Quaternionic Gabor Filters

The three different types of signal phase, as described by Bülow (1999) are global, instantaneous, and local. Global phase is the angular phase of the complex Fourier transform of a signal. It gives a real-valued number for each point in the frequency domain that is indicative of the relative position of the frequency components. Instead of giving the phase of a certain frequency component, instantaneous and local phase give the phase at a certain position in a real signal. The instantaneous and local phases, however, do this in different ways. The instantaneous phase is the angular phase of the complex value at each signal position. The instantaneous phase has the drawback that while it provides local information, that information depends on the entire signal. This causes the instantaneous phase at any point to be effected by changes at any other point in the image, regardless of separation distance. To overcome this problem quadrature filters are employed. The local phase is defined as angular phase of the quadrature filter response at a particular position of the signal. In this paper, in order to obtain a local quaternion phase, quaterernonic Gabor filters are used. While quaternionic Gabor filters are not exact quaternonic quadrature filters, they are a good approximation to them.

Gabor filters are linear time-invariant (LTI) filters that exhibit many useful properties, which have led to their use in a wide range of signal processing applications. The impulse response of a two-dimensional Gabor filter is  

$$h(x,y;u_0,v_0,\sigma,\epsilon)=g(x,y;\sigma,\epsilon)e^{i 2\pi(u_0 x + v_0 y)}$$ (13)

where $g(x,y;\sigma)$ is the Gaussian with aspect ratio $\epsilon$, 

$$g(x,y;\sigma, \epsilon)=e^{-\frac{x^2+(\epsilon y)^2}{\sigma^2}}.$$ (14)

Analogous to equation 13, a quaternionic Gabor filter is defined such that the impulse response to the filter is,

$$\begin{eqnarray} h^q(x,y;u_0,v_0,\sigma,\epsilon)=g(x,y;\sigma,\epsilon)e^{i 2 \... ...number\ =g(x,y;\sigma,\epsilon)e^{i\omega_1x}e^{j\omega_2y}~~~~~~\end{eqnarray}$$ (15)

where $g(x,y;\sigma,\epsilon)$ is defined as in equation 14. The quaternionic Gabor filter can be split into its even and odd symmetries, just as the QFT and the original image. In that case the filter h can be written as

$$h^q(x,y;u_0,v_0,\sigma,\epsilon)=(h^q_{ee}+ih^q_{oe}+jh^q_{eo}+kh^q_{oo}).$$ (16)

Note that h^q_ee, h^q_oe, h^q_eo, and h^q_oo are real-valued functions (see Figure ).

 

hs
Figure 1 The h_ee, h_oe, h_eo, and h_oo symmetries of a quaternionic Gabor filter