Difference_of_Gaussians

Difference of Gaussians

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Please improve this article if you can. (January 2008)

Comparison of Difference of Gaussian with Mexican hat wavelet

The Difference of Gaussians (DOG) is a wavelet mother function of null total sum which approximates the Mexican Hat wavelet by subtracting a wide Gaussian from a narrow Gaussian, as defined by this formula in one dimension:

$f(x;\mu,\sigma_1,\sigma_2) = \frac{1}{\sigma_1\sqrt{2\pi}} \, \exp \left( -\frac{(x- \mu)^2}{2\sigma_1^2} \right)-\frac{1}{\sigma_2\sqrt{2\pi}} \, \exp \left( -\frac{(x- \mu)^2}{2\sigma_2^2} \right).$

and for the centered two-dimensional case (see Gaussian blur):

$f(u,v,\sigma) = \frac{1}{2\pi \sigma^2} \exp ^{-(u^2 + v^2)/(2 \sigma^2)} - \frac{1}{2\pi K^2 \sigma^2} \exp ^{-(u^2 + v^2)/(2 K^2 \sigma^2)}$

In computer vision, images are convolved with this function as part of an edge detection algorithm; see also Marr–Hildreth algorithm. Differences of Gaussians have also been used for blob detection in the scale-invariant feature transform; please read the treatment of difference of Gaussian approach in the article on blob detection. In fact, the DOG as the difference of two Multivariate normal distribution has always a total null sum and convolving it with a uniform signal generates no response. It approximates well a second derivate of Gaussian (Laplacian of Gaussian) with K~1.6 and the receptive fields of ganglion cells in the retina with K~5. It may easily used in recursive schemes and is used as an operator in real-time algorithms for blob detection and automatic scale selection; see also scale-space and scale-invariant feature transform.

References

C. Enroth-Cugell and J. G. Robson (1966). "The Contrast Sensitivity of Retinal Ganglion Cells of the Cat.". Journal of Physiology 187: 517-23.

See also

References