Gabor filter

 ( Linear Filters ) 

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 mapping 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 can be modeled by Gabor functions. Thus, image analysis with Gabor filters is thought by some to be similar to perception in the Visual system.

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Definition Its impulse response is defined by a sine wave 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 number and an imaginary number component representing orthogonal directions.3D surface tracking and approximation using Gabor filters, Jesper Juul Henriksen, South Denmark University, March 28, 2007 The two components may be formed into a complex number or used individually.

Complex

$$V=1+1+2+4+8+16+32+64+128+256+...$$

Real

$V$

= 1 + 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + . . .

Imaginary

$g(x,y;\backslash lambda,\backslash theta,\backslash psi,\backslash sigma,\backslash gamma)\; =\; \backslash exp\backslash left(-\backslash frac\{x\text{'}^2+\backslash gamma^2y\text{'}^2\}\{2\backslash sigma^2\}\backslash right)\backslash sin\backslash left(2\backslash pi\backslash frac\{x\text{'}\}\{\backslash lambda\}+\backslash psi\backslash right)$

where $x\text{'}\; =\; x\; \backslash cos\backslash theta\; +\; y\; \backslash sin\backslash theta$ and $y\text{'}\; =\; -x\; \backslash sin\backslash theta\; +\; y\; \backslash cos\backslash theta$.

In this equation, $\backslash lambda$ represents the wavelength of the sinusoidal factor, $\backslash theta$ represents the orientation of the normal to the parallel stripes of a Gabor function, $\backslash psi$ is the phase offset, $\backslash sigma$ is the sigma/standard deviation of the Gaussian envelope and $\backslash gamma$ is the spatial aspect ratio, and specifies the ellipticity of the support of the Gabor function.

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Wavelet space Gabor filters are directly related to , 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.

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Time-causal analogue of the Gabor filter When processing temporal signals, data from the future cannot be accessed, which leads to problems if attempting to use Gabor functions for processing real-time signals that depend on the temporal dimension. A time-causal analogue of the Gabor filter has been developed in based on replacing the Gaussian kernel in the Gabor function with a time-causal and time-recursive kernel referred to as the time-causal limit kernel. In this way, time-frequency analysis based on the resulting complex-valued extension of the time-causal limit kernel makes it possible to capture essentially similar transformations of a temporal signal as the Gabor filter can, and as can be described by the Heisenberg group, see for further details.

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Extraction of features from images

A set of Gabor filters with different frequencies and orientations may be helpful for extracting useful features from an image.

(2026). 9783642402456 ISBN 9783642402456

In the discrete domain, two-dimensional Gabor filters are given by,

$$G\_ci,j\; =B*e^$$

$G\_si,j\; =\; C\; e^\{-\backslash frac\{(i^2+j^2)\}\{2\backslash sigma^2\}\}\; \backslash sin(2\backslash pi\; f(i\backslash cos\backslash theta+j\backslash sin\backslash theta))$

where B and C are normalizing factors to be determined.

2D Gabor filters have rich applications in image processing, especially in feature extraction for texture analysis and segmentation.

(2026). 9780780376168, IEEE. ISBN 9780780376168

$f$ defines the frequency being looked for in the texture. By varying $/theta$, we can look for texture oriented in a particular direction. By varying $/sigma$, we change the support of the basis or the size of the image region being analyzed.

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Applications of 2D Gabor filters in image processing In document image processing, Gabor features are ideal for identifying the script of a word in a multilingual document. Gabor filters with different frequencies and with orientations in different directions have been used to localize and extract text-only regions from complex document images (both gray and colour), since text is rich in high frequency components, whereas pictures are relatively smooth in nature.

(2026). 9780769520889, IEEE. ISBN 9780769520889

(2026). 9783540307501 . ISBN 9783540307501

S Sabari Raju, P B Pati and A G Ramakrishnan, “Text Localization and Extraction from Complex Color Images,” Proc. First International Conference on Advances in Visual Computing (ISVC05), Nevada, USA, LNCS 3804, Springer Verlag, Dec. 5-7, 2005, pp. 486-493. It has also been applied for facial expression recognition

(1998). 9780818683442 . ISBN 9780818683442

Gabor filters have also been widely used in pattern analysis applications. For example, it has been used to study the directionality distribution inside the porous spongy trabecular bone in the Vertebral column. The Gabor space is very useful in image processing applications such as optical character recognition, iris recognition and fingerprint recognition. Relations between activations for a specific spatial location are very distinctive between objects in an image. Furthermore, important activations can be extracted from the Gabor space in order to create a sparse object representation.

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Example implementations

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Python This is an example implementation in Python:

import numpy as np

def gabor(sigma, theta, Lambda, psi, gamma):

   """Gabor feature extraction."""
   sigma_x = sigma
   sigma_y = float(sigma) / gamma
     

   # Bounding box
   nstds = 3  # Number of standard deviation sigma
   xmax = max(
       abs(nstds * sigma_x * np.cos(theta)), abs(nstds * sigma_y * np.sin(theta))
   )
   xmax = np.ceil(max(1, xmax))
   ymax = max(
       abs(nstds * sigma_x * np.sin(theta)), abs(nstds * sigma_y * np.cos(theta))
   )
   ymax = np.ceil(max(1, ymax))
   xmin = -xmax
   ymin = -ymax
   (y, x) = np.meshgrid(np.arange(ymin, ymax + 1), np.arange(xmin, xmax + 1))
     

   # Rotation
   x_theta = x * np.cos(theta) + y * np.sin(theta)
   y_theta = -x * np.sin(theta) + y * np.cos(theta)
     

   gb = np.exp(
       -0.5 * (x_theta**2 / sigma_x**2 + y_theta**2 / sigma_y**2)
   ) * np.cos(2 * np.pi / Lambda * x_theta + psi)
   return gb
     

For an implementation on images, see [1].

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MATLAB This is an example implementation in MATLAB/GNU Octave:

function gb = gabor_fn(sigma, theta, lambda, psi, gamma)

sigma_x = sigma; sigma_y = sigma / gamma;

% Bounding box nstds = 3; xmax = max(abs(nstds * sigma_x * cos(theta)), abs(nstds * sigma_y * sin(theta))); xmax = ceil(max(1, xmax)); ymax = max(abs(nstds * sigma_x * sin(theta)), abs(nstds * sigma_y * cos(theta))); ymax = ceil(max(1, ymax)); xmin = -xmax; ymin = -ymax; x,y = meshgrid(xmin:xmax, ymin:ymax);

% Rotation x_theta = x * cos(theta) + y * sin(theta); y_theta = -x * sin(theta) + y * cos(theta);

gb = exp(-.5*(x_theta.^2/sigma_x^2+y_theta.^2/sigma_y^2)).*cos(2*pi/lambda*x_theta+psi);

Code for Gabor feature extraction from images in MATLAB can be found at http://www.mathworks.com/matlabcentral/fileexchange/44630.

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Haskell This is another example implementation in Haskell:

import Data.Complex gabor λ θ ψ σ γ x y = exp(-(x'^2 + γ^2 * y'^2) / (2*σ^2)) * exp(i * (2*pi*x'/λ + ψ))

   where x' =  x * cos θ + y * sin θ
         y' = -x * sin θ + y * cos θ
         i  = 0 :+ 1
     

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See also

• Gabor transform
• Gabor wavelet
• Gabor atom
• Log Gabor filter

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External links

• MATLAB code for Gabor filters and Gabor feature extraction
• 3D Gabor demonstrated with Mathematica
• python implementation of log-Gabors for still images
• Gabor filter for image processing and computer vision (demonstration)

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Further reading

• (1998). 9780817639594, Birkhäuser. ISBN 9780817639594
• Gröchenig, Karlheinz (2026). 9780817640224, Birkhäuser. ISBN 9780817640224
• Steerable Pyramids:
1. Eero Simoncelli's page on Steerable Pyramids
2. ( PDF ) ( Code )

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