Mathematics - 1D and 2D Gaussian Derivatives

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

\(G(x, y, \sigma) = \frac{1}{2 \pi \sigma^2} e^{- \frac{x^2 + y^2}{2 \sigma^2}}\)

\(\frac{\delta G(x, y, \sigma)}{\delta x} = - \frac{x}{2 \pi \sigma^4} e^{- \frac{x^2 + y^2}{2 \sigma^2}}\)

\(\frac{\delta G(x, y, \sigma)}{\delta y} = - \frac{y}{2 \pi \sigma^4} e^{- \frac{x^2 + y^2}{2 \sigma^2}}\)

\(\frac{\delta^2 G(x, y, \sigma)}{\delta^2 x} = (-1 + \frac{x^2}{\sigma^2}) \frac{e^{- \frac{x^2 + y^2}{2 \sigma^2}}}{2 \pi \sigma^4}\)

\(\frac{\delta^2 G(x, y, \sigma)}{\delta^2 y} = (-1 + \frac{y^2}{\sigma^2}) \frac{e^{- \frac{x^2 + y^2}{2 \sigma^2}}}{2 \pi \sigma^4}\)

\(\frac{\delta^2 G(x, y, \sigma)}{\delta xy} = \frac{xy}{2 \pi \sigma^6} e^{- \frac{x^2 + y^2}{2 \sigma^2}}\)