of a Gaussian kernel times a complex sinusoid, i.e.
where
\(w(t)=e^{-\pi t^{2}}\)
\(s(t)=e^{j(2\pi f_{0}t)}\)
Here \(k,\theta,f_{0}\) are filter parameters. We can think of the complex Gabor filter as two out of phase filters continently allocated in the real and complex part of a complex function, the real part holds the filter
and the imaginary part holds the filter
\[g_{i}(t)=w(t)cos(2\pi f_{0}t+\theta) \]1.1 Frequency Response
Taking the Fourier transform
\[\hat{g}(f)=ke^{j\theta}\int_{-\infty}^{\infty}e^{-j2\pi ft}w(at)s(t)dt=\frac{k}{a}e^{j\theta}\hat{w}(\frac{f-f_{0}}{a}) \]where
\(\hat{w}(f)=w(f)=e^{-\pi f^{2}}\)
1.2 Gabor Energy Filters
The real and imaginary components of a complex Gabor filter are phase sensitive, i.e., as a consequence their response to a sinusoid is another sinusoid (Fig 1.2).
In the frequency domain, the magnitude of the response to a particular frequency is simply the magnitude of the complex Fourier transform, i.e.
Note this is a Gaussion function centered at \(f_{0}\) and with width proportional to \(alpha\).
Fig 1. Top: An input signal. Second: Output of Gabor filter (cosine carrier). Third: Output of Gabor Filter in quadrature (sine carrier). Fourth: Output of Gabor Energy Filter.
1.2.1 Bandwidth and Peak Response
Thus the peak filter response is at \(f_{0}\). To get the half-magnitude bandwidth \(\Delta_{f}\) note
\[\hat{w}(\frac{f-f_{0}}{a})=e^{-\pi\frac{f-f_{0}}{\alpha^2}}=0.5 \]Thus the half peak magnitude is achieved for
\[f-f_{0}\pm\sqrt{\alpha^{2}log2\pi}=0.4697\alpha\approx 0.5\alpha \]Thus the half-magnitude bandwidth is approximately equal to \(\alpha\).
1.3 Eliminating the DC response
Depending on the value of \(f_{0}\) and \(alpha\) the filter may have a large DC response. A popular approach to get a zero DC response is to subtract the output of a low-pass Gaussian filter
\[h(t)=g(t)-cw(bt)=ke^{j\theta}w(at)s(t)-cw(bt) \]Thus
\[\hat{h}(f)=\hat{g}(f)-\frac{c}{b}\hat{w}(\frac{f}{b}) \]To get a zero DC response we need
\[\frac{c}{b}\hat{w}(0)=\hat{g}(0) \]\[c=b\hat{g}(0)=b\frac{k}{a}e^{j\theta}\hat{w}(\frac{f_{0}}{\alpha}) \]Thus
\[h(t)=ke^{j\theta}(w(\alpha t)s(t)-\frac{b}{a}\hat{w}(\frac{f_{0}}{\alpha})w(bt)) \]\[\hat{h}(f)=\frac{k}{a}e^{j\theta}(\hat{w}(\frac{f-f_{0}}{\alpha})-\hat{w}(\frac{f_{0}}{a})\hat{w}(\frac{f}{b})) \]It is convenient, to let \(b=a\), in which case
\[h(t)=ke^{j\theta}w(\alpha t)(s(t)-\hat{w}(\frac{f_{0}}{a}) \]\[\hat{h}(f)=\frac{k}{a}e^{j\theta}(\hat{w}(\frac{f-f_{0}}{\alpha})-\hat{w}(\frac{f_{0}}{a})\hat{w}(\frac{f}{a})) \]2 The Spatial (2-D) Gabor Filter
Here is the formula of a complex Gabor function in space domain
\[g(x,y)=s(x,y)w_{r}(x,y) \]where \(s\) is a complex sinusoid, known as the carrier, and \(w_{r}\) is a 2-D Gaussian-shaped function, known as the envelope.