Gabor filter

<h2 id="definition">Definition</h2>

<p>Its <a href="/facts/Impulse_response/l4vaE5er">impulse response</a> is defined by a <a href="/facts/Sine_wave/mD1CB09M">sinusoidal</a> wave (a <a href="/facts/Plane_wave/0aDUbNhQ">plane wave</a> for 2D Gabor filters) multiplied by a <a href="/facts/Gaussian_function/5bG8Kpp9">Gaussian function</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
Because of the multiplication-convolution property (<a href="/facts/Convolution_theorem/E7zhjIx2">Convolution theorem</a>), the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> of a Gabor filter's impulse response is the <a href="/facts/Convolution/PVPrdz9J">convolution</a> of the Fourier transform of the harmonic function (sinusoidal function) and the Fourier transform of the Gaussian function.  The filter has a <a href="/facts/Real_number/R02gw5Pb">real</a> and an <a href="/facts/Imaginary_number/6Xh9W3hq">imaginary</a> component representing <a href="/facts/Orthogonal/JVIjYPog">orthogonal</a> directions.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>  The two components may be formed into a <a href="/facts/Complex_number/2w7mqI7e">complex number</a> or used individually.
</p><p>Complex
</p>

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        (
        x
        ,
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        ;
        λ
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        =
        exp
        ⁡
        
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            −
            
              
                
                  
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        exp
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    {\displaystyle g(x,y;\lambda ,\theta ,\psi ,\sigma ,\gamma )=\exp \left(-{\frac {x'^{2}+\gamma ^{2}y'^{2}}{2\sigma ^{2}}}\right)\exp \left(i\left(2\pi {\frac {x'}{\lambda }}+\psi \right)\right)}

g
        (
        x
        ,
        y
        ;
        λ
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        ψ
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        =
        exp
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        cos
        ⁡
        
          (
          
            2
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                  x
                  ′
                
                λ
              
            
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            ψ
          
          )
        
      
    
    {\displaystyle g(x,y;\lambda ,\theta ,\psi ,\sigma ,\gamma )=\exp \left(-{\frac {x'^{2}+\gamma ^{2}y'^{2}}{2\sigma ^{2}}}\right)\cos \left(2\pi {\frac {x'}{\lambda }}+\psi \right)}

<p>Imaginary
</p>

g
        (
        x
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        y
        ;
        λ
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        exp
        ⁡
        
          (
          
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                    x
                    
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        sin
        ⁡
        
          (
          
            2
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                  x
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          )
        
      
    
    {\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)}

<p>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 }
  
.
</p><p>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 <a href="/facts/Gabor_function/WrnLmgtl">Gabor function</a>, 
  
    
      
        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.
</p>
<h2 id="wavelet-space">Wavelet space</h2>

<p>Gabor filters are directly related to <a href="/facts/Gabor_wavelet/1EPTtlQm">Gabor wavelets</a>, 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 <a href="/facts/Visual_cortex/qmF8F5Sb">visual cortex</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
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.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>
</p>
<h2 id="time-causal-analogue-of-the-gabor-filter">Time-causal analogue of the Gabor filter</h2>
<p>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 <a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> 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 <a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> for further details.
</p>
<h2 id="extraction-of-features-from-images">Extraction of features from images</h2>
<p>A set of Gabor filters with different frequencies and orientations may be helpful for extracting useful features from an image.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> In the discrete domain, two-dimensional Gabor filters are given by,
</p>

G
          
            c
          
        
        [
        i
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        =
        B
        
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        cos
        ⁡
        (
        2
        π
        f
        (
        i
        cos
        ⁡
        θ
        +
        j
        sin
        ⁡
        θ
        )
        )
      
    
    {\displaystyle G_{c}[i,j]=Be^{-{\frac {(i^{2}+j^{2})}{2\sigma ^{2}}}}\cos(2\pi f(i\cos \theta +j\sin \theta ))}

G
          
            s
          
        
        [
        i
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        j
        ]
        =
        C
        
          e
          
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                    i
                    
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                    j
                    
                      2
                    
                  
                  )
                
                
                  2
                  
                    σ
                    
                      2
                    
                  
                
              
            
          
        
        sin
        ⁡
        (
        2
        π
        f
        (
        i
        cos
        ⁡
        θ
        +
        j
        sin
        ⁡
        θ
        )
        )
      
    
    {\displaystyle G_{s}[i,j]=Ce^{-{\frac {(i^{2}+j^{2})}{2\sigma ^{2}}}}\sin(2\pi f(i\cos \theta +j\sin \theta ))}

<p>where <i>B</i> and <i>C</i> are normalizing factors to be determined.
</p><p>2D Gabor filters have rich applications in image processing, especially in <a href="/facts/Feature_extraction/BSMX5e8I">feature extraction</a> for texture analysis and segmentation.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> 
  
    
      
        f
      
    
    {\displaystyle f}
  
 defines the frequency being looked for in the texture. By varying 
  
    
      
        θ
      
    
    {\displaystyle \theta }
  
, we can look for texture oriented in a particular direction. By varying 
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
, we change the support of the basis or the size of the image region being analyzed.
</p>
<h2 id="applications-of-2d-gabor-filters-in-image-processing">Applications of 2D Gabor filters in image processing</h2>
<p>In document image processing, Gabor features are ideal for identifying the script of a word in a multilingual document.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> 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.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a><a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a><a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> It has also been applied for facial expression recognition <a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a>
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 <a href="/facts/Trabecular/3dfcqlwH">trabecular</a> <a href="/facts/Bone/kwJaR6IQ">bone</a> in the <a href="/facts/Vertebral_column/Nuw1kV7o">spine</a>.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> The Gabor space is very useful in <a href="/facts/Image_processing/zenalNCf">image processing</a> applications such as <a href="/facts/Optical_character_recognition/kdTyxa1l">optical character recognition</a>, <a href="/facts/Iris_recognition/8dt1HS19">iris recognition</a> and <a href="/facts/Fingerprint_recognition/5WLcd20T">fingerprint recognition</a>. 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.
</p>
<p class="note">See also: <a href="/facts/Morlet_wavelet/aTEPLUIM">Morlet wavelet</a></p>
<h2 id="example-implementations">Example implementations</h2>
<h3>Python</h3>
<p>This is an example implementation in <a href="/facts/Python_(programming_language)/YbuGqofa">Python</a>:
</p>
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

<p>For an implementation on images, see <a href="https://pythonhosted.org/LogGabor/">[1]</a>.
</p>
<h3>MATLAB</h3>
<p>This is an example implementation in <a href="/facts/MATLAB/qPjLISCk">MATLAB</a>/<a href="/facts/GNU_Octave/vVVq5evM">Octave</a>:
</p>
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);

<p>Code for Gabor feature extraction from images in <a href="/facts/MATLAB/qPjLISCk">MATLAB</a> can be found at <a href="http://www.mathworks.com/matlabcentral/fileexchange/44630">http://www.mathworks.com/matlabcentral/fileexchange/44630</a>.
</p>
<h3>Haskell</h3>
<p>This is another example implementation in <a href="/facts/Haskell_(programming_language)/4Htv9WLu">Haskell</a>:
</p>
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

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Gabor_transform/dbbHrA1l">Gabor transform</a></li>
<li><a href="/facts/Gabor_wavelet/1EPTtlQm">Gabor wavelet</a></li>
<li><a href="/facts/Gabor_atom/WrnLmgtl">Gabor atom</a></li>
<li><a href="/facts/Log_Gabor_filter/ExfeAI4f">Log Gabor filter</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.mathworks.com/matlabcentral/fileexchange/44630">MATLAB code for Gabor filters and Gabor feature extraction</a></li>
<li><a href="http://demonstrations.wolfram.com/Gabor3D/">3D Gabor demonstrated with Mathematica</a></li>
<li><a href="https://pythonhosted.org/LogGabor/">python implementation of log-Gabors for still images</a></li>
<li><a href="http://matlabserver.cs.rug.nl/edgedetectionweb/web/edgedetection_examples.html">Gabor filter for image processing and computer vision (demonstration)</a> <a href="https://web.archive.org/web/20180529034508/http://matlabserver.cs.rug.nl/edgedetectionweb/web/edgedetection_examples.html">Archived</a> 2018-05-29 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a></li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/Hans_Georg_Feichtinger/m7dXLsBj">Feichtinger, Hans G.</a>; Strohmer, Thomas, eds. (1998). <i>Gabor analysis and algorithms : theory and applications</i>. Boston: Birkhäuser. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8176-3959-4. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/97032252">97032252</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/37761814">37761814</a>. <a href="/facts/OL_(identifier)/A13NLNNh">OL</a> <a href="https://openlibrary.org/books/OL685385M">685385M</a>.</li>
<li>Gröchenig, Karlheinz (2001). <i>Foundations of time-frequency analysis : with 15 figures</i>. Applied and Numerical Harmonic Analysis. Boston: Birkhäuser. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4612-0003-1">10.1007/978-1-4612-0003-1</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8176-4022-3. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/00044508">00044508</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/44420790">44420790</a>. <a href="/facts/OL_(identifier)/A13NLNNh">OL</a> <a href="https://openlibrary.org/books/OL8074618M">8074618M</a>.</li>
<li>Daugman, J.G. (1988). <a href="http://www.cs.gmu.edu/~zduric/cs774/Papers/Daugman-GaborTransforms.pdf">"Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression"</a> (PDF). <i>IEEE Transactions on Acoustics, Speech, and Signal Processing</i>. 36 (7): 1169–1179. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.371.5847">10.1.1.371.5847</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2F29.1644">10.1109/29.1644</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0096-3518">0096-3518</a>.</li>
<li><a href="https://web.archive.org/web/20090615224334/http://matlabserver.cs.rug.nl/">"Online Gabor filter demo"</a>. Archived from <a href="http://matlabserver.cs.rug.nl">the original</a> on 2009-06-15. Retrieved 2009-05-25.</li>
<li>Movellan, Javier R. <a href="https://web.archive.org/web/20090419123314/http://mplab.ucsd.edu/tutorials/gabor.pdf">"Tutorial on Gabor Filters"</a> (PDF). Archived from <a href="http://mplab.ucsd.edu/tutorials/gabor.pdf">the original</a> (PDF) on 2009-04-19. Retrieved 2008-05-14.</li>
<li>Lagae, Ares; Lefebvre, Sylvain; Drettakis, George; Dutré, Philip (2009). <a href="http://www.cs.kuleuven.be/~graphics/publications/LLDD09PNSGC/">"Procedural Noise using Sparse Gabor Convolution"</a>. <i>ACM Transactions on Graphics</i>. 28 (3): 1. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.232.5566">10.1.1.232.5566</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1145%2F1531326.1531360">10.1145/1531326.1531360</a>. Retrieved 2009-09-12.</li>
<li>Steerable Pyramids:
<ol><li>Eero Simoncelli's page on <a href="http://www.cns.nyu.edu/~eero/STEERPYR/">Steerable Pyramids</a></li>
<li>Manduchi, R.; Perona, P.; Shy, D. (April 1998). <a href="https://authors.library.caltech.edu/2146/1/MANieeetsp98.pdf">"Efficient deformable filter banks"</a> (PDF). <i>IEEE Transactions on Signal Processing</i>. 46 (4): 1168–1173. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1998ITSP...46.1168M">1998ITSP...46.1168M</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2F78.668570">10.1109/78.668570</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/1053-587X">1053-587X</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/926890247">926890247</a>. (<a href="http://www.vision.caltech.edu/publications/ManduchiPeronaShy_efficient_deformable.pdf">PDF</a> <a href="https://web.archive.org/web/20211112033332/http://www.vision.caltech.edu/publications/ManduchiPeronaShy_efficient_deformable.pdf">Archived</a> 2021-11-12 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a>) (<a href="http://www.vision.caltech.edu/manduchi/def.tar.Z">Code</a> <a href="https://web.archive.org/web/20100613151506/http://www.vision.caltech.edu/manduchi/def.tar.Z">Archived</a> 2010-06-13 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a>)</li></ol></li>
<li>Fischer, Sylvain; Šroubek, Filip; Perrinet, Laurent; Redondo, Rafael; Cristóbal, Gabriel (2007). <a href="http://staff.utia.cas.cz/sroubekf/papers/gabor_07.pdf">"Self-Invertible 2D Log-Gabor Wavelets"</a> (PDF). <i>International Journal of Computer Vision</i>. 75 (2): 231–246. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.329.6283">10.1.1.329.6283</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs11263-006-0026-8">10.1007/s11263-006-0026-8</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0920-5691">0920-5691</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:1452724">1452724</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Gabor, D. (1946). "Theory of communication". J. Inst. Electr. Eng. 93. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Granlund G. H. (1978). "In Search of a General Picture Processing Operator". Computer Graphics and Image Processing. 8 (2): 155–173. doi:10.1016/0146-664X(78)90047-3. ISSN 0146-664X. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Olshausen, B. A. & Field, D. J. (1996). "Emergence of simple-cell receptive-field properties by learning a sparse code for natural images". Nature. 381 (6583): 607–609. Bibcode:1996Natur.381..607O. doi:10.1038/381607a0. PMID 8637596. S2CID 4358477.{{cite journal}}:  CS1 maint: multiple names: authors list (link) <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
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</ol>

Gabor filter open-in-new

Gabor filter