Gabor transform

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

General

Each local change of the signal causes a change in the Fourier transform (FT) on the entire frequency axis. For example, the graph of the FT of the delta function covers ( dirac function ), the entire frequency range. The FT therefore has no local information of the signal. This in turn means that the information of the frequency spectrum of the spatial domain, in the frequency occurs, without indicating directly. A way of localization of FT in the spatial domain is the Short-Time Fourier Transform ( WFT ), which describes the local frequency content in a window around the point. It is usually chosen for a fast falling to 0 function so that it acts as a window.

The Fourier transform window is therefore dependent on two parameters, the frequency and the center of the location. One therefore speaks of a representation in Orts-/Frequenzraum. The Fourier transform window is also referred to as short -time Fourier transform ( STFT )

The WFT with a Gaussian function as window function was used by Dennis Gabor 1946:

This special WFT called Gabor transform. Denoting the result of the Gabor transform with such results because of the symmetry of

In the spatial domain, therefore, the Gabor filtering up to a factor of a convolution This factor, however, only causes a phase shift and therefore, in applications which only consider the amplitude of the result, be neglected.

Since the Fourier transform of a Gaussian function is again a Gaussian function, represents the result of Gabor transformation both in the local as well as in the frequency domain local information is supplied, the filter can cover any elliptical region of the frequency or location of space. Furthermore, the Gabor transform achieved - regardless of the arrangement - maximum concurrent resolution in the spatial and frequency domain, ie, the Gaussian function achieved as the (sole ) window function the minimum of the uncertainty relation, the variance of the window function in the spatial domain ( position uncertainty ) and correspondingly in the frequency domain ( frequency blur) indicates. It follows directly the reciprocal relationship between the blur and thus an important trade-off. That is, to double the resolution in the spatial domain, a halved resolution in the frequency domain must be taken into account, and vice versa.

Filter with a low bandwidth in the frequency space are desirable because they allow a fine distinction between different textures. On the other hand necessary for accurate detection of texture limits filters having a small bandwidth in the spatial domain.

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