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| [[File:gabor filter.png|thumb|Right|''Example of a two-dimensional Gabor filter'']] | | Yesterday I woke up and noticed - At the moment I've been single for a little while and after much [http://Browse.deviantart.com/?q=intimidation intimidation] from pals [http://lukebryantickets.hamedanshahr.com luke bryan concert tickets 2014] I today find myself signed up for on line dating. They promised me that there are plenty of sweet, regular and entertaining people to fulfill, so here goes the pitch!<br>luke bryan tour date, [http://lukebryantickets.sgs-suparco.org lukebryantickets.sgs-suparco.org], I try to stay as physically healthy as possible coming to the gymnasium several-times weekly. I enjoy my athletics and make an effort to play or watch as several a possible. I shall regularly at Hawthorn matches being wintertime. Note: Supposing that you will considered buying an athletics I really don't brain, I've seen the carnage of fumbling [http://www.netpaw.org concerts for luke bryan] matches at stocktake revenue.<br>My household and friends are wonderful and spending some time with them at pub gigabytes or dishes is concert luke bryan; [http://lukebryantickets.omarfoundation.org Learn More Here], obviously imperative. As I realize that one can do not get a decent dialog together with the noise I haven't ever been in to night clubs. I also have two [http://En.Search.Wordpress.com/?q=unquestionably+cheeky unquestionably cheeky] and quite cute dogs who are almost always enthusiastic to meet fresh folks.<br><br> |
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| 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.
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| [[John Daugman|J. G. Daugman]] discovered that simple cells in the [[visual cortex]] of [[mammalian brain]]s can be modeled by Gabor functions.<ref>[[John Daugman|J. G. Daugman]]. Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters. ''Journal of the Optical Society of America A'', 2(7):1160–1169, July 1985.</ref> Thus, [[image analysis]] by the Gabor functions is similar to perception in the [[human visual system]].
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| == Definition ==
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| {{Disputed-section|date=February 2013}}
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| Its [[impulse response]] is defined by a [[sine wave|sinusoidal]] wave (a [[plane wave]] for 2D Gabor filters) multiplied by a [[Gaussian function]].<ref name="FogelSagi1989">{{cite journal|last1=Fogel|first1=I.|last2=Sagi|first2=D.|title=Gabor filters as texture discriminator|journal=Biological Cybernetics|volume=61|issue=2|year=1989|issn=0340-1200|doi=10.1007/BF00204594}}</ref>
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| 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.<ref>3D surface tracking and approximation using
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| Gabor filters, Jesper Juul Henriksen, South Denmark University, March 28, 2007</ref> The two components may be formed into a [[complex number]] or used individually.
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| Complex
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| :<math>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>
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| Real
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| :<math>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>
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| Imaginary
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| :<math>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>
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| where
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| :<math>x' = x \cos\theta + y \sin\theta\,</math>
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| and
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| :<math>y' = -x \sin\theta + y \cos\theta\,</math>
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| In this equation, <math>\lambda</math> represents the wavelength of the sinusoidal factor, <math>\theta</math> represents the orientation of the normal to the parallel stripes of a [[Gabor function]], <math>\psi</math> is the phase offset, <math>\sigma</math> is the sigma/standard deviation of the Gaussian envelope and <math>\gamma</math> is the spatial aspect ratio, and specifies the ellipticity of the support of the Gabor function.
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| ==Feature extraction==
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| A set of Gabor filters with different frequencies and orientations may be helpful for extracting useful features from an image.
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| ==Wavelet space== <!-- This section is linked from redirect "[[Gabor Wavelet]]" -->
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| [[File:Gabor-ocr.png|thumb|Demonstration of a Gabor filter applied to Chinese OCR. Four orientations are shown on the right 0°, 45°, 90° and 135°. The original character picture and the superposition of all four orientations are shown on the left.]]
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| Gabor filters are directly related to Gabor [[wavelet]]s, 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]].<ref>{{citation |last=Daugman |first=J.G. |authorlink=John Daugman |title=Two-dimensional spectral analysis of cortical receptive field profiles |year=1980 |journal=Vision Res. |volume=20 |issue=10 |pages=847–56 |pmid=7467139}}</ref>
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| 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.<ref>J.P. Jones and L.A. Palmer. An evaluation of the two-dimensional gabor filter model of simple receptive fields in cat striate cortex. J. Neurophysiol., 58(6):1233-1258, 1987</ref>
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| 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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| {{see also|Morlet wavelet}}
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| ==Example implementation==
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| This is an example implementation in [[MATLAB]]/[[GNU Octave|Octave]]:
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| <source lang="matlab">
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| function gb=gabor_fn(sigma,theta,lambda,psi,gamma)
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| sigma_x = sigma;
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| sigma_y = sigma/gamma;
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| % Bounding box
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| nstds = 3;
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| xmax = max(abs(nstds*sigma_x*cos(theta)),abs(nstds*sigma_y*sin(theta)));
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| xmax = ceil(max(1,xmax));
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| ymax = max(abs(nstds*sigma_x*sin(theta)),abs(nstds*sigma_y*cos(theta)));
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| ymax = ceil(max(1,ymax));
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| xmin = -xmax; ymin = -ymax;
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| [x,y] = meshgrid(xmin:xmax,ymin:ymax);
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| % Rotation
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| x_theta=x*cos(theta)+y*sin(theta);
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| y_theta=-x*sin(theta)+y*cos(theta);
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| gb= exp(-.5*(x_theta.^2/sigma_x^2+y_theta.^2/sigma_y^2)).*cos(2*pi/lambda*x_theta+psi);
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| </source>
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| == See also ==
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| * [[Gabor transform]]
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| * [[Gabor atom]]
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| == References ==
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| <references/>
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| == External links ==
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| *[http://demonstrations.wolfram.com/Gabor3D/ 3D Gabor demonstrated with Mathematica]
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| *[http://www.nuhag.eu NuHAG homepage]
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| *[http://covil.sdu.dk/publications/jespermaster07.pdf Jesper Juul Henriksen Thesis]
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| ==Further reading==
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| *Hans G. Feichtinger, Thomas Strohmer: "Gabor Analysis and Algorithms", Birkhäuser, 1998; ISBN 0-8176-3959-4
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| *Karlheinz Gröchenig: "Foundations of Time-Frequency Analysis", Birkhäuser, 2001; ISBN 0-8176-4022-3
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| *[http://www.cs.gmu.edu/~zduric/cs774/Papers/Daugman-GaborTransforms.pdf John Daugman: "Complete Discrete 2-D Gabor Transforms by Neural Networks for Image Analysis and Compression", IEEE Trans on Acoustics, Speech, and Signal Processing. Vol. 36. No. 7. July 1988, pp. 1169–1179]
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| *{{cite web |url=http://matlabserver.cs.rug.nl |title=Online Gabor filter demo |accessdate=2009-05-25}}
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| *{{cite web |url=http://mplab.ucsd.edu/tutorials/gabor.pdf |title=Tutorial on Gabor Filters |last=Movellan |first=Javier R. |accessdate=2008-05-14}}
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| *{{cite web |url=http://www.cs.kuleuven.be/~graphics/publications/LLDD09PNSGC/ |title=Procedural Noise using Sparse Gabor Convolution |accessdate=2009-09-12}}
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| * Steerable Pyramids:
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| *# Eero Simoncelli's page on [http://www.cns.nyu.edu/~eero/STEERPYR/ Steerable Pyramids]
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| *# R. Manduchi, P. Perona and D. Shy. Efficient Deformable Filter Banks ([http://www.vision.caltech.edu/publications/ManduchiPeronaShy_efficient_deformable.pdf PDF]) ([http://www.vision.caltech.edu/manduchi/def.tar.Z Code])
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| {{DEFAULTSORT:Gabor Filter}}
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| [[Category:Linear filters]]
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