In image processing, a Gabor filter, named after Dennis Gabor, who first proposed it as a 1D filter, is a linear filter used for texture analysis, which essentially means that it analyzes whether there is any specific frequency content in the image in specific directions in a localized region around the point or region of analysis. The Gabor filter was first generalized to 2D by Gösta Granlund, by adding a reference direction. Frequency and orientation representations of Gabor filters are claimed by many contemporary vision scientists to be similar to those of the human visual system. They have been found to be particularly appropriate for texture representation and discrimination. In the spatial domain, a 2D Gabor filter is a Gaussian kernel function modulated by a sinusoidal plane wave (see Gabor transform). Some authors claim that simple cells in the visual cortex of mammalian brains can be modeled by Gabor functions. Thus, image analysis with Gabor filters is thought by some to be similar to perception in the human visual system.
Definition
Its impulse response is defined by a sinusoidal wave (a plane wave for 2D Gabor filters) multiplied by a Gaussian function. Because of the multiplication-convolution property (Convolution theorem), the Fourier transform of a Gabor filter's impulse response is the convolution of the Fourier transform of the harmonic function (sinusoidal function) and the Fourier transform of the Gaussian function. The filter has a real and an imaginary component representing orthogonal directions. The two components may be formed into a complex number or used individually. Complex
V = 1 + 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + . . . {\displaystyle V=1+1+2+4+8+16+32+64+128+256+...}
Real
V = 1 + 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + . . . {\textstyle V=1+1+2+4+8+16+32+64+128+256+...}
Imaginary
g ( x , y ; λ , θ , ψ , σ , γ ) = exp ( − x ′ 2 + γ 2 y ′ 2 2 σ 2 ) sin ( 2 π x ′ λ + ψ ) {\displaystyle g(x,y;\lambda ,\theta ,\psi ,\sigma ,\gamma )=\exp \left(-{\frac {x'^{2}+\gamma ^{2}y'^{2}}{2\sigma ^{2}}}\right)\sin \left(2\pi {\frac {x'}{\lambda }}+\psi \right)}
where x ′ = x cos θ + y sin θ {\displaystyle x'=x\cos \theta +y\sin \theta } and y ′ = − x sin θ + y cos θ {\displaystyle y'=-x\sin \theta +y\cos \theta } . In this equation, λ {\displaystyle \lambda } represents the wavelength of the sinusoidal factor, θ {\displaystyle \theta } represents the orientation of the normal to the parallel stripes of a Gabor function, ψ {\displaystyle \psi } is the phase offset, σ {\displaystyle \sigma } is the sigma/standard deviation of the Gaussian envelope and γ {\displaystyle \gamma } is the spatial aspect ratio, and specifies the ellipticity of the support of the Gabor function.
Wavelet space
Gabor filters are directly related to Gabor wavelets, since they can be designed for a number of dilations and rotations. However, in general, expansion is not applied for Gabor wavelets, since this requires computation of bi-orthogonal wavelets, which may be very time-consuming. Therefore, usually, a filter bank consisting of Gabor filters with various scales and rotations is created. The filters are convolved with the signal, resulting in a so-called Gabor space. This process is closely related to processes in the primary visual cortex. Jones and Palmer showed that the real part of the complex Gabor function is a good fit to the receptive field weight functions found in simple cells in a cat's striate cortex.
… excerpt ends here. Continue reading the full article.
