学步园

http://matlabserver.cs.rug.nl/edgedetectionweb/web/edgedetection_params.html

N. Petkov  and
M.B. Wieling,
University of  Groningen,
Department of Computing Science,

Intelligent Systems

This page contains explanations concerning a
simulation  program available on internet.

On this site you can: visualize Gabor functions, use a Gabor filter for edge  detection and extraction of texture features, simulate simple and complex cells  (visual cortex), simulate non-classical receptive field inhibition or surround  suppression and
use it for object contour detection, and explain certain visual  perception effects.

Explanation of parameters

Gabor filtering

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)".

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.

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").

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.

Thinning and thresholding

These are post-processing techniques  standardly used in image processing.

Thinning thins edges in the output to one-pixel wide edges by non-maxima  suppression.

Hysteresis thresholding results in a binary output image. If it is enabled,  two threshold values must be specified:
T-high and T-low. These  are given as fractions (between 0 and 1) of the maximum response value.

Pixels with responses higher than T-high are assigned the binary value 1 in  the output, while pixels with responses below T-low are assigned the binary  value 0. Pixels with responses between T-low and T-high are assigned the value 1  in the binary output
if they can be connected to any pixel with a response  larger than T-high through a chain of other pixels with responses larger than  T-low.

References

1. J.G. Daugman: Uncertainty relations for resolution in space, spatial   frequency, and orientation optimized by two-dimensional visual cortical   filters, Journal of the Optical Society of America A, 1985, vol. 2, pp.   1160-1169.  
2. N. Petkov: Biologically motivated computationally intensive approaches to   image pattern recognition,
   Future Generation Computer Systems,   11 (4-5), 1995, 451-465.
   [bibTex],  [pdf
   1.5 MB © Elsevier - original publication available at
   doi:10.1016/0167-739X(95)00015-K].
3. N. Petkov and P. Kruizinga: Computational models of visual neurons specialised in the   detection of periodic and aperiodic oriented visual stimuli: bar and grating   cells,
   Biological Cybernetics, 76 (2), 1997,   83-96.
   [bibTex], [pdf
   470 KB© Springer]
4. P. Kruizinga and N. Petkov: Non-linear operator for oriented texture,   IEEE Trans. on Image Processing,
   8 (10), 1999,   1395-1407.
   [bibTex],  [pdf
   1775   KB © IEEE]
5. S.E. Grigorescu, N. Petkov and P. Kruizinga: Comparison of texture   features based on Gabor filters,
   IEEE Trans. on Image Processing,   11 (10), 2002, 1160-1167.
   [bibTex], [pdf
   286   KB © IEEE]
6. N. Petkov and M. A. Westenberg: Suppression of contour perception by   band-limited noise and its relation to non-classical receptive field   inhibition,
   Biological Cybernetics, 88, 2003,   236-246.
   [bibTex], [pdf
   636 KB© Springer - original publication available at http://link.springer.de   DOI 10.1007/s00422-002-0378-2]
7. C. Grigorescu, N. Petkov and M. A. Westenberg: Contour detection based on   nonclassical receptive field inhibition,
   IEEE Trans. on Image   Processing, 12 (7), 2003, 729-739.
     [bibTex],    [pdf  
   843 KB © IEEE]