一维高斯公式及其导数
(G(x, sigma) = frac{1}{sigma sqrt{2 pi}} e^{- frac{x^2}{2 sigma^2}})
(frac{delta G(x, sigma)}{delta x} = - frac{x}{sigma^3 sqrt{2 pi}} e^{- frac{x^2}{2 sigma^2}})
(frac{delta G(x, sigma)}{delta^2 x} = - frac{sigma^2 - x^2}{sigma^5 sqrt{2 pi}} e^{- frac{x^2}{2 sigma^2}})
二维高斯公式及其导数
(G(x, y, sigma) = frac{1}{2 pi sigma^2} e^{- frac{x^2 + y^2}{2 sigma^2}})
(frac{delta G(x, y, sigma)}{delta x} = - frac{x}{2 pi sigma^4} e^{- frac{x^2 + y^2}{2 sigma^2}})
(frac{delta G(x, y, sigma)}{delta y} = - frac{y}{2 pi sigma^4} e^{- frac{x^2 + y^2}{2 sigma^2}})
(frac{delta^2 G(x, y, sigma)}{delta^2 x} = (-1 + frac{x^2}{sigma^2}) frac{e^{- frac{x^2 + y^2}{2 sigma^2}}}{2 pi sigma^4})
(frac{delta^2 G(x, y, sigma)}{delta^2 y} = (-1 + frac{y^2}{sigma^2}) frac{e^{- frac{x^2 + y^2}{2 sigma^2}}}{2 pi sigma^4})
(frac{delta^2 G(x, y, sigma)}{delta xy} = frac{xy}{2 pi sigma^6} e^{- frac{x^2 + y^2}{2 sigma^2}})