Difference of Gaussians

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Comparison of Difference of Gaussian with Mexican hat wavelet
Comparison of Difference of Gaussian with Mexican hat wavelet

The Difference of Gaussians (DOG) is a wavelet mother function 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 the general multidimensionnal case, the DOG is the difference of two Multivariate normal distribution with a total null sum.

In the early days of computer vision, images were often convolved with this function as part of an edge detection algorithm; see also Marr-Hildreth_algorithm. Today, however, there are much better edge detectors. Nevertheless, differences of Gaussians are still used for approximating the Laplacian of the Gaussian operator in real-time algorithms for blob detection and automatic scale selection; see also scale-space and scale-invariant feature transform. This form of approximation is, however, not always necessary.

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