Gabor Filter

A Gabor Filter is a linear band-pass filter of the form [math]\displaystyle{ g(x,y;\lambda,\theta,\psi,\sigma,\gamma) = \exp\left(-\frac{x'^2+\gamma^2y'^2}{2\sigma^2}\right)\exp\left(i\left(2\pi\frac{x'}{\lambda}+\psi\right)\right) }[/math], where [math]\displaystyle{ x' = x \cos\theta + y \sin\theta\, }[/math] and : [math]\displaystyle{ y' = -x \sin\theta + y \cos\theta\, }[/math] In this equation, [math]\displaystyle{ \lambda }[/math] represents the wavelength of the sinusoidal factor, [math]\displaystyle{ \theta }[/math] represents the orientation of the normal to the parallel stripes, [math]\displaystyle{ \psi }[/math] is the phase offset, [math]\displaystyle{ \sigma }[/math] is the sigma/standard deviation of the Gaussian envelope and [math]\displaystyle{ \gamma }[/math] is the spatial aspect ratio.

• AKA: Gabor Function.
• Context:
  • It can range from being a 1-D Gabor Function to being a 2-D Gabor Function, to being an n-D Gabor Function.
  • It can minimize the space (time)-uncertainty product.
  • It can be used to obtain localised frequency information (both spatial and frequency information).
  • …
• Counter-Example(s):
  • a Kalman Filter.
  • a Gaussian Filter.
• See: Gabor Transform, Image Processing, Visual Cortex, Image Analysis, Human Visual System, Edge Detection.

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References

2017

• (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Gabor_filter#Definition Retrieved:2017-6-5.
  • 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 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 : [math]\displaystyle{ g(x,y;\lambda,\theta,\psi,\sigma,\gamma) = \exp\left(-\frac{x'^2+\gamma^2y'^2}{2\sigma^2}\right)\exp\left(i\left(2\pi\frac{x'}{\lambda}+\psi\right)\right) }[/math]
    • Real : [math]\displaystyle{ g(x,y;\lambda,\theta,\psi,\sigma,\gamma) = \exp\left(-\frac{x'^2+\gamma^2y'^2}{2\sigma^2}\right)\cos\left(2\pi\frac{x'}{\lambda}+\psi\right) }[/math]
    • Imaginary : [math]\displaystyle{ g(x,y;\lambda,\theta,\psi,\sigma,\gamma) = \exp\left(-\frac{x'^2+\gamma^2y'^2}{2\sigma^2}\right)\sin\left(2\pi\frac{x'}{\lambda}+\psi\right) }[/math] where : [math]\displaystyle{ x' = x \cos\theta + y \sin\theta\, }[/math] and : [math]\displaystyle{ y' = -x \sin\theta + y \cos\theta\, }[/math] In this equation, [math]\displaystyle{ \lambda }[/math] represents the wavelength of the sinusoidal factor, [math]\displaystyle{ \theta }[/math] represents the orientation of the normal to the parallel stripes of a Gabor function, [math]\displaystyle{ \psi }[/math] is the phase offset, [math]\displaystyle{ \sigma }[/math] is the sigma/standard deviation of the Gaussian envelope and [math]\displaystyle{ \gamma }[/math] is the spatial aspect ratio, and specifies the ellipticity of the support of the Gabor function.

2014

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Gabor_filter Retrieved:2014-7-25.
  • In image processing, a Gabor filter, named after Dennis Gabor, is a linear filter used for edge detection. Frequency and orientation representations of Gabor filters are similar to those of the human visual system, and 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.

    Simple cells in the visual cortex of mammalian brains can be modeled by Gabor functions. Thus, image analysis with Gabor filters is thought to be similar to perception in the human visual system.

2003

• (Jianwei et al., 2003) ⇒ Yang Jianwei, Lifeng Liu, Tianzi Jiang, and Yong Fan. (2003). “A modified Gabor filter design method for fingerprint image enhancement.” In: Pattern Recognition Letters 24, no. 12.
  • QUOTE: The Gabor function has been recognized as a very useful tool in computer vision and image processing, especially for texture analysis, due to its optimal localization properties in both spatial and frequency domain. There are lots of papers published on its applications since Gabor (1946) proposed the 1-D Gabor function. The family of 2-D Gabor filters was originally presented by Daugman (1980) as a framework for understanding the orientation-selective and spatial–frequency-selective receptive field properties of neurons in the brains visual cortex, and then was further mathematically elaborated (Daugman, 1985).
    The 2-D Gabor function is a harmonic oscillator, composed of a sinusoidal plane wave of a particular frequency and orientation, within a Gaussian envelope. A complex 2-D Gabor filter over the image domain …

1990

• (Jain & Farrokhnia, 1990) ⇒ Anil K. Jain, and Farshid Farrokhnia. "Unsupervised texture segmentation using Gabor filters.” In: Proceedings of the International Conference on Systems, Man and Cybernetics.

1987

• (Jones & Palmer, 1987) ⇒ Judson P. Jones , and Larry A. Palmer. (1987). “An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex.” In: Journal of neurophysiology, 58(6).

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