Gabor filter for image processing and computer vision

介绍

我们已经知道，傅里叶变换是一种信号处理中的有力工具，可以帮助我们将图像从空域转换到频域，并提取到空域上不易提取的特征。但是经过傅里叶变换后，图像在不同位置的频度特征往往混合在一起，但是Gabor滤波器却可以抽取空间局部频度特征，是一种有效的纹理检测工具。
Figure 1: A sinusoid and it's Fourier spectrum

如何生成一个Gabor滤波器

在二维空间中，使用一个三角函数(如正弦函数)与一个高斯函数叠加我们就得到了一个Gabor滤波器[1]，如下图。

Figure 2: Gabor filter composition: (a) 2D sinusoid oriented at 30◦ with the x-axis, (b) a Gaussian kernel, (c) the corresponding Gabor filter. Notice how the sinusoid becomes spatially localized.

Gabor核函数

二维Gabor核函数由一个高斯函数和一个余弦函数相乘得出，其中$\theta ,\varphi ,\gamma ,\lambda ,\sigma$为参数。

在OpenCV中的getGaborKernel函数里需要传入的参数除了上述5个外，还需要传入卷积核的大小。

cv::Mat getGaborKernel(Size ksize, double sigma, double theta, double lambd, double gamma, double psi=CV_PI*0.5, int ktype=CV_64F );

Figure 3: The Gabor Filter in frequency with the orientation of 0°, 45°, 90°.

参数

This block implements one or multiple convolutions of an input image with a two-dimensional Gabor function:

To visualize a Gabor function select the option "Gabor function" under "Output image". The Gabor function for the specified values of the parameters "wavelength", "orientation", "phase offset", "aspect ratio", and "bandwidth" will be calculated and displayed as an intensity map image in the output window. (Light and dark gray colors correspond to positive and negative function values, respectively.) The image in the output widow has the same size as the input image: select, for instance, input image octagon.jpg to get an output image of size 100 by 100. If lists of values are specified under "orientation(s)" and "phase offset(s)", only the first values in these lists will be used.

Two-dimensional Gabor functions were proposed by Daugman [1] to model the spatial summation properties (of the receptive fields) of simple cells in the visual cortex. They are widely used in image processing, computer vision, neuroscience and psychophysics. The parametrisaton used in Eq.(1) follows references [2-7] where further details can be found.

Wavelength (λ)

This is the wavelength of the cosine factor of the Gabor filter kernel and herewith the preferred wavelength of this filter. Its value is specified in pixels. Valid values are real numbers equal to or greater than 2. The value λ=2 should not be used in combination with phase offset φ=-90 or φ=90 because in these cases the Gabor function is sampled in its zero crossings. In order to prevent the occurence of undesired effects at the image borders, the wavelength value should be smaller than one fifth of the input image size.

The images (of size 100 x 100) on the left show Gabor filter kernels with values of the wavelength parameter of 5, 10 and 15, from left to right, respectively. The values of the other parameters are as follows: orientation 0, phase offset 0, aspect ratio 0.5, and bandwidth 1.

Orientation(s) (θ)

This parameter specifies the orientation of the normal to the parallel stripes of a Gabor function. Its value is specified in degrees. Valid values are real numbers between 0 and 360.

The images (of size 100 x 100) on the left show Gabor filter kernels with values of the orientation parameter of 0, 45 and 90, from left to right, respectively. The values of the other parameters are as follows: wavelength 10, phase offset 0, aspect ratio 0.5, and bandwidth 1.

For one single convolution, enter one orientation value and set the value of the last parameter in the block "number of orientations" to 1. 
If "number of orientations" is set to an integer value N, N >= 1, then N convolutions will be computed. The orientations of the corresponding Gabor functions are equidistantly distributed between 0 and 360 degrees in increments of 360/N, starting from the value specified under "orientation(s)". An alternative way of computing multiple convolutions for different orientations is to specify under "orientation(s)" a list of values separated by commas (e.g. 0,45,110). In this case, the value of the parameter "number of orientations" is ignored. 

Phase offset(s) (φ)

The phase offset φ in the argument of the cosine factor of the Gabor function is specified in degrees. Valid values are real numbers between -180 and 180. The values 0 and 180 correspond to center-symmetric 'center-on' and 'center-off' functions, respectively, while -90 and 90 correspond to anti-symmetric functions. All other cases correspond to asymmetric functions.

The images (of size 100 x 100) on the left show Gabor filter kernels with values of the phase offset parameter of 0, 180, -90 and 90 dgerees, from left to right, respectively. The values of the other parameters are as follows: wavelength 10, orientation 0, aspect ratio 0.5, and bandwidth 1.

If one single value is specified, one convolution per orientation will be computed. If a list of values is given (e.g. 0,90 which is default), multiple convolutions per orientation will be computed, one for each value in the phase offset list. 

Aspect ratio (γ)

This parameter, called more precisely the spatial aspect ratio, specifies the ellipticity of the support of the Gabor function. For γ = 1, the support is circular. For γ < 1 the support is elongated in orientation of the parallel stripes of the function. Default value is γ = 0.5.

The images (of size 100 x 100) on the left show Gabor filter kernels with values of the aspect ratio parameter of 0.5 and 1, from left to right, respectively. The values of the other parameters are as follows: wavelength 10, orientation 0, phase offset 0, and bandwidth 1.

Bandwidth (b)

The half-response spatial frequency bandwidth b (in octaves) of a Gabor filter is related to the ratio σ / λ, where σ and λ are the standard deviation of the Gaussian factor of the Gabor function and the preferred wavelength, respectively, as follows:

The value of σ cannot be specified directly. It can only be changed through the bandwidth b. The bandwidth value must be specified as a real positive number. Default is 1, in which case σ and λ are connected as follows: σ = 0.56 λ. The smaller the bandwidth, the larger σ, the support of the Gabor function and the number of visible parallel excitatory and inhibitory stripe zones.

The images (of size 100 x 100) on the left show Gabor filter kernels with values of the bandwidth parameter of 0.5, 1, and 2, from left to right, respectively. The values of the other parameters are as follows: wavelength 10, orientation 0, phase offset 0, and aspect ratio 0.5.

Number of orientations

Default value is 1. If an integer value N, N >= 1, is specified then N convolutions will computed. The orientations of the corresponding Gabor functions are equidistantly distributed between 0 and 360 degrees, with increments of 360/N, starting from the value specified in "orientation(s)". For this option to work, one single value (without a comma present) must be specified for the parameter "orientation(s)".

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Half-wave rectification (HWR)

Enable HWR

If this option is enabled, all values in the convolution results below a certain threshold value will be set to zero (HWR is disabled by default).

HWR threshold (%)

The threshold value can be specified as a percentage of the maximum value in a given convolution result. If this percentage is set to 0, all negative values in that convolution result will be changed to 0.

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Superposition of phases

If a list of multiple values is entered under parameter "Phase offset(s)" of the "Gabor filtering" block, multiple convolutions will be computed for each orientation value specified, one convolution for each phase offset value in the list. The convolution results for the different phase offset values of a given orientation can be combined in one single output image for that orientation. This combination can be done in different ways, using the L2, L1 or L-infinity norms. If the L2 norm is used, the squared values of the convolution results for the concerned orientation will be added together pixel-wise and followed by a pixel-wise square root computation to produce the combined result. The L1 and the L-infinity norms correspond to the pixel-wise sum and maximum of the absolute values, respectively. Default is the L2 norm. This choice, together with the default (0,90) of the "Phase offset(s)" of the "Gabor filtering" block, implements the Gabor energy filter that is widely uses in image processing and computer vision. One can also choose not to apply superposition of phases ("None").

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Surround inhibition

The Gabor filter can be augmented with surround inhibition which suppresses texture edges while leaving relativley unaffected the contours of objects and region boundaries. This biologically motivated mechanism introduced in  [6,7]  is particularly useful for contour-based object recognition. In that case, texture edges play the role of noise that obscures object contours and region boundaries and should preferably be eliminated. One can best observe the effect of surround inhibition on different types of oriented features, such as edges in texture vs. isolated edges and lines, by taking the default input image "synthetic1.png".

Select inhibition type

Default is "no surround inhibition".

If "isotropic surround inhibition" is selected, edges in the surroundings of a given edge have a suppression effect on that edge. The relative orientation of these edges has no influence on the suppression effect.

If "anisotropic surround inhibition" is selected, the suppression effect of edges surrounding a given edge depends on their relative orientation: edges parallel to the considered edge have stronger suppression effect than oblique edges, and orthogonal edges have no such effect.

Superposition for isotropic inhibition

If "isotropic inhibition" is selected, a superposition of the convolution results for all used orientations is computed and deployed for surround suppression. Different types of superposition can be used: L1, L2 and L-infinity norms (see the explanations of these terms under "Superposition of phases" in the "Gabor filtering" block). 

Alpha (α)

This parameter controls the strength of surround suppression. Default is 1 but one may need larger values in order to completely suppress texture edges.

K₁ and K₂

The surround that has a suppression effect on an edge in a given point has annular form with inner radius controlled by the combination of values of the parameters K₁ and K₂. The contribution of points in the annular surround is defined by a weighting function which is a half-wave rectified difference of Gaussian functions with standard deviations of K₁σ and K₂σ where σ is the standard deviation of the Gaussian factor of the Gabor function(s) used. One can visualize the weighting function by selecting option "inhibition kernel" under parameter "Output image".

The inner radius of the annular surround increases with K₁. The size of the annual surround which has substantial contribution to the suppression increases with K₂.

Default values are K₁ = 1 and K₂ = 4.