Difference of Gaussians - Mathematics of Difference of Gaussians

Mathematics of Difference of Gaussians

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} e^{-(u^2 + v^2)/(2 \sigma^2)} - \frac{1}{2\pi K^2 \sigma^2} e^{-(u^2 + v^2)/(2 K^2 \sigma^2)}

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